JP4993061B2 - Optimization device, optimization method, and optimization program - Google Patents

Description

  The present invention relates to an optimization device, an optimization method, and an optimization program for optimizing a variable to be optimized.

  There are various optimization problems in engineering. For example, in the optimization of the engine control parameter, the control parameter is optimized so that the evaluation value of the engine state quantity is the highest, and the optimized control parameter is obtained as the optimum solution. For example, the fuel injection timing in the engine is optimized so that the concentration of a specific component of the exhaust gas is reduced. In an optimization problem, a variable to be optimized is called a decision variable. In the above example, the fuel injection timing that is the control parameter is the determining variable, and the concentration of the specific component of the exhaust gas is the state quantity.

  There have been many studies on applying evolutionary algorithms such as genetic algorithms (GA) to optimization problems.

  Genetic algorithms are computational methods that mimic the adaptive evolution of organisms. In this genetic algorithm, solution candidates are called individuals. In the optimization process using the genetic algorithm, first, an initial individual set is generated, and the fitness is evaluated using the fitness function for each individual. Individuals with good fitness are considered good individuals. In the following, considering the minimization problem, it is assumed that the decision variable takes a continuous value. Therefore, the smaller the fitness value, the better the fitness.

  A predetermined number of individuals are selected as parent individuals from the individual set based on the fitness of each individual. A child individual is generated by performing a crossover that partially combines the selected parent individual, and the fitness of the child individual is evaluated. An individual set of the next generation is generated by replacing a child individual with a parent individual based on the evaluation result. By repeating the generation change in this way, the individual with the best fitness is obtained as the optimum individual. The optimal individual is the optimal solution for the optimization problem. That is, the genetic algorithm performs a multipoint search for generating and evaluating a plurality of individuals by crossover.

  In the field of engineering, there is an optimization problem in which evaluation values change periodically due to changes in decision variables. For example, in an apparatus having a rotation mechanism such as an engine or a motor, the state quantity periodically changes every rotation of the rotation shaft. In the wide area control of traffic signals, the traffic signals blink periodically. Furthermore, in pattern matching in image processing, when the target image is rotated with respect to the template and it is determined whether or not they match, the matching amount (degree of matching) between the template and the target image changes periodically. To do.

  In addition, the dihedral angle between two atoms bonded by a covalent bond in a protein is variable and coincides with the original dihedral angle by rotating the bond by 360 degrees. Furthermore, when the link mechanism of the robot can rotate 360 degrees, the link mechanism returns to the original posture by rotating 360 degrees.

  Therefore, optimization of the timing of a device having a rotation mechanism, pattern matching in image processing, optimization of dihedral angle in estimation of protein structure, and optimization of posture of a robot link mechanism are evaluated values by changing the decision variables. Is an optimization problem that changes periodically.

  An optimization problem in which the objective function (fitness function) changes periodically can be expressed by the following equation.

  In the above equation (1), θ is a decision variable belonging to the real number R, f (θ) is a periodic objective function (hereinafter referred to as a periodic function), and N is an arbitrary integer. As shown in the above equation, the periodic function f (θ) has a period 2π. The optimal solution is a decision variable that minimizes the value of the periodic function f (θ).

Here, it is considered to optimize the decision variable in the periodic function f (θ) using the above genetic algorithm. Let θ _{min be} the minimum value of the decision variable and θ _{max be} the maximum value of the decision variable. When the domain [θ _min , θ _max ] of the periodic function f (θ) is [−∞, + ∞], there are an infinite number of optimal solutions (optimal individuals) having exactly the same fitness. Therefore, a genetic algorithm that performs a multipoint search by crossover may perform a useless search.

Therefore, it is conceivable to set the domain [θ _min , θ _max ] of the periodic function f (θ) within an appropriate range and optimize the decision variable within the domain [θ _min , θ _max ]. In this case, the definition area [θ _min , θ _max ] is set to be in the range of 2π. For example, the domain [θ _min , θ _max ] is set to [−π, π]. Thereby, the above equation (1) can be replaced by the following equation (2).

  The above equation (2) means that the decision variable θ that minimizes the value of the periodic function f (θ) in the domain [−π, π] is obtained as an optimal solution. In the genetic algorithm, the value of the periodic function f (θ) corresponds to the fitness. That is, the smaller the value of the periodic function f (θ), the better the fitness.

  A genetic algorithm in which a decision variable is represented by a binary code is called a bit string GA, and a genetic algorithm in which a decision variable is represented by a real number code (floating point representation) is called a real value GA.

  In the bit string GA, the decision variable of the child individual generated by the crossing of the parent individual does not deviate from the domain of definition. On the other hand, in the real value GA, the decision variable of the child individual generated by the crossing of the parent individual may be out of the domain.

  For example, in the crossing method called UNDX (Unimodal Normal Disribution Crossover) and the crossing method called BLX-α (Blend Crossover), a child individual is also outside the region between two parent individuals. Is generated. That is, UNDX and BLX-α are crossover methods having extrapolation. In the real value GA, when the crossover method having extrapolation is used, the decision variable of the child individual generated by the crossover of the parent individual is out of the domain. For this reason, it is necessary to perform some processing on the offspring that is out of the domain.

On the other hand, since the real value GA provides higher search performance than the bit string GA, it is desirable to use the real value GA.
Tomobe, Ono, Kobayashi: Experimental study on protein structure determination by GA, 25th Intelligent System Symposium presentation material, pp. 43-48 (1998) T. Back: Evolutionary Algorithms in Theory and Practice, Oxford (1996)

  However, when the decision variable in the periodic function is optimized using the real value GA, the nature of the problem varies depending on the setting range of the domain as shown below.

  FIG. 35 is a schematic diagram showing an example of the setting range of the definition area of the periodic function. FIG. 36 is a schematic diagram showing another example of the setting range of the definition area of the periodic function. The horizontal axis in FIGS. 35 and 36 is a decision variable, and the vertical axis is fitness (a value of a periodic function). In FIGS. 35 and 36, black circles represent individuals.

  In FIGS. 35 and 36, the value of the decision variable that minimizes the value of the periodic function is searched for as the optimum solution.

In the example of FIG. 35, the value of the periodic function is in the domain [−π , Π ] At the center. As a result, the periodic function has a unimodal shape and the domain [−π, π ] Is searched for in the center.

On the other hand, in the example of FIG. , Π ] Is the minimum in the vicinity of one of the boundaries. As a result, the periodic function has a bimodal shape and the domain [−π, π ] Is searched for in the vicinity of one boundary of , Π ], The local solution accompanying the domain setting appears at the other boundary.

  In the case as shown in FIG. 36, the real value GA has two problems of sampling bias and evolutionary stagnation.

  FIG. 37 is a schematic diagram showing the occurrence probability density distribution of offspring by crossing the real value GA. The horizontal axis in FIG. 37 is the decision variable, and the vertical axis is the occurrence probability density of the child individual. In FIG. 37, P (θ) is an occurrence probability density distribution of a child individual. When crossover is repeated using parent individuals uniformly distributed in the domain [−π, π], as shown in FIG. 37, a child individual is generated at the center of the domain [−π, π]. The probability is high, and the probability that a child individual is generated near the boundary of the domain [−π, π] is low. This phenomenon is called sampling bias.

  Therefore, when the setting range of the domain is not appropriate, it is difficult to find an optimal solution existing near the boundary of the domain [−π, π].

  FIG. 38 is a schematic diagram for explaining the evolutionary stagnation. The horizontal axis in FIG. 38 is the decision variable, and the vertical axis is the fitness (the value of the periodic function).

In the example of FIG. 38, the domain [−π , Π ], There is an optimal solution in the vicinity of one boundary, and the domain [−π , Π], a local solution appears at the other boundary. In this case, when crossover is performed using the parent individual P1 near the optimal solution and the parent individual P2 near the local solution, the domain [−π , Π], child individuals C1 to C7 are generated in a region having poor fitness (region having a large periodic function value). Thus, the domain with good fitness [−π , Π], it is difficult for a child individual to be generated near the boundary, and the search for the optimal solution is stagnant, resulting in poor search efficiency.

  In order to reduce the sampling bias, a boundary method represented by Tsutsui's BEM (Boundary Extension by Mirroring) has been proposed.

  FIG. 39 is a schematic diagram for explaining the boundary method. The horizontal axis of FIG. 39 is a decision variable, and the vertical axis is fitness (a value of a periodic function). In FIG. 39, black circles represent individuals.

  As shown in FIG. 39, in the boundary method, the domain in the vicinity of the domain boundary is folded around the domain boundary based on a predetermined expansion rate, so that the domain is expanded. As a result, the influence of the sampling bias near the boundary of the domain is mitigated. As a result, the possibility of finding the optimal solution is improved.

  However, even when the boundary method is used, the local solution still remains, so the problem of evolutionary stagnation cannot be solved.

  Therefore, when the real value GA is used to optimize the decision variable in the periodic function, it is necessary to devise crossover in consideration of the phase connection. Non-Patent Document 1 proposes a method for generating a child individual in a complementary angle region when an angle between parent individuals is equal to or larger than π in a domain of determination variable [−π, π]. .

  However, according to the method of Non-Patent Document 1, it is necessary to determine the complementary angle region for the parent individual. In addition, this method is difficult to extend to crossover using a large number of parent individuals. Therefore, the process of optimizing the decision variable in the periodic function becomes complicated.

  An object of the present invention is to provide an optimization device, an optimization method, and an optimization program capable of efficiently performing variable optimization.

  (1) The optimization apparatus according to the first aspect of the invention provides N variables to be optimized when there is a constraint that the sum of squares of m variables among N variables to be optimized is 1. An optimization device that gives a solution candidate having, and receives a state quantity corresponding to an objective function for N variables from an optimization target, and points on an N-dimensional variable space defined by a set of N variables A solution candidate storage unit that stores a set of variable vectors to be represented as a solution candidate set, a solution candidate selection unit that selects a predetermined number of solution candidates from the solution candidate set stored in the solution candidate storage unit, and a solution candidate selection unit A new solution candidate set is generated on the N-dimensional variable space including the (m−1) -dimensional unit sphere based on the predetermined number of solution candidates selected by the above, and the generated solution candidate set is optimized and It is output from the solution candidate generator to be given to the solution candidate storage unit and the optimization target. An evaluation value calculation unit that calculates an evaluation value of the state quantity; and a solution candidate set update unit that updates a set of solution candidates stored in the solution candidate storage unit based on the evaluation value calculated by the evaluation value calculation unit , N is a natural number of 2 or more, and m is a natural number of N or less.

  In the optimization apparatus, when there is a constraint that the sum of the squares of m variables out of N variables to be optimized is 1, solution candidates having N variables are given as optimization targets. A state quantity corresponding to the objective function for N variables is output from the optimization target. In the solution candidate storage unit, a set of variable vectors representing points on the N-dimensional variable space defined by a set of N variables is stored as a set of solution candidates.

  A predetermined number of solution candidates are selected by the solution candidate selection unit from the set of solution candidates stored in the solution candidate storage unit. Based on the selected predetermined number of solution candidates, a new solution candidate set is generated in the N-dimensional variable space including the (m−1) -dimensional unit sphere, and the generated solution candidate set is generated. The optimization target and the solution candidate storage unit are given.

  The evaluation value of the state quantity output from the optimization target is calculated by the evaluation value calculation unit, and the solution candidate set stored in the solution candidate storage unit is updated by the solution candidate set update unit based on the calculated evaluation value. The An optimal solution can be obtained by repeating the above processing.

  In this case, since a set of new solution candidates is generated on the N-dimensional variable space including the (m−1) -dimensional unit sphere, the generation of new solution candidates does not depend on the setting of the domain. Thereby, the optimum solution can be stably obtained regardless of the setting of the domain. In addition, since the definition area can be arbitrarily set, the optimization process is not complicated.

  It is also possible to search for solution candidates while maintaining continuity on the (m−1) -dimensional unit sphere. Thus, sampling bias and evolutionary stagnation are prevented from occurring.

  As a result, it is possible to efficiently optimize variables in the optimization target.

  (2) The optimization device according to the second aspect of the present invention provides a solution candidate having N variables to be optimized as an optimization target, and includes N variables including an objective function having periodicity with respect to m variables. An optimization device that receives a state quantity corresponding to an objective function of a target from an optimization target, and a set of variable vectors representing a point on an N-dimensional variable space defined by a set of N variables as a set of solution candidates Based on the solution candidate storage unit to be stored, the solution candidate selection unit that selects a predetermined number of solution candidates from the set of solution candidates stored in the solution candidate storage unit, and the predetermined number of solution candidates selected by the solution candidate selection unit Thus, by associating m variables with m points on the one-dimensional unit sphere, the N-dimensional variable space is expanded to the [(N−m) + 2m] dimensional variable space, and the expanded [(N−m) + 2m ] A set of solution candidates is generated in the dimensional variable space and generated A solution candidate generation unit that generates a new solution candidate set by re-associating the set of solution candidates with a point on the N-dimensional variable space and provides the generated solution candidate set to the optimization target and the solution candidate storage unit And an evaluation value calculation unit for calculating an evaluation value of the state quantity output from the optimization target, and a set of solution candidates stored in the solution candidate storage unit based on the evaluation value calculated by the evaluation value calculation unit And N is a natural number greater than or equal to 1, and m is a natural number less than or equal to N.

