CN104021431B - Robust Optimal methods based on average gradient value and improvement multi-objective particle swarm optimization - Google Patents
Robust Optimal methods based on average gradient value and improvement multi-objective particle swarm optimization Download PDFInfo
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Abstract
The present invention relates to a kind of robust Optimal methods based on average gradient value and improvement multi-objective particle swarm optimization, comprise the following steps：1）Technological parameter to be optimized is determined according to engineering practice and optimization aim；2）The constraints of technological parameter to be optimized is determined according to engineering practice；3）Establish the target function model of optimization aim and technological parameter to be optimized；4）The discrete relationship obtained between optimization aim and welding condition is tested by limited number of time；5）According to discrete relationship, calculating target function；6）The uncertain domain of setting, using the average gradient value of uncertain domain sample point, calculates robustness；7）Using object function and robustness as two optimization sub-goals, biobjective scheduling solution is carried out.The evaluation of robustness and the calculating of former object function at the same time as the sub-goal of two optimization, are enable designer to select rational compromise solution between the amplitude of variation that performance indicator and uncertain domain produce according to actual needs by the method for the present invention.
Description
Technical Field
The invention relates to the field of computer data processing, in particular to a robust optimization method based on average gradient values and improved multi-objective particle swarm optimization.
Background
Robust optimization is a new optimization method for solving uncertain environments of internal structures (such as parameters) and external environments (such as disturbance), uncertainty of an optimization model is considered at the beginning of optimization, and the optimization result is insensitive to uncertain factors and optimally unifies performance indexes through the optimization method. The classical robust optimization method is mainly used in operational research, and researches convex problems such as a linear programming problem, a quadratic programming problem and a semi-definite programming problem with different forms of data uncertainty. However, real-world engineering problems are often non-convex, even without mathematical expressions, and classical methods in operations research are not suitable in engineering design. In the existing engineering problem, how to solve the robust optimization solution of a series of practical problems such as no analytic expression, high nonlinearity, high decision space dimension and the like becomes a problem which is urgently needed to be solved in the field of robust optimization.
Disclosure of Invention
The invention aims to solve the technical problem of providing a robust optimization method based on average gradient values and improved multi-target particle swarm optimization aiming at the defects in the prior art.
The technical scheme adopted by the invention for solving the technical problem is as follows:
the robust optimization method based on the average gradient value and the improved multi-objective particle swarm optimization comprises the following steps of:
1) Determining process parameters to be optimized according to the actual engineering situation and the optimization target;
2) Determining constraint conditions of process parameters to be optimized according to the actual engineering conditions;
3) Establishing an optimization target and an objective function model of a process parameter to be optimized;
4) Obtaining a discrete relation between an optimization target and welding process parameters through a limited number of tests; calculating an objective function according to the discrete relation between the objective function model and the process parameters;
5) Setting an uncertain domain, and calculating robustness by using the average gradient value of the uncertain domain samples;
6) And taking the objective function and the robustness as two optimization sub-objectives, and performing dual-objective optimization solution.