  In the optimization apparatus, a solution candidate having a variable having N variables to be optimized is given to an optimization target, and an objective function for N variables including an objective function having periodicity with respect to m variables. The state quantity corresponding to is output from the optimization target. In the solution candidate storage unit, a set of variable vectors representing points on the N-dimensional variable space defined by a set of N variables is stored as a set of solution candidates.

  A predetermined number of solution candidates are selected by the solution candidate selection unit from the set of solution candidates stored in the solution candidate storage unit. The N-dimensional variable space is expanded to the [(N−m) + 2m] -dimensional variable space by associating m variables with m points on the one-dimensional unit sphere based on the selected predetermined number of solution candidates. Then, a set of solution candidates is generated on the expanded [(N−m) + 2m] dimensional variable space, and the generated solution candidate set is again associated with a point on the N dimensional variable space, thereby creating a new one. A set of solution candidates is generated, and the generated set of solution candidates is given to the optimization target and the solution candidate storage unit.

  The evaluation value of the state quantity output from the optimization target is calculated by the evaluation value calculation unit, and the solution candidate set stored in the solution candidate storage unit is updated by the solution candidate set update unit based on the calculated evaluation value. The An optimal solution can be obtained by repeating the above processing.

  In this case, since m variables are associated with m points on the one-dimensional unit sphere, the generation of new solution candidates does not depend on the setting of the domain. Thereby, the optimum solution can be stably obtained regardless of the setting of the domain. In addition, since the definition area can be arbitrarily set, the optimization process is not complicated.

  In addition, it is possible to search for solution candidates while maintaining continuity on m one-dimensional unit spheres. Thus, sampling bias and evolutionary stagnation are prevented from occurring.

As a result, it is possible to efficiently optimize the variables of the objective function in the optimization target.

(3) When each of the m variables is θ and the first and second coordinates of the point on the two-dimensional plane are x and y, respectively, the solution candidate generation unit is selected by the solution candidate selection unit The variable θ of the predetermined number of solution candidates is replaced with the first and second coordinates (x, y) by the calculation of (x, y) = (cos θ, sin θ), and the first and second coordinates of the generated solution candidates are replaced. The coordinates (x, y) of 2 may be converted into a variable θ ^C by the following equation.

  In the above equation, sign (y) becomes +1 when y is positive, and becomes -1 when y is negative.

  In this case, each of the m variables θ of a predetermined number of solution candidates can be easily replaced with the first and second coordinates (x, y) by calculation using a trigonometric function, and an arctangent function is used. By calculation, the first and second coordinates (x, y) of the new solution candidate can be easily converted into the variable θ.

(4) solution candidate selecting unit sets the domain of the objective function to the period of the objective function, domain that is configured to expand or contract in the direction of the variable axis an objective function to be 2 [pi, the domain The objective function may be shifted so that the center is located at the coordinate 0 of the variable axis.

In this case, even when the period of the objective function is different from 2π, it is possible to efficiently optimize the variable of the objective function in the optimization target by converting the domain of the objective function to 2π.

  (5) The optimization target includes an evaluation system for evaluating the performance of the device, the variable vector includes a control parameter for the evaluation system, and the state quantity is the performance of the device obtained by the evaluation of the evaluation system. Also good.

  In this case, a control parameter is given to the evaluation system. The performance of the device is evaluated by the evaluation system based on the control parameters, and a state quantity corresponding to the performance is output. According to this optimization apparatus, the control parameter can be efficiently optimized when the performance of the device has periodicity with respect to the control parameter.

  (6) The evaluation system may be a device evaluation device that controls a device based on a control parameter and outputs the performance of the device generated by the operation of the device as a state quantity.

  In this case, control parameters are given to the device evaluation apparatus. By evaluating the operation of the device based on the control parameters by the device evaluation apparatus, the performance of the device is output as a state quantity. According to this optimization apparatus, the control parameter can be efficiently optimized when the performance of the device has periodicity with respect to the control parameter.

  (7) The evaluation system may be a device simulator that evaluates the performance of the device by simulating the operation of the device based on the control parameter and outputs the evaluated performance as a state quantity.

  In this case, control parameters are given to the equipment simulator. The device performance is output as a state quantity by simulating the operation of the device based on the control parameters by the device simulator. According to this optimization apparatus, the control parameter can be efficiently optimized when the performance of the device has periodicity with respect to the control parameter.

  (8) The device may be an internal combustion engine, the control parameter may be a parameter based on a crank rotation angle of the internal combustion engine, and the state quantity may be the performance of the internal combustion engine that periodically changes with respect to the parameter.

  In this case, a parameter based on the crank rotation angle of the internal combustion engine is given to the device evaluation system. By evaluating the operation of the internal combustion engine based on the parameters by the device evaluation system, the performance of the internal combustion engine is output as a state quantity. According to this optimization apparatus, when the performance of the internal combustion engine has periodicity with respect to the parameter based on the crank rotation angle, the parameter can be efficiently optimized.

  (9) The internal combustion engine may have a fuel injection device, and the parameter based on the crank rotation angle may include the fuel injection timing of the fuel injection device.

  In this case, the fuel injection timing of the fuel injection device is given to the device evaluation system as a control parameter for the internal combustion engine. The performance of the internal combustion engine is output as a state quantity by controlling the fuel injection timing of the fuel injection device based on the control parameter by the device evaluation system. According to this optimization device, the parameters can be efficiently optimized when the performance of the internal combustion engine has periodicity with respect to the fuel injection timing of the fuel injection device.

  (10) The internal combustion engine has a fuel injection device, and the parameter based on the crank rotation angle is a fuel injection pattern by the fuel injection device within a certain period including a plurality of sections, and the fuel injection pattern is a fuel injection by the fuel injection device. It is a vector composed of the fuel injection amount of each section in which the total injection amount is constant, and the state amount may be the performance of the internal combustion engine that changes periodically with respect to the parameters.

  In this case, the fuel injection pattern of the fuel injection device is given to the device evaluation system as a control parameter for the internal combustion engine. The performance of the internal combustion engine is output as a state quantity by controlling the fuel injection pattern of the fuel injection device based on the control parameter by the device evaluation system. According to this optimization device, parameters can be efficiently optimized when the performance of the internal combustion engine has periodicity with respect to the fuel injection pattern of the fuel injection device.

  (11) The state quantity may include at least one of a concentration of a predetermined component in the exhaust gas, fuel consumption, output torque, throttle valve opening, and crank rotation speed.

  In this case, the device evaluation system evaluates the operation of the internal combustion engine based on the parameters based on the crank rotation angle, so that the concentration of predetermined components in the exhaust gas, fuel consumption, output torque, throttle valve opening, and crank At least one of the rotation speeds is output as a state quantity. According to this optimization device, at least one of the concentration of the predetermined component in the exhaust gas, the fuel consumption, the output torque, the opening degree of the throttle valve, and the crank rotation speed has periodicity with respect to the parameter based on the crank rotation angle. In this case, the parameters can be optimized efficiently.

  (12) The device may be a robot link mechanism, the variable vector may be a parameter indicating an angle formed by a joint of the link mechanism, and the state quantity may be the performance of the robot link mechanism that periodically changes with respect to the parameter.

  In this case, a parameter indicating the angle formed by the joint of the robot link mechanism is given to the device evaluation system. By evaluating the operation of the robot link mechanism based on the parameters by the device evaluation system, the performance of the robot link mechanism is output as a state quantity. According to this optimization apparatus, when the performance of the robot link mechanism has periodicity with respect to the parameter indicating the angle formed by the joint, the parameter can be efficiently optimized.

  (13) The optimization target is a traffic simulator that simulates wide area control of a traffic system including a plurality of traffic signals arranged on the road, the variable vector is at least one lighting timing of the traffic signal, and the state quantity is the road It may include information about cars passing through.

  In this case, at least one traffic signal lighting timing is given to the traffic simulator. The traffic simulator simulates wide-area control of the traffic system based on the lighting timing of at least one traffic signal, and outputs information related to vehicles passing through the road. According to this optimizing device, at least one lighting timing of the traffic signal can be efficiently optimized when the information regarding the vehicle passing through the road has periodicity with respect to the lighting timing of the traffic signal.

  (14) The optimization device may be a protein simulator that simulates a protein structure, the variable vector may be a direction vector representing a dihedral angle between covalent bonds of the protein, and the state quantity may be information on energy.

  In this case, a direction vector representing the dihedral angle between the covalent bonds of the protein is given to the protein simulator. The protein simulator simulates the structure of the protein based on the direction vector and outputs information about energy. According to this optimization apparatus, the direction vector can be efficiently optimized when the information about energy has periodicity with respect to the direction vector representing the dihedral angle between the covalent bonds of the protein.

  (15) The optimization target is an image processing device that performs a matching process between the target image and the reference image, the variable vector is a direction vector of the target image with respect to the reference image, and the state quantity is an error between the target image and the reference image. It may be information indicating.

  In this case, the direction vector of the target image with respect to the reference image is given to the image processing apparatus. The image processing device performs matching processing between the target image and the reference image based on the direction vector, and outputs information indicating an error between the target image and the reference image. According to this optimization apparatus, the direction vector can be efficiently optimized when the information indicating the error between the target image and the reference image has periodicity with respect to the direction vector of the target image with respect to the reference image.

  (16) In the optimization method according to the third aspect of the invention, when there is a constraint that the sum of squares of m variables out of N variables to be optimized is 1, N variables are to be optimized. An optimization method for providing a solution candidate having a state quantity corresponding to an objective function for N variables from an optimization target, wherein a point on an N-dimensional variable space defined by a set of N variables A step of storing a set of variable vectors to be represented in the storage unit as a set of solution candidates, a step of selecting a predetermined number of solution candidates from the stored set of solution candidates, and based on the selected predetermined number of solution candidates ( m-1) generating a new set of solution candidates on the N-dimensional variable space including the dimensional unit sphere, and providing the generated set of solution candidates to the optimization target and the storage unit; and outputting from the optimization target Calculating the evaluation value of the state quantity And a step of updating the set of candidate solutions stored in the storage unit on the basis of the evaluation value, N is a natural number of 2 or more, m is a natural number equal to or less than N.

  In the optimization method, a solution candidate having a variable having N variables to be optimized is given as an optimization target, and N includes an objective function having a constraint that the sum of squares of m variables is 1. A state quantity corresponding to the objective function for each variable is output from the optimization target. In the storage unit, a set of variable vectors representing points on the N-dimensional variable space defined by a set of N variables is stored as a set of solution candidates.

  A predetermined number of solution candidates are selected from a set of solution candidates stored in the storage unit. A new solution candidate set is generated on the N-dimensional variable space including the (m−1) -dimensional unit sphere based on the selected predetermined number of solution candidates, and the generated solution candidate set is optimized and stored. Given to the department.

  An evaluation value of the state quantity output from the optimization target is calculated, and a set of solution candidates stored in the storage unit is updated based on the calculated evaluation value. An optimal solution can be obtained by repeating the above processing.

  In this case, since a set of new solution candidates is generated on the N-dimensional variable space including the (m−1) -dimensional unit sphere, the generation of new solution candidates does not depend on the setting of the domain. Thereby, the optimum solution can be stably obtained regardless of the setting of the domain. In addition, since the definition area can be arbitrarily set, the optimization process is not complicated.

  It is also possible to search for solution candidates while maintaining continuity on the (m−1) -dimensional unit sphere. Thus, sampling bias and evolutionary stagnation are prevented from occurring.

  As a result, it is possible to efficiently optimize variables in the optimization target.

  (17) An optimization method according to a fourth aspect of the present invention provides a solution candidate having a variable having N variables to be optimized, and includes N objective functions having periodicity with respect to m variables. An optimization method for receiving from an optimization target a state quantity corresponding to an objective function for a variable of a variable, wherein a set of variable vectors representing a point on an N-dimensional variable space defined by a set of N variables is a solution candidate A step of storing in the storage unit as a set of m, a step of selecting a predetermined number of solution candidates from the set of stored solution candidates, and m variables based on the selected predetermined number of solution candidates. The N-dimensional variable space is expanded to a [(N−m) + 2m] -dimensional variable space by associating it with a point on the dimensional unit sphere, and a set of solution candidates on the expanded [(N−m) + 2m] -dimensional variable space And a set of generated solution candidates A new solution candidate set is generated by re-associating with a point on the dimensional variable space, the generated solution candidate set is given to the optimization target and the storage unit, and the state quantity output from the optimization target A step of calculating an evaluation value; and a step of updating a set of solution candidates stored in the storage unit based on the calculated evaluation value, wherein N is a natural number of 1 or more, and m is a natural number of N or less. is there.

  In the optimization method, a solution candidate having a variable having N variables to be optimized is given to an optimization target, and an objective function for N variables including an objective function having periodicity with respect to m variables. The state quantity corresponding to is output from the optimization target. In the storage unit, a set of variable vectors representing points on the N-dimensional variable space defined by a set of N variables is stored as a set of solution candidates.

  A predetermined number of solution candidates are selected from a set of solution candidates stored in the storage unit. The N-dimensional variable space is expanded to the [(N−m) + 2m] -dimensional variable space by associating m variables with m points on the one-dimensional unit sphere based on the selected predetermined number of solution candidates. Then, a set of solution candidates is generated on the expanded [(N−m) + 2m] dimensional variable space, and the generated solution candidate set is again associated with a point on the N dimensional variable space, thereby creating a new one. A set of solution candidates is generated, and the generated set of solution candidates is given to the optimization target and the storage unit.

  An evaluation value of the state quantity output from the optimization target is calculated, and a set of solution candidates stored in the storage unit is updated based on the calculated evaluation value. An optimal solution can be obtained by repeating the above processing.

  In this case, since m variables are associated with m points on the one-dimensional unit sphere, the generation of new solution candidates does not depend on the setting of the domain. Thereby, the optimum solution can be stably obtained regardless of the setting of the domain. In addition, since the definition area can be arbitrarily set, the optimization process is not complicated.

  In addition, it is possible to search for solution candidates while maintaining continuity on m one-dimensional unit spheres. Thus, sampling bias and evolutionary stagnation are prevented from occurring.

As a result, it is possible to efficiently optimize the variables of the objective function in the optimization target.

  (18) The optimization program according to the fifth aspect of the present invention provides N variables as optimization targets when there is a constraint that the sum of squares of m variables out of N variables to be optimized is 1. An N-dimensional variable defined by a set of N variables, which is an optimization program that can be executed by a computer that gives a solution candidate having a state quantity corresponding to an objective function for N variables from an optimization target A process of storing a set of variable vectors representing points in space as a set of solution candidates in the storage unit, a process of selecting a predetermined number of solution candidates from the stored set of solution candidates, and a predetermined number of selected solutions A process for generating a new set of solution candidates on an N-dimensional variable space including the (m−1) -dimensional unit sphere based on the candidates, and providing the generated set of solution candidates to the optimization target and the storage unit; Of the state quantity output from the target The computer executes a process of calculating a value and a process of updating a set of solution candidates stored in the storage unit based on the calculated evaluation value, N is a natural number of 2 or more, and m is N or less It is a natural number.

  In the optimization program, a solution candidate having a variable having N variables to be optimized is given as an optimization target, and an N including an objective function having a constraint that the sum of squares of m variables is 1. A state quantity corresponding to the objective function for each variable is output from the optimization target. In the storage unit, a set of variable vectors representing points on the N-dimensional variable space defined by a set of N variables is stored as a set of solution candidates.

  A predetermined number of solution candidates are selected from a set of solution candidates stored in the storage unit. A new solution candidate set is generated on the N-dimensional variable space including the (m−1) -dimensional unit sphere based on the selected predetermined number of solution candidates, and the generated solution candidate set is optimized and stored. Given to the department.

  An evaluation value of the state quantity output from the optimization target is calculated, and a set of solution candidates stored in the storage unit is updated based on the calculated evaluation value. An optimal solution can be obtained by repeating the above processing.

  In this case, since a set of new solution candidates is generated on the N-dimensional variable space including the (m−1) -dimensional unit sphere, the generation of new solution candidates does not depend on the setting of the domain. Thereby, the optimum solution can be stably obtained regardless of the setting of the domain. In addition, since the definition area can be arbitrarily set, the optimization process is not complicated.

  It is also possible to search for solution candidates while maintaining continuity on the m-dimensional unit sphere. Thus, sampling bias and evolutionary stagnation are prevented from occurring.

  As a result, it is possible to efficiently optimize variables in the optimization target.

  (19) An optimization program according to a sixth aspect of the present invention provides a solution candidate having a variable having N variables to be optimized as an optimization target, and includes N objective functions having periodicity with respect to m variables. A variable vector representing a point on an N-dimensional variable space defined by a set of N variables, which is an executable program that can be executed by a computer that receives a state quantity corresponding to an objective function for a variable of Storing a set of solution candidates as a set of solution candidates in the storage unit, processing for selecting a predetermined number of solution candidates from the stored solution candidate set, and m variables based on the selected predetermined number of solution candidates Is associated with m points on the one-dimensional unit sphere to expand the N-dimensional variable space to the [(N−m) + 2m] -dimensional variable space, and on the expanded [(N−m) + 2m] -dimensional variable space Set of solution candidates Processing for generating a new solution candidate set by re-associating the generated solution candidate set with a point on the N-dimensional variable space, and providing the generated solution candidate set to the optimization target and storage unit And a process for calculating an evaluation value of the state quantity output from the optimization target and a process for updating a set of solution candidates stored in the storage unit based on the calculated evaluation value, and N Is a natural number of 1 or more, and m is a natural number of N or less.

  In the optimization program, a solution candidate having a variable having N variables to be optimized is given as an optimization target, and an objective function for N variables including an objective function having periodicity with respect to m variables. The state quantity corresponding to is output from the optimization target. In the storage unit, a set of variable vectors representing points on the N-dimensional variable space defined by a set of N variables is stored as a set of solution candidates.

  A predetermined number of solution candidates are selected from a set of solution candidates stored in the storage unit. The N-dimensional variable space is expanded to the [(N−m) + 2m] -dimensional variable space by associating m variables with m points on the one-dimensional unit sphere based on the selected predetermined number of solution candidates. Then, a set of solution candidates is generated on the expanded [(N−m) + 2m] dimensional variable space, and the generated solution candidate set is again associated with a point on the N dimensional variable space, thereby creating a new one. A set of solution candidates is generated, and the generated set of solution candidates is given to the optimization target and the storage unit.

  An evaluation value of the state quantity output from the optimization target is calculated, and a set of solution candidates stored in the storage unit is updated based on the calculated evaluation value. An optimal solution can be obtained by repeating the above processing.

  In this case, since m variables are associated with m points on the one-dimensional unit sphere, the generation of new solution candidates does not depend on the setting of the domain. Thereby, the optimum solution can be stably obtained regardless of the setting of the domain. In addition, since the definition area can be arbitrarily set, the optimization process is not complicated.

  In addition, it is possible to search for solution candidates while maintaining continuity on m one-dimensional unit spheres. Thus, sampling bias and evolutionary stagnation are prevented from occurring.

As a result, it is possible to efficiently optimize the variables of the objective function in the optimization target.

  According to the first, third, and fifth inventions, a new set of solution candidates is generated on the N-dimensional variable space including the (m−1) -dimensional unit sphere, so that generation of new solution candidates is defined. It does not depend on the area setting. Thereby, the optimum solution can be stably obtained regardless of the setting of the domain. In addition, since the definition area can be arbitrarily set, the optimization process is not complicated.

  It is also possible to search for solution candidates while maintaining continuity on the (m−1) -dimensional unit sphere. Thus, sampling bias and evolutionary stagnation are prevented from occurring.

  As a result, it is possible to efficiently optimize variables in the optimization target.

  According to the second, fourth, and sixth inventions, m variables are associated with m points on the one-dimensional unit sphere, so that generation of new solution candidates does not depend on the setting of the domain. Thereby, the optimum solution can be stably obtained regardless of the setting of the domain. In addition, since the definition area can be arbitrarily set, the optimization process is not complicated.

  In addition, it is possible to search for solution candidates while maintaining continuity on m one-dimensional unit spheres. Thus, sampling bias and evolutionary stagnation are prevented from occurring.

As a result, it is possible to efficiently optimize the variables of the objective function in the optimization target.

(1) First Embodiment In this embodiment, crossover on a hypersphere, which will be described later, is used for individual crossover.

(A) Functional Configuration of Optimization Device First, the optimization device according to the first embodiment of the present invention will be described with reference to FIG. FIG. 1 is a block diagram showing a functional configuration of the optimization apparatus according to the first embodiment of the present invention.

  The optimization apparatus 1 of FIG. 1 calculates the optimal solution of the decision variable in a periodic function using a genetic algorithm (GA) as an evolutionary algorithm. The optimization apparatus 1 is connected to the optimization target 8. In the genetic algorithm, the optimal solution is called the optimal individual.

  The optimization target 8 is an evaluation system that evaluates the performance of the device. The evaluation system is an evaluation device or a simulator that evaluates an actual system. In the present embodiment, the actual system is an engine, and the optimization target 8 is an engine evaluation device.

  The optimization apparatus 1 includes an individual generation unit 2, a generation change unit 3, an individual set storage unit 4, an individual selection unit 5, an fitness calculation unit 6, and an input / output interface 7. The individual generation unit 2 includes a crossover unit 21 and a normalization calculation unit 22.

  The individual generation unit 2, the generation change unit 3, the individual selection unit 5, and the fitness calculation unit 6 are realized by a CPU 101 (FIG. 2) described later executing an optimization program. The individual set storage unit 4 includes an external storage device 106 (FIG. 2) described later.

  The user 10 sets an fitness function (objective function) in the optimization device 1. The fitness function includes CO (carbon monoxide) concentration, HC (hydrocarbon) concentration, NOx (nitrogen oxide) concentration, fuel consumption, output torque, throttle valve opening, engine speed in exhaust gas. Etc. can be set. In the present embodiment, HC concentration and NOx concentration in engine exhaust gas are used as the fitness function. The fitness is a value of the fitness function.

  Here, an individual of a genetic algorithm is a candidate for a solution to an optimization problem, and has one or more decision variables (control parameters) and one or more fitness. A decision variable is a variable to be optimized. One or more decision variables are represented by a decision variable vector. In this case, each decision variable is an element of a decision variable vector. In genetic algorithms, the decision variable is called a gene. In the present embodiment, the decision variable vector has a constraint that the sum of squares of a plurality of decision variables is 1. In the example described later, fuel injection amounts in a plurality of sections on the time axis are used as the decision variable.

  The individual set storage unit 4 stores a plurality of individuals. Here, each individual consists of a decision variable vector and one or more fitness values. The individual selection unit 5 selects a predetermined number of individuals used for generating child individuals from a plurality of individuals stored in the individual set storage unit 4 as parent individuals.

  The individual generation unit 2 generates a plurality of child individuals from a predetermined number of parent individuals selected by the individual selection unit 5, gives a decision variable vector of the child individuals to the individual set storage unit 4, and via the input / output interface 7. The optimization object 8 is given. Details of the functions of the crossover unit 21 and the normalization calculation unit 22 of the individual generation unit 2 will be described later.

  The optimization target 8 outputs a state quantity based on the individual decision variable vector given from the optimization device 1. The state quantity is, for example, a measured value such as CO concentration, HC concentration, NOx concentration, fuel consumption, output torque, throttle valve opening, and engine speed in the exhaust gas. In the present embodiment, the state quantity is a measured value of the HC concentration and NOx concentration in the exhaust gas.

  The state quantity output from the optimization target 8 is input to the fitness calculation unit 6 via the input / output interface 7. The fitness calculation unit 6 calculates fitness based on the state quantity, and gives the calculated fitness to the individual set storage unit 4. In the example described later, the fitness calculation unit 6 calculates the average value of the HC concentration and the NOx concentration in the exhaust gas in each section as the fitness. The individual set storage unit 4 stores the fitness set and the decision variable vector calculated by the fitness calculation unit 6 as individuals.

  The generation change unit 3 updates the individual set stored in the individual set storage unit 4 based on the fitness of a plurality of individuals stored in the individual set storage unit 4. As a result, generational changes are made.

  In this way, the individual set storage unit 4 stores an individual set including the optimum individual. The optimization apparatus 1 presents to the user 10 an individual set including the optimal individual.

(B) Hardware Configuration of Optimization Device FIG. 2 is a block diagram showing a hardware configuration of the optimization device 1 of FIG.

  The optimization device 1 includes a CPU (Central Processing Unit) 101, a ROM (Read Only Memory) 102, a RAM (Random Access Memory) 103, an input device 104, a display device 105, an external storage device 106, a recording medium driving device 107, and An input / output interface 7 is included.

  The input device 104 includes a keyboard, a mouse, and the like, and is used for inputting various commands and various data. The ROM 102 stores a system program. The recording medium driving device 107 includes a CD (compact disk) drive, a DVD (digital versatile disk) drive, a flexible disk drive, and the like, and reads / writes data from / to a recording medium 109 such as a CD, DVD, or flexible disk.

  An optimization program is recorded on the recording medium 109. The external storage device 106 includes a hard disk device or the like, and stores an optimization program and various data read from the recording medium 109 via the recording medium driving device 107. The CPU 101 executes the optimization program stored in the external storage device 106 on the RAM 103.

  The display device 105 includes a liquid crystal display panel, a CRT (cathode ray tube), and the like, and displays various images such as an individual set. An optimization target 8 is connected to the input / output interface 7 by wireless communication or wired communication. The input / output interface 7 transfers the state quantity output from the optimization target 8 to the external storage device 106 and provides the optimization target 8 with a set of individual fitness values generated by the optimization program.

  As the recording medium 109 for recording the optimization program, various recording media such as a semiconductor memory such as a ROM and a hard disk can be used. Alternatively, the optimization program may be downloaded to the external storage device 106 via a communication medium such as a communication line and executed on the RAM 103.

  Here, as long as the recording medium 109 is a computer-readable recording medium, the recording medium 109 includes an electronic reading method, a magnetic reading method, an optical reading method, or any other reading method. For example, in addition to the above-mentioned CD, DVD and flexible disk, optical reading type recording medium such as CDV (compact disk video), semiconductor recording medium such as RAM and ROM, magnetic recording type recording medium such as hard disk, MO (magneto-optical) A magnetic storage / optical reading type recording medium such as a disk) can be used.

(C) Application to Optimization of Fuel Injection Pattern FIG. 3 is a diagram for explaining a method of applying the optimization device to optimization of the fuel injection pattern.

  Consider performing fuel injection in various patterns within a certain time while keeping the fuel injection amount constant. Due to the difference in the fuel injection pattern, the adhesion of the fuel to the wall surface and the combustion state change, resulting in a difference in the amount of harmful components in the exhaust gas.

  In the example of FIG. 3, the fuel injection timing is constant and the fuel injection start time 0 to the fuel injection end time t2 are equally divided into five sections. Here, it is assumed that the total fuel injection amount is constant.

  If optimization is performed by a method of normalizing the weight of each section each time, the solutions exist infinitely. Therefore, it is considered that the efficiency of searching for an individual is deteriorated.

If the fuel injection amount in a certain section is x _i ² (i = 1,..., 5), the total fuel injection amount in all sections is x ₁ ² +... + X ₅ ² = 1. Thereby, it is understood that the fuel injection amount vector x is a point distributed on the four-dimensional unit spherical surface. Therefore, the hyperspherical crossover method (UNDX-H) described in detail below can be applied to the optimization of the fuel injection pattern. As a result, the fuel injection pattern can be optimized efficiently.

(D) Overall Processing of Optimization Device FIG. 4 is a flowchart showing overall processing of the optimization device 1 of FIG. As shown in FIG. 4, first, the individual generation unit 2 generates an initial individual set by randomly generating individuals within a predetermined decision variable range (step S1), and the generated initial individual set. Are sequentially output to the optimization target 8 (step S2). Thereby, state quantities are sequentially output from the optimization target 8.

  The fitness calculation unit 6 acquires a state quantity corresponding to each individual output from the optimization target 8 (step S3), and calculates the fitness of each individual based on the acquired state quantity (step S4). The calculated fitness of each individual is stored in the individual set storage unit 4 together with the decision variable vector.

  Next, the individual selection unit 5 selects a predetermined number of individuals used for generating a new child individual from a plurality of individuals stored in the individual set storage unit 4 as a parent individual (step S5). In the present embodiment, three parent individuals are selected.

  Furthermore, the individual generation unit 2 generates a plurality of new child individuals by crossover using a predetermined number of parent individuals selected by the individual selection unit 5 (step S6). Here, since the decision variable vector having the fuel injection amount of each section as a decision variable has a constraint that the sum of the squares of the decision variables is 1, a crossover method on a super plane described later is used.

  The individual generation unit 2 sequentially outputs the decision variable vector of each child individual to the optimization target 8 (step S7). Thereby, state quantities are sequentially output from the optimization target 8.

  The fitness calculation unit 6 acquires a state quantity corresponding to each child individual output from the optimization target 8 (step S8), and calculates the fitness of each child individual based on the acquired state quantity (step S9). ). The calculated fitness of each child individual is stored in the individual set storage unit 4 together with the decision variable vector.

  The generation change unit 3 updates the individual set stored in the individual set storage unit 4 by replacing the child individual with the parent individual based on the fitness of each child individual and each parent individual (step S10). In this case, the generation change unit 3 updates the individual set by at least one method selected from selection based on a comparison of fitness levels, selection based on the probability determined by the high fitness level, and random selection. As a result, generational changes are made.

  Here, the generation includes a selection step for selecting a parent individual from an individual set, a crossover step for generating a new child individual by crossover, and a generation change step for replacing the parent individual with the child individual. Due to the generation change, individuals with poor fitness belonging to the individual set are deceived.

  Thereafter, the individual generation unit 2 determines whether or not the number of generations has reached a predetermined end condition (step S11).

  If it is determined that the number of generations has not reached the predetermined end condition, the process proceeds to step S5. If it is determined that the number of generations has reached a predetermined end condition, the individual generation unit 2 presents the individual set including the optimum individual stored in the individual set storage unit 4 to the user 10 and ends the process. .

(E) Example of Optimization Process FIG. 5 is a schematic diagram illustrating an example of the optimization process performed by the optimization apparatus 1 of FIG. In the optimization process of FIG. 5, MGG (Minimal Generation Gap) is used as a generation change model.

  As shown in FIG. 5, when UNDX (Unimodal Normal Distribution Crossover) is used as a crossover operator, three parent individuals p1, p2, and p3 are selected from the individual set P. A plurality of child individuals c1, c2, c3, and c4 are generated by crossing the three selected parent individuals. A family individual set F is composed of the parent individuals p1, p2, p3 and the child individuals c1, c2, c3, c4. The fitness of each individual in the family individual set F is obtained.

  An individual having the best fitness is selected from the family individual set F by elite selection, and the selected individual is replaced with a parent individual in the individual set P. In the example of FIG. 5, the child individual c2 is selected as an elite individual and is replaced with the parent individual p1 of the individual set P. In addition, roulette wheel selection is performed with a probability proportional to the fitness of each individual in the family individual set F, and the selected individual is replaced with a parent individual in the individual set P. In the example of FIG. 5, the child individual c3 is selected by roulette wheel selection and replaced with the parent individual p2 of the individual set P.

(F) Basic Concept of Crossover Method on Hypersphere The crossover method on the hypersphere will be described. In the n-dimensional real space R ⁿ , the (n−1) -dimensional unit sphere S ⁿ⁻¹ is defined as follows:

S ^n-1 = {(x ₁ , x ₂ ,..., X _n ) ∈R ⁿ | x ₁ ² + x ₂ ² + ... x _n ² = 1}
In the above equation, n is an arbitrary natural number equal to or greater than 2, and x ₁ , x ₂ ,..., X _n are n decision variables and are elements of an n-dimensional decision variable vector. That is, the n-dimensional real space R ⁿ is defined by n decision variables. The (n−1) -dimensional unit spherical surface S ⁿ⁻¹ satisfies the condition of x ₁ ² + x ₂ ² +... + X _n ² = 1.

For example, in the two-dimensional real space R ² , the one-dimensional unit spherical surface S ¹ is a unit circle that satisfies the condition x ₁ ² + x ₂ ² = 1. In the three-dimensional real space R ³ , the two-dimensional unit spherical surface S ² is a unit sphere that satisfies the condition x ₁ ² + x ₂ ² + x ₃ ² = 1.

The problem of ^obtaining the unit direction vector of the n-dimensional real space R ⁿ can be generalized as a problem in which individuals are distributed on the (n−1) -dimensional unit sphere S ⁿ⁻¹ .

  Here, the (n−1) -dimensional unit sphere is called a hypersphere.

Here, the crossover method on the hypersphere will be described. 6a, 6b and 6c are schematic diagrams for explaining a crossover method on a hypersphere. In this example, UNDX is used as the crossover operator. 6A, 6B, and 6C, the horizontal axis is the decision variable x ₁ , and the vertical axis is the decision variable x ₂ .

In FIG. 6a, a unit circle UC that satisfies the condition x ₁ ² + x ₂ ² = 1 is set as a hypersphere in the two-dimensional real space R ² defined by the decision variable x ₁ and the decision variable x ₂ .

As shown in FIG. 6a, three parent individuals P ¹ , P ² and P ³ are set on the unit circle UC. Parent individuals P ¹ , P ² , and P ³ are two-dimensional unit direction vectors having the decision variable x ₁ and the decision variable x ₂ as elements. In UNDX, child individuals are generated according to normal distribution random numbers determined based on the positional relationship between the ^three parent individuals P ¹ , P ² , and P ³ .

As shown in FIG. 6b, a plurality of child individuals C ¹ ′ to C ⁵ ′ are generated from the ^three parent individuals P ¹ , P ² , P ³ . The child individuals C ¹ ′ to C ⁵ ′ are two-dimensional decision variable vectors having the decision variable x ₁ and the decision variable x ₂ as elements.

Next, as shown in FIG. 6c, a plurality of child individuals C ^{1 to} C ⁵ are obtained in the unit circle UC by normalizing the plurality of child individuals C ¹ ′ to C ⁵ ′, respectively. The child individuals C ^{1 to} C ⁵ are two-dimensional unit direction vectors having the decision variable x ₁ and the decision variable x ₂ as elements.

In the present embodiment, the n-dimensional real space R ⁿ or the two-dimensional real space (two-dimensional real plane) R ² corresponds to the N-dimensional variable space, and the (n−1) -dimensional unit spherical surface S ⁿ⁻¹ , hypersphere or A unit circle corresponds to an (m−1) -dimensional unit sphere, and n decision variables x ₁ , x ₂ ,..., X _{n correspond} to N variables and m variables. That is, in this embodiment, N = m = n. Note that the N variables may include variables other than n decision variables. In that case, N> m = n.

(H) Procedure for Crossover Method on Hypersphere FIG. 7 is a flowchart showing the crossover process on the hypersphere. The crossover process of FIG. 7 is performed by the individual selection unit 5, the crossover unit 21, and the normalization calculation unit 22 of FIG.

First, the crossing unit 21 places the individual set on the (n−1) -dimensional unit spherical surface S ^n-1 (hyperspherical surface) (step S21). The individual set distributed on the hypersphere is stored in the individual set storage unit 4.

Next, the individual selection unit 5 selects a predetermined number of individuals as a parent individual from the individual set distributed on the (n−1) -dimensional unit sphere S ^n-1 (hypersphere) (step S22).

Next, the crossover unit 21 generates a plurality of child individuals by applying a crossover operator as points on the n-dimensional real space R ⁿ to a predetermined number of parent individuals selected from the hypersphere (step S23). ). Each child individual is represented by a decision variable vector X ^C ′ on the n-dimensional real space R ⁿ .

Furthermore, the normalization calculation unit 22 determines whether or not the magnitude of the decision variable vector X ^C ′ of each child individual is 0, that is, whether ‖X ^{C ′} ‖ = 0 (step S24).

If ‖X ^{C ′} ‖ = 0, the process returns to step S21 to generate a child individual again. If ‖X ^{C ′}で な い = 0 is not true, the normalization operation unit 22 performs the following normalization operation on the decision variable vector X ^C ′ of the child individual (step S25).

From the above equation, the decision variable vector X ^C of the child individual is obtained on the hypersphere.

  Thereafter, the individual selection unit 5 determines whether or not the necessary number of individuals has been generated (step S26). If the required number of individuals has not been generated, the process returns to step S21, and the processes of steps S21 to S26 are repeated. When the required number of individuals is generated, the crossover process is terminated.

  A hypersphere crossover method using UNDX as a crossover operator is referred to as UNDX-H (UNDX for Hypersphere). Child individuals generated from a parent individual on the hypersphere by UNDX-H are distributed on the hypersphere. As a result, it is possible to search for individuals using the real value GA while maintaining continuity on the hypersphere.

  Note that the crossover method on the hypersphere is not limited to UNDX, and other crossover operators with real values GA can be used. For example, BLX-α may be used as the crossover operator. In BLX-α, child individuals are generated with a uniform distribution determined from the positional relationship between two parent individuals.

(H) Generation of Initial Individual Set on (n-1) Dimensional Unit Spherical FIGS. 8a and 8b are diagrams illustrating an example of processing for generating an initial individual set on an (n-1) dimensional unit sphere. In FIGS. 8a and 8b, the n-dimensional real space R ⁿ is a two-dimensional real space R ² and the (n−1) -dimensional unit sphere S ^n-1 is a one-dimensional unit sphere S ¹ (unit circle). Indicated. 8A and 8B, the horizontal axis is the decision variable x ₁ , and the vertical axis is the decision variable x ₂ .

As shown in FIG. 8a, a plurality of individuals in a two-dimensional real space R ² are randomly generated. In this case, the plurality of individuals have a two-dimensional normal distribution.

Next, as shown in FIG. 8b, the normalization operation of the plurality of individuals in the two-dimensional real space R ² is converted to an individual on the unit circle.

(I) Effects of the First Embodiment In the optimization apparatus 1 according to the present embodiment, crossing on a hypersphere is used for crossing individuals. In this case, child individuals generated from parent individuals on the hypersphere are distributed on the hypersphere. As a result, it is possible to search for individuals using the real value GA while maintaining continuity on the hypersphere.

  In addition, since crossover with continuity is performed by an operation of setting an individual to a hypersphere and normalization calculation, optimization processing of a decision variable vector having a constraint condition that the sum of squares of all decision variables is 1 is performed. It is not complicated.

  As a result, it is possible to efficiently optimize a decision variable vector having a constraint that the sum of squares of all decision variables is 1.

(2) Second Embodiment In this embodiment, crossover for a periodic function, which will be described later, is used for individual crossover.

(A) Functional Configuration of Optimization Device Next, an optimization device according to the second embodiment of the present invention will be described with reference to FIG. FIG. 9 is a block diagram showing a functional configuration of the optimization apparatus according to the second embodiment of the present invention.

  The optimization device 1 in FIG. 9 is different from the optimization device 1 in FIG. 1 in the configuration of the individual generation unit 2. The configuration of the other parts of the optimization apparatus 1 in FIG. 9 is the same as the configuration of the corresponding part of the optimization apparatus 1 in FIG. Further, the hardware configuration of the optimization apparatus 1 in FIG. 9 is the same as the configuration shown in FIG.

  The user 10 sets an fitness function (objective function) in the optimization device 1. The fitness function includes CO (carbon monoxide) concentration, HC (hydrocarbon) concentration, NOx (nitrogen oxide) concentration, fuel consumption, output torque, throttle valve opening, engine speed in exhaust gas. Etc. can be set. In the present embodiment, HC concentration and NOx concentration in engine exhaust gas are used as the fitness function. The fitness is a value of the fitness function.

  In this embodiment, fuel injection timing, ignition timing, intake valve timing, and exhaust valve timing are used as decision variables. That is, the order of the decision variable vector is 4. Here, the fitness function having the fuel injection timing as a determining variable is a periodic function.

  The state quantity output from the optimization target 8 is, for example, measured values such as CO concentration, HC concentration, NOx concentration, fuel consumption, output torque, throttle valve opening, and engine speed in the exhaust gas. . In the present embodiment, the state quantity is a measured value of the HC concentration and NOx concentration in the exhaust gas.

  As shown in FIG. 9, the individual generation unit 2 includes a crossover unit 21, a trigonometric function calculation unit 23, and an arctangent function calculation unit 24. Detailed operation of the individual generation unit 2 will be described later. In the optimization apparatus 1 according to the present embodiment, crossover on a unit circle is performed as crossover of individuals having a periodic function.

  In the present embodiment, the fuel injection timing, the ignition timing, the intake valve timing, and the exhaust valve timing correspond to N variables, and the fuel injection timing corresponds to m variables. That is, in this embodiment, N = 4 and m = 1.

(B) Configuration of Optimization Target FIG. 10 is a block diagram illustrating an example of the configuration of the optimization target 8. The optimization target 8 in FIG. 10 is an engine evaluation device.

  The optimization target 8 includes an ECU (engine control unit) 81, an engine 82, and an exhaust gas analyzer 83.

  A fuel injection device 86 and an intake valve 87 are provided in the intake port 84 of the engine 82, and an exhaust valve 88 is provided in the exhaust port 85. The cylinder head 89 is provided with a spark plug 90.

  The ECU 81 receives the decision variable vector from the optimization device 1 by serial communication. In this example, the decision variable vector includes fuel injection timing, ignition timing, intake valve timing, and exhaust valve timing as elements. The ECU 81 receives a signal indicating the rotation angle (crank rotation angle) of the crankshaft 91, and based on the decision variable vector, the fuel injection timing of the fuel injection device 86, the ignition timing of the spark plug 90, and the opening / closing timing of the intake valve 87 (intake air) Valve timing) and the opening / closing timing (exhaust valve timing) of the exhaust valve 88 are controlled.

  The exhaust gas analyzer 83 analyzes the components in the exhaust gas from the engine 82, and outputs the exhaust gas data indicating the HC concentration and the NOx concentration to the optimization device 1 by serial communication as state quantities.

  The HC concentration and NOx concentration with the fuel injection timing as the determining variables are periodic functions with a period of 720 °.

  10 receives fuel injection timing, ignition timing, intake valve timing and exhaust valve timing as individual decision variable vectors from the optimization device 1, and outputs HC concentration and NOx concentration as state quantities.

(C) Overall Processing of Optimization Device The overall processing of the optimization device 1 in FIG. 9 will be described using the flowchart in FIG. As shown in FIG. 4, first, the individual generation unit 2 generates an initial individual set by randomly generating individuals within a predetermined decision variable range (step S1), and the generated initial individual set. Are sequentially output to the optimization target 8 (step S2). Thereby, state quantities are sequentially output from the optimization target 8.

  The fitness calculation unit 6 acquires a state quantity corresponding to each individual output from the optimization target 8 (step S3), and calculates the fitness of each individual based on the acquired state quantity (step S4). The calculated fitness of each individual is stored in the individual set storage unit 4 together with the decision variable vector.

  Next, the individual selection unit 5 selects a predetermined number of individuals used for generating a new child individual from a plurality of individuals stored in the individual set storage unit 4 as a parent individual (step S5). In the present embodiment, three parent individuals are selected.

  Furthermore, the individual generation unit 2 generates a plurality of new child individuals by crossover using a predetermined number of parent individuals selected by the individual selection unit 5 (step S6). Here, since the fitness function having the fuel injection timing as a determining variable is a periodic function, the crossover method described later is used as an element of the fuel injection timing, and the three elements of ignition timing, intake valve timing, and exhaust valve timing are used. In general, a general crossover method such as UNDX is used.

  Here, the fuel injection timing is a decision variable represented by an angle. As will be described later, the angle is decomposed into two orthogonal new decision variables in the two-dimensional real space. Therefore, the order of the decision variable vector is 4 before the crossover, the order of the decision variable vector is 5 at the time of crossover, and the order of the decision variable vector returns to 4 after the crossover.

  The individual generation unit 2 sequentially outputs the decision variable vector of each child individual to the optimization target 8 (step S7). Thereby, state quantities are sequentially output from the optimization target 8.

  The fitness calculation unit 6 acquires a state quantity corresponding to each child individual output from the optimization target 8 (step S8), and calculates the fitness of each child individual based on the acquired state quantity (step S9). ). The calculated fitness of each child individual is stored in the individual set storage unit 4 together with the decision variable vector.

  The generation change unit 3 updates the individual set stored in the individual set storage unit 4 by replacing the child individual with the parent individual based on the fitness of each child individual and each parent individual (step S10). In this case, the generation change unit 3 updates the individual set by at least one method selected from selection based on a comparison of fitness levels, selection based on the probability determined by the high fitness level, and random selection. As a result, generational changes are made. Due to the generation change, individuals with poor fitness belonging to the individual set are deceived.

  Thereafter, the individual generation unit 2 determines whether or not the number of generations has reached a predetermined end condition (step S11).

  If it is determined that the number of generations has not reached the predetermined end condition, the process proceeds to step S5. If it is determined that the number of generations has reached a predetermined end condition, the individual generation unit 2 presents the individual set including the optimum individual stored in the individual set storage unit 4 to the user 10 and ends the process. .

(D) Basic Concept of Crossover Method for Periodic Function Next, a crossover method for a periodic function that applies the above-described crossover method on a hypersphere will be described. In the two-dimensional real space (two-dimensional real plane) R ² , the one-dimensional unit spherical surface S ¹ (unit circle) is defined as follows:

S ¹ = {(x, y) ∈R ² | x ² + y ² = 1}
In the above equation, x and y are the coordinates of the two-dimensional real plane R ² . The unit circle S ¹ satisfies the condition x ² + y ² = 1.

Here, it is considered that the angle θ is optimized as a point on the unit circle S ¹ for the periodic function f (θ) having the angle θ∈ [−π, π] as a decision variable. In other words, consider the decision variable as a unit direction vector in the two-dimensional real plane R ^2.

  11a, 11b, and 11c are schematic diagrams for explaining a crossing method on a unit circle. In this example, UNDX is used as the crossover operator.

In FIG. 11a, a unit circle UC that satisfies the condition of x ² + y ² = 1 is set in the two-dimensional real plane R ² having x and y coordinates. The angle θ of the decision variable vector of each individual is decomposed into the x coordinate and the y coordinate. In this case, the coordinates (x, y) on the unit circle are calculated by the following formula for each individual having the angle θ as a decision variable.

(X, y) = (cos θ, sin θ)
That is, consider a new decision variable vector whose elements are a variable represented by an x coordinate and a variable represented by a y coordinate. As a result, the search point of each individual is set on the two-dimensional real plane R ² based on the calculated coordinates (x, y).

In the example of FIG. 11a, three parent individuals P ¹ , P ² and P ³ are set on the unit circle UC. The decision variables of the ^three parent individuals P ¹ , P ² , P ³ are the angles θ ¹ , θ ² , θ ³ . In UNDX, child individuals are generated according to normal distribution random numbers determined based on the positional relationship between the ^three parent individuals P ¹ , P ² , and P ³ .

As shown in FIG. 11b, a plurality of child individuals C ^{1 to} C ⁵ are generated from the x and y coordinates of the ^three parent individuals P ¹ , P ² and P ³ .

Next, as shown in FIG. 11c, the angles θ ^{C1 to} θ ^C5 that are decision variables for the x-coordinate and y-coordinate of each of the child individuals C ^{1 to} C ⁵ on the two-dimensional real number plane R ² by the arctangent function of the following equation: Get each. In the following equation, angles θ ^C1 to θ ^C5 of a plurality of offspring C ^{1 to} C ⁵ are defined as an angle θ ^C.

  In the above equation, sign (y) becomes +1 when y is positive, and becomes -1 when y is negative.

  The crossover method of the second embodiment is equivalent to the case where a unit circle is used as a hypersphere in the crossover method of the first embodiment. In the present embodiment, when a new individual (search point) having an angle θ is finally obtained as a decision variable, the decision variable is directly determined using an arctangent function without performing the normalization operation in the first embodiment. Is obtained.

(E) Procedure of Periodic Function Crossover Method FIG. 12 is a flowchart showing the crossover process on a unit circle. The crossover process of FIG. 12 is performed by the individual selection unit 5, the crossover unit 21, the trigonometric function calculation unit 23, and the arctangent function calculation unit 24 of FIG.

First, the trigonometric function calculation unit 23 decomposes the angle θ, which is a decision variable of each individual, into x and y coordinates on the two-dimensional real plane R ² by calculation using a trigonometric function (step S31), It arrange | positions on a unit circle (step S32). The individual set distributed on the unit circle is stored in the individual set storage unit 4.

  Next, the individual selection unit 5 selects a predetermined number of individuals as a parent individual from the set of individuals distributed on the unit circle (step S33).

Next, the crossover unit 21 generates a plurality of child individuals by applying a crossover operator as points on the two-dimensional real plane R ² to a predetermined number of parent individuals selected from the unit circle (step S34). ).

The arc tangent function calculation unit 24 converts the x coordinate and y coordinate of the child individual into an angle by calculation using the arc tangent function of the above equation (step S35). Thereby, an angle θ ^C that is a decision variable of the child individual is obtained.

  Thereafter, the individual selection unit 5 determines whether or not the required number of individuals has been generated (step S36). If the necessary number of individuals has not been generated, the process returns to step S31, and the processes of steps S31 to S36 are repeated. When the required number of individuals is generated, the crossover process is terminated.

  A crossover method on a unit circle using UNDX as a crossover operator is called UNDX-P (UNDX for Periodic function).

  Note that the crossover method on the unit circle is not limited to UNDX, and other crossover operators with real values GA can be used. For example, BLX-α may be used as the crossover operator.

(F) Extension of crossover method on unit circle to high-dimensional periodic function n-dimensional periodic function for n-dimensional angle vector (n-dimensional decision variable vector) θ = (θ ₁ , θ ₂ ,..., Θ _n ) A method using a crossover method on a unit circle for f (θ) will be described. n is an arbitrary natural number. Here, θ ₁ , θ ₂ ,..., Θ _n are angles that are decision variables, and are elements of an n-dimensional decision variable vector.

The x _i and y _i coordinates on the unit circle are calculated from (x _i , y _i ) = (cos θ _i , sin θ _i ) for the angle θ _i that is the decision variable. Here, i = 1, 2,..., N. Thereby, each individual is arranged on the 2n-dimensional real number plane R ²ⁿ . A parent individual is selected from the individuals distributed in the 2n-dimensional real number plane R ²ⁿ , UNDX is performed on the selected parent individual, and the x _i and y _i coordinates of each child individual are calculated. Thereafter, the x _i and y _i coordinates of each child individual are converted into an angle θ ^Ci which is a decision variable of each child individual by an operation using an arctangent function.

In this case, the unit circle corresponds to a one-dimensional unit sphere, θ ₁ , θ ₂ ,..., Θ _n correspond to a set of m variables, the n-dimensional real space R ⁿ corresponds to an N-dimensional variable space, The 2n-dimensional real number plane R ^{2n corresponds} to a [(N−m) + 2m] -dimensional variable space, and the n-dimensional angle vector (n-dimensional decision variable vector) θ = (θ ₁ , θ ₂ ,..., Θ _n ) is a variable vector. It corresponds to. That is, in this example, N = m = n.

  FIG. 13 is a schematic diagram showing a method of extending the crossover method on the unit circle to a decision variable vector composed of an n-dimensional angle vector. In this example, n = 3.

As shown in FIG. 13, the x ₁ coordinate and the y ₁ coordinate on the unit circle are calculated from (x ₁ , y ₁ ) = (cos θ ₁ , sin θ ₁ ) for the angle θ ₁ that is a decision variable. The x ₁ coordinate and y ₁ coordinate of each child individual are calculated by the crossing method on the unit circle. Thereafter, an angle θ ^C1 that is a decision variable of each child individual is calculated from the x ₁ coordinate and y ₁ coordinate of each child individual using an arctangent function.

For the angle θ ₂ that is a decision variable, the x ₂ coordinate and the y ₂ coordinate on the unit circle are calculated from (x ₂ , y ₂ ) = (cos θ ₂ , sin θ ₂ ), respectively. The x ₂ coordinate and y ₂ coordinate of each child individual are calculated by the crossing method on the unit circle. Then, to convert the x ₂ coordinate and y ₂ coordinate of each child individual by arctan angle theta ^C2 is a decision variable for each child individual.

For the angle θ ₃ that is a decision variable, the x ₃ coordinate and the y ₃ coordinate on the unit circle are calculated from (x ₃ , y ₃ ) = (cos θ ₃ , sin θ ₃ ), respectively. The x ₃ coordinate and y ₃ coordinate of each child individual are calculated by the crossing method on the unit circle. Thereafter, the x ₃ coordinate and y ₃ coordinate of each child individual are converted into an angle θ ^C3 which is a decision variable of each child individual by an arctangent function.

(3) Other Embodiments (a) Domain Conversion The domain representing one cycle is not [−π, π], and the periodic function f (φ) of the following expression is changed to the above-described first and second embodiments. A case where the crossover method is applied will be described.

f (φ), φ∈ [φ _min , φ _max ]
In the above equation, φ is a decision variable, φ _min indicates the minimum value of the domain, and φ _max indicates the maximum value of the domain. Hereinafter, a method for converting the domain [φ _min , φ _max ] to the domain [−π, π] will be described.

  14A and 14B are schematic diagrams for explaining the domain conversion method. The horizontal axis of FIG. 14a is the decision variable φ, and the vertical axis is the value of the periodic function f (φ). The horizontal axis of FIG. 14b is the decision variable θ, and the vertical axis is the value of the periodic function f (θ).

The individual selection unit 5 adjusts the scale of the decision variable φ and adjusts the periodic function f (θ) having the domain [φ _min , φ _max ] of FIG. 14 a so that the size of the domain is 2π. The periodic function is translated along the horizontal axis so that the center of the definition area of the subsequent periodic function becomes zero. As shown in FIG. 14b, the determination variable θ after the scale adjustment and the parallel movement is used. Thereby, a periodic function f (θ) whose domain is [−π, π] is obtained. In this way, conversion from the decision variable φ to the decision variable θ is performed.

  Thereafter, crossover for a periodic function is performed, and after the crossover, inverse conversion from the decision variable θ to the decision variable φ is performed.

(B) Application Example to Engine Simulator FIG. 15 is a block diagram showing an example in which the optimization device is applied to the engine simulator.

  The optimization target 8a in FIG. 15 is an engine simulator. The engine simulator is composed of a personal computer, for example. The optimization object 8a performs a simulation of engine operation based on the decision variable vector given from the optimization device 1, and outputs the simulation result to the optimization device 1 as a state quantity.

  For example, fuel injection timing, ignition timing, intake valve timing, and exhaust valve timing are set as the decision variables. As the fitness function, for example, the HC concentration and NOx concentration contained in the exhaust gas of the engine are set. The HC concentration and NOx concentration with respect to the fuel injection timing are periodic functions.

  In the present embodiment, the simulation results of the HC concentration and NOx concentration contained in the exhaust gas are output as the exhaust gas data from the optimization target 8a.

  According to the optimization device 1 of FIG. 15, the fuel injection timing can be optimized efficiently by setting the periodic function as the fitness function.

(C) Application Example to Robot Evaluation Device FIG. 16 is a block diagram showing an example in which the optimization device is applied to the robot evaluation device.

  The optimization target 8b in FIG. 16 is a robot evaluation apparatus. The robot evaluation device includes an ECU 91 and a robot 92.

  The robot 91 has, for example, redundant arms having four or more joints in a two-dimensional plane. Further, the robot 91 may have a redundant arm having seven or more joints in the three-dimensional space.

  When the hand position of the arm of the robot 91 is determined, the angle of each joint cannot be uniquely determined. Therefore, it is necessary to optimize the target joint angle. When the link mechanism of the arm of the robot 91 can make one rotation, the optimization of the target joint angle can be considered as a periodic function optimization problem.

  For example, the target joint angle of the link mechanism of the robot 91 is set as the decision variable vector. As the fitness function, for example, the power consumption for posture maintenance, the integrated value of the power consumption from the start posture to the end posture, or the number of times of contact when an obstacle exists in the movement path of the arm of the robot 91 is set. .

  The ECU 91 of the optimization target 8b in FIG. 16 receives the target joint angle from the optimization device 1 as the individual decision variable vector, and controls the link mechanism of the robot 92. The power consumption with the target joint angle as a decision variable vector is a periodic function. The robot 92 outputs, for example, power consumption data for a specific arm tip position to the optimization device 1 as a state quantity.

  According to the optimization device 1 of FIG. 16, the target joint angle of the link mechanism of the robot 92 can be optimized efficiently by setting a periodic function as the fitness function.

(D) Application Example to Robot Simulator FIG. 17 is a block diagram showing an example in which the optimization apparatus is applied to the robot simulator.

  The optimization target 8c in FIG. 17 is a robot simulator. The robot simulator is composed of a personal computer, for example. The optimization target 8c performs a simulation of the robot operation based on the decision variable vector given from the optimization device 1, and outputs the simulation result to the optimization device 1 as a state quantity.

  As the decision variable vector, for example, the target joint angle of the link mechanism of the robot is set. As the fitness function, for example, power consumption for posture maintenance, an integrated value of power consumption from the start posture to the end posture, or the number of times of contact when an obstacle exists in the movement path of the robot arm is set.

  In the present embodiment, a simulation result of power consumption for a specific arm tip position is output from the optimization target 8c as power consumption data.

  According to the optimization apparatus 1 of FIG. 17, the target joint angle of the robot link mechanism can be optimized efficiently by setting the periodic function as the fitness function.

(E) Application Example to Robot Simulator FIG. 18 is a block diagram showing an example in which the optimization apparatus is applied to a robot simulator.

  The optimization target 8d in FIG. 18 is a robot simulator. The robot simulator is composed of a personal computer, for example. The optimization target 8d performs a simulation of the robot operation based on the decision variable vector given from the optimization device 1, and outputs the simulation result to the optimization device 1 as a state quantity.

  Simulates the motion of a biped robot swinging an arm at the same cycle as walking. The optimization of the timing of swinging the arm when the time when the free leg has landed is 0 and the time when the opposite free leg is landed is T1 is considered as an optimization problem of the periodic function of the period T1. Can do.

  Further, by adding the arm swing speed to the decision variable vector, it is possible to optimize the arm swing timing that maximizes the walking stabilization index and the arm swing timing that improves the walking efficiency.

  In the present embodiment, the timing and speed of swinging the arm as a decision variable vector are input to the optimization target 8d, and the simulation result of the walking stabilization index and the walking efficiency is obtained from the optimization target 8d as the stabilization index data and the efficiency. Output as data.

  According to the optimization apparatus 1 of FIG. 18, the arm swing timing can be optimized efficiently by setting a periodic function as the fitness function.

(F) Application Example to Protein Simulator FIG. 19 is a block diagram showing an example in which the optimization device is applied to a protein simulator.

  The optimization target 8e in FIG. 19 is a protein simulator. The protein simulator is composed of, for example, a personal computer. The optimization object 8e performs a simulation of the protein structure based on the decision variable vector given from the optimization device 1, and outputs the simulation result to the optimization device 1 as a state quantity.

  In the present embodiment, the dihedral angle of the protein is input to the optimization target 8e as a decision variable vector, and the energy simulation result is output from the optimization target 8e as energy data.

  According to the optimization apparatus 1 of FIG. 19, the dihedral angle of the protein can be optimized efficiently by setting the periodic function as the fitness function.

(G) Application Example to Traffic Simulator FIG. 20 is a block diagram showing an example in which the optimization device is applied to a traffic simulator. FIG. 21 is a schematic diagram for explaining wide-area control of traffic signals.

  The optimization target 8f in FIG. 20 is a traffic simulator. The traffic simulator is composed of a personal computer, for example. This optimization object 8f performs a simulation of traffic signal wide-area control based on the decision variable vector given from the optimization device 1, and outputs the simulation result to the optimization device 1 as a state quantity.

  In wide area control of traffic signals, traffic flow is controlled by the lighting timing of signals installed on a certain road. Congestion can be minimized by optimizing the lighting timing of the signal for the traffic volume that changes from moment to moment.

  On the road RT shown in FIG. 21, the traffic flow is considered to be controlled by the lighting timing of the signals A and B. At this time, the traffic flow can be optimally controlled by optimizing the lighting timing of the signal B that maximizes the number or average speed of the vehicles that have passed the road RT.

  The lighting colors of the signals A and B change in the order of blue, yellow and red, and return from red to blue again. At this time, the time required for one lighting cycle of the signals A and B is defined as t1. When the blue lighting start timing of the signal A is set to time 0, the blue lighting start timing of the signal B is changed within the definition range from the section 0 to t1. Also, setting the blue lighting start timing of signal B to time 0 is completely the same as setting the blue lighting start timing of signal B to time t1. Therefore, the above traffic flow control can be considered as an optimization problem of a periodic function using the blue lighting start timing of the signal B as a decision variable vector.

  Furthermore, when the traffic volume is different between the up lane and the down lane, optimizing the lighting timing of the signals in both lanes can be considered as a multipurpose periodic function optimization problem.

  In the present embodiment, the lighting start timing of the signal B is input to the optimization target 8e as a decision variable vector, and a traffic volume simulation result is output from the optimization target 8f as traffic volume data.

  According to the optimization apparatus 1 of FIG. 20, the signal lighting timing can be optimized efficiently by setting the periodic function as the fitness function.

(H) Application Example to Image Processing Device FIG. 22 is a block diagram showing an example in which the optimization device is applied to an image processing device. FIG. 23 a is a schematic diagram for explaining two-dimensional pattern matching. FIG. 23b is a schematic diagram for explaining the three-dimensional pattern matching.

  The optimization target 8g in FIG. 22 is an image processing apparatus. The image processing apparatus is composed of a personal computer, for example. The optimization target 8g performs pattern matching based on the decision variable vector given from the optimization device 1, and outputs the matching amount to the optimization device 1 as a state quantity.

  In FIG. 23a, (x, y) is a two-dimensional unit direction vector obtained at the intersection on the hypersphere. This two-dimensional unit direction vector represents the direction of the template PI. In the two-dimensional pattern matching, first, the edge of the target image PO is extracted by a differentiation method or the like.

  Next, the coordinates of the template PI, the enlargement ratio, and the two-dimensional unit direction vector are input to the optimization target 8e as decision variable vectors, and the matching amount between the target image PO and the template PI based on the extracted edge of the target image PO. (Degree of match) is output.

  In FIG. 23b, (x, y, z) is a three-dimensional unit direction vector obtained at the intersection on the hypersphere. This three-dimensional unit direction vector represents the direction of the template Pi. In the three-dimensional pattern matching, first, the edge of the target image Po is extracted by a differentiation method or the like.

  Next, the coordinates of the template Pi, the enlargement factor, and the three-dimensional unit direction vector are input to the optimization target 8e as the decision variable vector, and the matching amount between the target image Po and the template Pi based on the extracted edge of the target image Po. (Coincidence) is output.

  According to the optimization apparatus 1 of FIG. 22, by setting a periodic function as the fitness function, it is possible to efficiently estimate the posture of the target image by pattern matching.

(I) Other Examples of Evolutionary Algorithm In the above embodiment, the genetic algorithm (GA) is used as the evolutionary algorithm. However, the present invention is not limited to this, and instead of the genetic algorithm, an evolution strategy (ES: Calculation methods based on similar ideas such as (Evolution Strategy) may be used.

  Note that calculation methods such as GA and ES are collectively referred to as evolutionary algorithms (EAs) or evolutionary computation (Evolutionary Computation).

(J) Realization method of each unit In the above embodiment, the individual generation unit 2, the generation change unit 3, the individual selection unit 5, and the fitness calculation unit 6 are realized by the CPU 101 and the program. Part or all of the unit 3, the individual selection unit 5, and the fitness calculation unit 6 may be realized by hardware such as an electronic circuit.

(4) Example 1 and Comparative Example 1
In order to confirm the effectiveness of the crossover method on the unit circle in the second embodiment, a numerical experiment using a periodic test function was performed.

  In Example 1 below, optimization using a periodic test function was performed by the optimization apparatus 1 according to the second embodiment. In Comparative Example 1, optimization using a periodic test function was performed using an optimization device similar to the optimization device according to the second embodiment except for the crossing operator. In Example 1, the crossover method (UNDX-P) on the unit circle was used as the crossover operator. In Comparative Example 1, UNDX was used as the crossover operator, and a method of regenerating an individual when the individual deviated from the definition range was adopted. First, the periodic test function and the evaluation method will be described.

  In Example 1 and Comparative Example 1, three types of periodic test functions TF-1, TF-2 and Fletcher and Powell functions (hereinafter referred to as FP) were used.

24A is a diagram illustrating a schematic shape of the periodic test function TF-1, FIG. 24B is a diagram illustrating a schematic shape of the periodic test function TF-2, and FIG. 24C is a diagram illustrating a schematic shape of the periodic test function FP. In FIG. 24a to FIG. 24c, two orthogonal horizontal axes are variables x ₁ and x ₂ , and a vertical axis is fitness (value of fitness function f (x)). Here, x = (x ₁ , x ₂ ).

  FIG. 25 is a diagram showing details of the three types of periodic test functions used in Example 1 and Comparative Example 1.

The periodic test function TF-1 has a unimodal shape, and the periodic test function TF-2 has a multimodal shape. The periodic test function TF-1 and the periodic test function TF-2 are periodic functions of period 2 that take the minimum value 0 when x _i = 2N + 1, N = 0, ± 1, ± 2,. Let the variable domain be x _i ∈ [0 + L, 2 + L].

The periodic test function FP is a periodic function with a period of 2π having a plurality of optimal solutions that cannot be accurately distinguished from the optimal solution α = (α ₁ , α ₂ ,..., Α _n ) having the minimum value 0, and a large number of local solutions. is there. Here, the values described in Non-Patent Document 2 were used as the constants a, b, and α of the periodic test function FP shown in FIG. Let the domain of the variable be x _i ∈ [−π + Lπ, π + Lπ].

  Here, L is a domain offset, and the range of the offset L is [0, 1]. In the periodic test function TF-1 and the periodic test function TF-2, the optimum solution is positioned at the center of the domain when L = 0, and the optimum solution approaches the boundary of the domain as the offset L approaches 1. Become. In the periodic test function FP, the positional relationship between the optimal solution and the local solution is expected to change in a complicated manner.

  In Example 1 and Comparative Example 1, the dimension n of the periodic test function TF-1 is 10 and 20, the dimension n of the periodic test function TF-2 is 4 and 6, and the dimension n of the periodic test function FP is 6 and 9 In the case where the offset L is 0, 0.3, 0.6, and 0.9, 30 optimizations are performed (30 trials), and the number of times the optimal solution is found and the state of convergence are evaluated. .

Ideally, an individual whose fitness is 0 is the optimal solution. Here, it is assumed that the optimal solution is found when the fitness of the individual becomes less than 1.0 × 10 ⁻⁷ .

As conditions for the genetic algorithm, an individual set size is set to 100 and the number of generated child individuals is set to 50. The MGG shown in FIG. 5 is used as a generation change model, and elite selection and roulette wheel selection are used as selection methods. Assume that the number of evaluations of the periodic test function TF-1 and the periodic test function TF-2 is 5.0 × 10 ⁵ times, and the number of evaluations of the periodic test function FP is 7.5 × 10 ⁵ times. Here, the number of evaluations is obtained by (number of generations × number of child individuals + number of individuals in the initial individual set).

  FIG. 26 is a diagram showing the number of times of finding the optimum solution related to the periodic test functions TF-1 and TF-2 in Example 1 and Comparative Example 1.

  In FIG. 26, the smallest value among the optimal solution discovery times at the four offsets L is underlined. In both of the periodic test functions TF-1 and TF-2, in the first comparative example, the optimal solution discovery count decreases as the offset L approaches 1 (that is, as the position of the optimal solution approaches the domain boundary). On the other hand, in the first embodiment, even when the offset L approaches 1 (that is, the position of the optimal solution approaches the boundary of the domain), the number of times of finding the optimal solution does not decrease, and in all 30 trials An optimal solution has been found.

  With respect to the periodic test function FP (n = 6), in the first comparative example, the optimal solution discovery count decreases as the offset L approaches 1. On the other hand, in the first embodiment, the optimal solution discovery count is not affected at all by the offset L, and the optimal solution is found in all 30 trials.

  Regarding the periodic test function FP (n = 9), in Comparative Example 1, the number of times of finding the optimum solution varies greatly due to the offset L. On the other hand, in the first embodiment, the optimum solution discovery frequency is hardly affected by the offset L, and the optimum solution is found in about half of the 30 trials.

  As a result, the stability of optimization in Example 1 could be confirmed.

  FIGS. 27a and 27b are diagrams showing how the periodic test function TF-1 converges in Example 1 and Comparative Example 1, respectively. FIGS. 27 a and 27 b show the results when dimension n is 20. FIG. 28a and 28b are diagrams showing how the periodic test function TF-2 converges in Example 1 and Comparative Example 1, respectively. 28a and 28b show the results when dimension n is 6. FIG. FIGS. 29a and 29b are diagrams showing how the periodic test function FP converges in Example 1 and Comparative Example 1, respectively. Figures 29a and 29b show the results for dimension n = 9.

  27a to 29b, the horizontal axis indicates the number of evaluations, and the vertical axis indicates the fitness. The fitness indicates the average fitness in the trial in which the optimal solution was found. A solid line indicates that the offset L is 0.0, a dotted line indicates that the offset L is 0.3, an alternate long and short dash line indicates that the offset L is 0.6, and a broken line indicates that the offset L is 0.9.

As shown in FIGS. 27b and 28b, with respect to the periodic test functions TF-1 and TF-2, in Comparative Example 1, the fitness of an individual generated as the offset L approaches 1 is the optimal solution (1.0 the rate of convergence in less than × 10 ^-7) is getting late. On the other hand, as shown in FIGS. 27a and 28a, in the first embodiment, the fitness of individuals generated is the same regardless of the offset L (that is, the domain) of the periodic test functions TF-1 and TF-2. It converges to the optimal solution (less than 1.0 × 10 ⁻⁷ ) at speed.

As for the periodic test function FP, in Comparative Example 1, as shown in FIG. 29b, the convergence speed greatly varies depending on the difference in the optimum solution obtained for each trial. On the other hand, in the first embodiment, as shown in FIG. 29a, the fitness of the individuals generated regardless of the offset L (that is, the domain) is the same speed up to the vicinity of the evaluation number 2.0 × 10 ⁵ times. It has converged. Thereby, it is considered that a stable search is performed until the fitness of the generated individual converges near the fitness of any optimal solution.

  FIG. 30 is a diagram illustrating a typical solution position and the number of times of discovery regarding the periodic test function FP (n = 6) in the first embodiment and the first comparative example. FIG. 31 is a diagram illustrating a representative solution position and the number of times of discovery regarding the periodic test function FP (n = 9) in the first embodiment and the first comparative example.

  In the optimization of the periodic test function FP, two optimal solutions and three local solutions were obtained in 30 × 8 trials when n = 6. In addition, when n = 9, 8 optimal solutions and 18 local solutions were obtained in 30 × 8 trials.

  FIG. 30 representatively shows two optimum solutions Opt1, Opt2 and one local solution Loc1 in the case of n = 6. FIG. 31 representatively shows eight optimal solutions Opt1 to Opt8 and three local solutions Loc1 to Loc3 in the case of n = 9.

  Even in the same solution, the position differs depending on the offset L (domain), but solutions having substantially the same position are regarded as the same solution. The number of times of finding the most generated solutions at each offset L is underlined.

  As shown in FIGS. 30 and 31, in the first embodiment, the position of the solution found regardless of the offset L (definition area) is stable. On the other hand, in the first comparative example, the number of times the optimal solution is found depends greatly on the offset L (definition area), and an optimal solution is found at a higher rate than in the first embodiment at a certain offset L (definition area). However, an optimal solution is found at a lower rate than in the first embodiment at other offsets L (domain). This is considered to be because an optimum solution near the center of the domain is easily obtained by the sampling bias.

When n = 9 and L = 0.6, the local solution Loc2 has an element (x ₃ = −1.255637) equal to the lower limit of the domain (−π + 0.6π = −1.255637). The optimal solution Opt2 is a trivial optimal solution α and has an element α ₃ = −1.283410 <x ₃ . As a result, the local solution Loc2 is a strong local solution near the optimal solution Opt2. Therefore, the local solution Loc2 is considered to be a local solution generated by phase cutting. Similarly, 2 local solutions out of 3 local solutions when n = 6 and 13 local solutions out of 18 local solutions when n = 9 are equal to domain boundaries. It has elements and is generated by phase cutting.

  As described above, when the conventional crossover method is used to optimize the decision variable vector in the periodic function, the positional relationship between the optimal solution and the local solution changes in a complicated manner depending on the setting of the domain. In addition, a large number of new local solutions resulting from phase cutting are generated. As a result, it can be seen that the setting of the domain has a great influence on the difficulty of optimization.

  On the other hand, when the crossover method for the periodic function is used to optimize the decision variable vector in the periodic function, the positional relationship between the optimal solution and the local solution is not affected by the setting of the domain. In addition, a new local solution is not generated by phase cutting. Therefore, it is understood that the crossover method for the periodic function is effective for optimizing the decision variable vector in the periodic function.

(5) Example 2
In Example 2, the fuel injection timing was optimized by the multipurpose genetic algorithm using the optimization device 1 according to the second embodiment.

As decision variables, ignition timing, fuel injection timing, intake valve timing and exhaust valve timing are set, and NOx concentration and HC concentration are set as fitness functions. That is, the optimization problem of the second embodiment is a multi-objective optimization problem with two objectives and four variables. The HC concentration and the NOx concentration have a trade-off relationship. That is, when the HC concentration is lowered, NOx Concentration increases, NOx Lowering the concentration increases the HC concentration. Such a multi-objective optimization problem generally does not have a single optimal solution, but has an optimal solution set defined by a concept called a Pareto optimal solution.

  The fuel injection timing is a parameter that determines at which timing fuel injection is performed with respect to the rotation angle of the crankshaft. The rotation angle of the crankshaft is defined in the range of 0 to 720 °. At this time, the influence of the fuel injection timing on the exhaust gas is the same when the rotation angle of the crankshaft is 0 ° and when the rotation angle is 720 °. Therefore, it can be seen that the HC concentration and NOx concentration with the fuel injection timing as a decision variable vector are periodic functions with a period of 720 °.

  In the second embodiment, the crossover method (UNDX-P) on the unit circle in the second embodiment is applied only to the optimization of the fuel injection timing.

  32 and 33 are diagrams showing the optimum individuals obtained by the optimization apparatus 1 of FIG. The horizontal axis in FIG. 32 indicates the ignition timing, and the vertical axis indicates the fuel injection timing. The horizontal axis in FIG. 33 indicates the fuel injection timing, and the vertical axis indicates the intake valve timing.

  In the second embodiment, since multi-objective optimization is performed, a plurality of Pareto optimal individuals can be obtained. 32 and 33, the fuel injection timing is 0 and the fuel injection timing is 1.

  As shown in FIGS. 32 and 33, it can be seen that the fuel injection timings are distributed so as to cross between 0 and 1. As a result, it can be seen that the optimum individual near the boundary of the domain can be found.

FIG. 34 shows the HC concentration and NOx obtained in Example 2. It is a figure which shows the relationship of a density | concentration. As shown in FIG. 34, it can be seen that the Pareto optimal population is obtained for the HC concentration and the NOx concentration having a trade-off relationship.

(6) Correspondence between constituent elements of claim and each part of embodiment In the above embodiment, the individual set storage unit 4 corresponds to a solution candidate storage unit, the individual selection unit 5 corresponds to a solution candidate selection unit, The individual generation unit 2 corresponds to a solution candidate generation unit, the fitness calculation unit 6 corresponds to an evaluation value calculation unit, and the generation change unit 3 corresponds to a solution candidate set update unit.

  Also, the decision variable corresponds to the variable to be optimized, the individual corresponds to the solution candidate, the predetermined number of parent individuals corresponds to the predetermined number of solution candidates, and the plurality of child individuals correspond to new solution candidates, The degree corresponds to the evaluation value.

In particular, in the first embodiment, the n-dimensional real space R ⁿ or the two-dimensional real space (two-dimensional real plane) R ² corresponds to the N-dimensional variable space, and the (n−1) -dimensional unit spherical surface S ^n−1. , The hypersphere or unit circle corresponds to the (m−1) -dimensional unit sphere, and n decision variables x ₁ , x ₂ ,..., X _{n correspond} to N variables and m variables. That is, in the first embodiment, N = m = n.

In the second embodiment, the unit circle UC corresponds to a one-dimensional unit sphere, the two-dimensional real space (two-dimensional real plane) R ² corresponds to a two-dimensional plane, and θ ₁ , θ ₂ ,. θ _n corresponds to a set of m variables.

  Further, the fuel injection timing or the fuel injection pattern corresponds to a parameter based on the crank rotation angle, the target images PO and Po correspond to the target image, the templates PI and Pi correspond to the reference image, and the matching amount corresponds to an error. .

  The present invention can be used for optimization of variables in a periodic function.

It is a block diagram which shows the functional structure of the optimization apparatus which concerns on the 1st Embodiment of this invention. It is a block diagram which shows the hardware constitutions of the optimization apparatus of FIG. It is a figure for demonstrating the method of applying an optimization apparatus to optimization of a fuel injection pattern. It is a flowchart which shows the whole process of the optimization apparatus of FIG. It is a schematic diagram which shows an example of the optimization process by the optimization apparatus of FIG. It is a schematic diagram for demonstrating the crossing method on a hypersphere. It is a schematic diagram for demonstrating the crossing method on a hypersphere. It is a schematic diagram for demonstrating the crossing method on a hypersphere. It is a flowchart which shows the cross process on a hypersphere. It is a figure which shows an example of the production | generation process of the initial individual set on a (n-1) dimension unit spherical surface. It is a figure which shows an example of the production | generation process of the initial individual set on a (n-1) dimension unit spherical surface. It is a block diagram which shows the functional structure of the optimization apparatus which concerns on the 2nd Embodiment of this invention. It is a block diagram which shows an example of a structure of the optimization object. It is a schematic diagram for demonstrating the crossing method on a unit circle. It is a schematic diagram for demonstrating the crossing method on a unit circle. It is a schematic diagram for demonstrating the crossing method on a unit circle. It is a flowchart which shows the crossover process on a unit circle. It is a schematic diagram which shows the method of extending the crossing method on a unit circle to the decision variable vector which consists of an n-dimensional angle vector. It is a schematic diagram for demonstrating the adjustment method of a definition area. It is a schematic diagram for demonstrating the adjustment method of a definition area. It is a block diagram which shows the example which applied the optimization apparatus to the engine simulator. It is a block diagram which shows the example which applied the optimization apparatus to the robot evaluation apparatus. It is a block diagram which shows the example which applied the optimization apparatus to the robot simulator. It is a block diagram which shows the example which applied the optimization apparatus to the robot simulator. It is a block diagram which shows the example which applied the optimization apparatus to the protein simulator. It is a block diagram which shows the example which applied the optimization apparatus to the traffic simulator. It is a schematic diagram for demonstrating wide area control of a traffic signal. It is a block diagram which shows the example which applied the optimization apparatus to the image processing apparatus. It is a schematic diagram for demonstrating two-dimensional pattern matching. It is a schematic diagram for demonstrating three-dimensional pattern matching. It is a figure which shows schematic shape of periodic test function TF-1. It is a figure which shows the schematic shape of periodic test function TF-2. It is a figure which shows the schematic shape of the period test function FP. It is a figure which shows the detail of three types of periodic test functions used in Example 1 and Comparative Example 1. FIG. It is a figure which shows the optimal individual discovery frequency regarding the periodic test function TF-1, TF-2, and FP in Example 1 and Comparative Example 1. FIG. It is a figure which shows the mode of convergence of the periodic test function TF-1 in Example 1 and Comparative Example 1. FIG. It is a figure which shows the mode of convergence of the periodic test function TF-1 in Example 1 and Comparative Example 1. FIG. It is a figure which shows the mode of convergence of the periodic test function TF-2 in Example 1 and Comparative Example 1. FIG. It is a figure which shows the mode of convergence of the periodic test function TF-2 in Example 1 and Comparative Example 1. FIG. It is a figure which shows the mode of convergence of the periodic test function FP in Example 1 and Comparative Example 1. FIG. It is a figure which shows the mode of convergence of the periodic test function FP in Example 1 and Comparative Example 1. FIG. It is a figure which shows the position of the typical individual regarding the periodic test function FP (n = 6) in Example 1 and Comparative Example 1, and the number of times of discovery. It is a figure which shows the position of the typical individual regarding the periodic test function FP (n = 9) in Example 1 and Comparative Example 1, and the number of times of discovery. It is a figure which shows the optimal individual | organism | solid obtained by the optimization apparatus of FIG. It is a figure which shows the optimal individual | organism | solid obtained by the optimization apparatus of FIG. FIG. 6 is a graph showing the relationship between HC concentration and NO _x concentration obtained in Example 2. It is a schematic diagram which shows an example of the setting range of the definition area of a periodic function. It is a schematic diagram which shows the other example of the setting range of the definition area of a periodic function. It is a schematic diagram which shows distribution probability density distribution of the child individual by crossing of a real value. It is a schematic diagram for demonstrating evolutionary stagnation. It is a schematic diagram for demonstrating the boundary method.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Optimization apparatus 2 Individual generation part 3 Generation change part 4 Individual set storage part 5 Individual selection part 6 Fitness calculation part 7 Input / output interface 8 Optimization object 10 User 21 Crossover part 22 Normalization calculation part 23 Trigonometric function calculation part 23 24 Inverse Tangent Function Calculation Unit 81 ECU (Engine Control Unit)
82 Engine 83 Exhaust gas analyzer 84 Intake port 85 Exhaust port 86 Fuel injection device 87 Intake valve 88 Exhaust valve 89 Cylinder head 90 Spark plug 101 CPU (Central processing unit)
102 ROM (read only memory)
103 RAM (Random Access Memory)
DESCRIPTION OF SYMBOLS 104 Input device 105 Display device 106 External storage device 107 Recording medium drive device 109 Recording medium

Claims (19)

When there is a constraint that the sum of squares of m variables out of N variables to be optimized is 1, a solution candidate having N variables is given to the optimization target, and an objective function for the N variables An optimization device that receives from the optimization object a state quantity corresponding to
A solution candidate storage unit that stores a set of variable vectors representing points on an N-dimensional variable space defined by a set of N variables as a set of solution candidates;
A solution candidate selection unit that selects a predetermined number of solution candidates from a set of solution candidates stored in the solution candidate storage unit;
Based on a predetermined number of solution candidates selected by the solution candidate selection unit, a new solution candidate set is generated on the N-dimensional variable space including the (m−1) -dimensional unit sphere, and the generated solution candidate set is generated. A solution candidate generator that provides the optimization target and the solution candidate storage unit with
An evaluation value calculator that calculates an evaluation value of the state quantity output from the optimization target;
A solution candidate set update unit that updates a set of solution candidates stored in the solution candidate storage unit based on the evaluation value calculated by the evaluation value calculation unit,
N is a natural number equal to or greater than 2, and m is a natural number equal to or less than N. A solution candidate having N variables to be optimized is given to the optimization target, and a state quantity corresponding to the objective function for the N variables including an objective function having periodicity with respect to m variables is set as the optimization target. An optimization device received from
A solution candidate storage unit that stores a set of variable vectors representing points on an N-dimensional variable space defined by a set of N variables as a set of solution candidates;
A solution candidate selection unit that selects a predetermined number of solution candidates from a set of solution candidates stored in the solution candidate storage unit;
By associating m variables with m points on the one-dimensional unit sphere based on a predetermined number of solution candidates selected by the solution candidate selection unit, the N-dimensional variable space is [(N−m) + 2m] dimensions. By expanding into the variable space, generating a set of solution candidates on the expanded [(N−m) + 2m] -dimensional variable space, and re-associating the generated set of solution candidates with points on the N-dimensional variable space A solution candidate generation unit that generates a set of new solution candidates and gives the generated solution candidate set to the optimization target and the solution candidate storage unit;
An evaluation value calculator that calculates an evaluation value of the state quantity output from the optimization target;
A solution candidate set update unit that updates a set of solution candidates stored in the solution candidate storage unit based on the evaluation value calculated by the evaluation value calculation unit,
N is a natural number of 1 or more, and m is a natural number of N or less. When each of the m variables is θ and the first and second coordinates of the point on the two-dimensional plane are x and y, respectively,
The solution candidate generation unit calculates the variable θ of the predetermined number of solution candidates selected by the solution candidate selection unit by calculating the first and second coordinates (x, y) = (cos θ, sin θ). 3. The optimization apparatus according to claim 2, wherein the first and second coordinates (x, y) of each solution candidate generated by replacing with y) are converted into a variable θ ^C by the following equation.

The solution candidate selecting unit sets the domain of the objective function with the period of the objective function, to increase or decrease the objective function as set domain is 2π in the direction of the variable axis, defined 4. The optimization apparatus according to claim 2, wherein the objective function is shifted so that the center of the region is located at the coordinate 0 of the variable axis. The optimization target includes an evaluation system for evaluating the performance of the device, the variable vector includes a control parameter for the evaluation system, and the state quantity is the performance of the device obtained by the evaluation of the evaluation system. The optimization device according to claim 1, wherein 6. The device evaluation apparatus according to claim 5, wherein the evaluation system is a device evaluation device that controls the device based on the control parameter and outputs the performance of the device generated by the operation of the device as a state quantity. Optimization device. The evaluation system is a device simulator that evaluates the performance of the device by simulating the operation of the device based on the control parameter, and outputs the evaluated performance as a state quantity. 5. The optimization device according to 5. The device is an internal combustion engine, the control parameter is a parameter based on a crank rotation angle of the internal combustion engine, and the state quantity is a performance of the internal combustion engine that changes periodically with respect to the parameter. The optimization apparatus according to claim 6 or 7. 9. The optimization device according to claim 8, wherein the internal combustion engine has a fuel injection device, and the parameter based on the crank rotation angle includes a fuel injection timing of the fuel injection device. The internal combustion engine includes a fuel injection device, and the parameter based on the crank rotation angle is a fuel injection pattern by the fuel injection device within a certain period including a plurality of sections, and the fuel injection pattern is the fuel injection device. 2. The vector of fuel injection amounts in each section such that the sum of fuel injection amounts due to is constant, and the state amount is a performance of the internal combustion engine that periodically changes with respect to the parameter. 8. The optimization device according to 8. 11. The state quantity includes at least one of a concentration of a predetermined component in exhaust gas, fuel consumption, output torque, throttle valve opening, and crank rotation speed. Optimization device. The device is a robot link mechanism, the variable vector is a parameter indicating an angle formed by a joint of the link mechanism, and the state quantity is a performance of the robot link mechanism that periodically changes with respect to the parameter. The optimization device according to claim 6 or 7. The optimization object is a traffic simulator for simulating wide area control of a traffic system including a plurality of traffic signals arranged on a road, the variable vector is at least one traffic signal lighting timing, and the state quantity is 6. The optimization apparatus according to claim 5, further comprising information on a vehicle passing through the road. The optimization device is a protein simulator for simulating a protein structure, the variable vector is a direction vector representing a dihedral angle between covalent bonds of the protein, and the state quantity is information on energy. The optimization apparatus according to claim 5. The optimization target is an image processing device that performs a matching process between a target image and a reference image, the variable vector is a direction vector of the target image with respect to the reference image, and the state quantity is the target image and the reference 6. The optimization apparatus according to claim 5, wherein the optimization apparatus is information indicating an error from an image. An objective function for the N variables including an objective function having a constraint that gives a solution candidate having a variable having N variables to be optimized and has a sum of squares of m variables being 1. An optimization method for receiving from the optimization target a state quantity corresponding to
Storing a set of variable vectors representing points on an N-dimensional variable space defined by a set of N variables in a storage unit as a set of solution candidates;
Selecting a predetermined number of solution candidates from the set of stored solution candidates;
Based on the selected predetermined number of solution candidates, a new solution candidate set is generated on an N-dimensional variable space including an (m−1) -dimensional unit sphere, and the generated solution candidate set is the optimization target. And giving to the storage unit;
Calculating an evaluation value of a state quantity output from the optimization target;
Updating a set of solution candidates stored in the storage unit based on the calculated evaluation value,
N is a natural number of 2 or more, and m is a natural number of N or less. A solution candidate having a variable having N variables to be optimized is given as an optimization target, and a state quantity corresponding to the objective function for the N variables including an objective function having periodicity with respect to m variables is set as the state quantity. An optimization method received from an optimization target,
Storing a set of variable vectors representing points on an N-dimensional variable space defined by a set of N variables in a storage unit as a set of solution candidates;
Selecting a predetermined number of solution candidates from the set of stored solution candidates;
The N-dimensional variable space is expanded to a [(N−m) + 2m] -dimensional variable space by associating m variables with m points on the one-dimensional unit sphere based on the selected predetermined number of solution candidates. Then, a solution candidate set is generated on the expanded [(N−m) + 2m] dimensional variable space, and the generated solution candidate set is re-associated with a point on the N dimensional variable space to generate a new solution candidate. Generating a set and providing the generated set of solution candidates to the optimization target and the storage unit;
Calculating an evaluation value of a state quantity output from the optimization target;
Updating a set of solution candidates stored in the storage unit based on the calculated evaluation value,
N is a natural number of 1 or more, and m is a natural number of N or less. An objective function for the N variables including an objective function having a constraint that gives a solution candidate having a variable having N variables to be optimized and has a sum of squares of m variables being 1. An optimization program that can be executed by a computer that receives from the optimization object a state quantity corresponding to
Storing a set of variable vectors representing points on an N-dimensional variable space defined by a set of N variables in a storage unit as a set of solution candidates;
A process of selecting a predetermined number of solution candidates from the stored solution candidate set;
Based on the selected predetermined number of solution candidates, a new solution candidate set is generated on an N-dimensional variable space including an (m−1) -dimensional unit sphere, and the generated solution candidate set is the optimization target. And processing to be given to the storage unit,
Processing for calculating an evaluation value of a state quantity output from the optimization target;
Processing to update the set of solution candidates stored in the storage unit based on the calculated evaluation value,
N is a natural number equal to or greater than 2, and m is a natural number equal to or less than N. A solution candidate having a variable having N variables to be optimized is given as an optimization target, and a state quantity corresponding to the objective function for the N variables including an objective function having periodicity with respect to m variables is set as the state quantity. An optimization program executable by a computer received from an optimization target,
Storing a set of variable vectors representing points on an N-dimensional variable space defined by a set of N variables in a storage unit as a set of solution candidates;
A process of selecting a predetermined number of solution candidates from the stored solution candidate set;
The N-dimensional variable space is expanded to a [(N−m) + 2m] -dimensional variable space by associating m variables with m points on the one-dimensional unit sphere based on the selected predetermined number of solution candidates. Then, a solution candidate set is generated on the expanded [(N−m) + 2m] dimensional variable space, and the generated solution candidate set is re-associated with a point on the N dimensional variable space to generate a new solution candidate. A process of generating a set and providing the generated set of solution candidates to the optimization target and the storage unit;
Processing for calculating an evaluation value of a state quantity output from the optimization target;
Processing to update the set of solution candidates stored in the storage unit based on the calculated evaluation value,
N is a natural number equal to or greater than 1, and m is a natural number equal to or less than N.

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