According to the scheme, the dual-objective optimization solving method in the step 6) comprises the following steps:
6.1 Set the population size, the maximum algebra of iteration, the maximum inertia weight and the minimum inertia weight and random variables R1, R2 between [0,1 ];
6.2 Initializing a population, initializing positions of particles and initializing velocities of the particles with a CVT method;
6.3 Calculating a fitness function of each particle, taking the position of each particle as a decision variable, calculating an objective function according to a discrete relation between an objective function model and a process parameter (decision variable), and calculating the robustness of each particle by using the average gradient value of a sample point in a particle uncertainty domain;
6.4 According to Pareto dominance principle), storing the positions of non-dominating particles in the population in an external archive file;
wherein the non-dominant particles are defined as follows: let p and q be two different individuals in the population, when p dominates q, the following two conditions must be met: (1) for all sub-targets, p is not worse than q; (2) there is at least one sub-target, such that p is better than q; q is called dominant particle, when no other particle in the population dominates p, p is called non-dominant particle;
6.5 Initializing a memory file of each particle, and recording the memory file of the particle to the optimal position of the particle so far;
wherein the optimal position of the particles up to now is defined as follows: if the current evolutionary generation of the particle is matched with the previous generation, the optimal position of the particle is the current generation; if the previous generation of the particles dominates the current generation, the optimal position of the particles is the previous generation; if the current generation and the previous generation of the particles are not dominant, the optimal position of the particles is one of the two randomly selected;
6.6 Selecting an optimal location of the population from an external archive file;
the concrete implementation is as follows: dividing a decision space into a plurality of hypercubes, determining the fitness value of each hypercube according to the number of particles contained in the hypercube, firstly selecting the hypercube by a roulette method according to the fitness value, and then randomly determining an optimal individual from the selected hypercube;
6.7 Calculate the velocity of each particle;
the concrete implementation is as follows: respectively calculating the product of the inertia weight and the previous generation speed of the particle, the product of the distance from the optimal position of the particle to the current position and R1, and the product of the distance from the optimal position of the population to the current position of the particle and R2, and setting the sum of the three products as the speed of the particle; the inertia weight linearly decreases from the maximum inertia weight to the minimum inertia weight along with the increase of evolution algebra;
6.8 Update the position of the particle, add the position of the previous generation of the particle to the velocity of the particle, and keep the particle in the search space;
6.9 ) and updating the fitness function value of the particle, wherein the calculation method is the same as the step 6.3);
6.10 Updating the external archive file and the particle representations in the divided hypercubes; inserting the current non-dominated solution into the external archive file, the dominated solution to be deleted from the external archive file; when the external archive file is full, preferentially storing the solution of the region with few particles in the target space;
6.11 Update the memory file of the particle;
according to the Pareto domination mechanism, when the current position of the particle is better than the position in the memory archive file, the best position of the particle needs to be updated; when the current position of the particle is different from the memory file, the best position of the particle does not need to be updated; if the two are not dominant, one is randomly selected;
6.12 If the iteration reaches the maximum iteration algebra, stopping and outputting an optimal solution; otherwise, return to step 6.6).
According to the scheme, in the step 6.2), initializing the position of the particle comprises the following steps:
step 6.2 a), a density function rho (x), a positive integer q and a constant alpha are given 1 ，α 2 ，β 1 And beta 2 Wherein: alpha (alpha) ("alpha") 2 >0，β 2 >0，α 1 +α 2 ＝1，β 1 +β 2 ＝1；
Point setRepresents a population to be initialized, where i = 1.., N; for i =1, say, N, set j i The initial values are all 1; selecting an initial point set in decision space by Monte Carlo method
Step 6.2 b), selecting q points in decision space according to the density function rho (x)
Step 6.2 c) for setsCombination of Chinese herbsEach point in (1) will be collectedNeutral z i The nearest point (i.e., point z) i Points in the Voronoi region) into a set w i In, if set w i Is empty, z i Keeping the same; otherwise, calculate w i Average position u of points within the set i And updating z according to the following expression i ；
j i ＝j i +1；
Step 6.2 d) if the updated particle set meets the convergence criterion, the updating is stopped; otherwise, go to step 6.2 b).
According to the scheme, in the step 6.3), calculating the robustness of the particles comprises the following steps:
6.3 a), if the dimension of a decision variable of the current particle with disturbance is c, the dimension of the uncertain domain of the particle is also c, each dimension of the uncertain domain is uniformly divided into t sections, c and t respectively represent a factor number and a level number, a uniform design table is selected according to the factor number and the level number to carry out factor level data arrangement, and a sample point set is selected according to the uniform design table in the uncertain domain of the current particle; the particle uncertain domain is preset manually;
6.3 b), calculating the absolute difference between the objective function value of each sample point and the objective function value of the current particle and the Euclidean distance between the decision variable of each sample point and the decision variable of the current particle, and recording the ratio of the absolute difference of the objective function to the Euclidean distance of the decision variable;
6.3 c) calculating the average value of the ratio of the absolute difference of the objective functions of all the sample points to the Euclidean distance of the decision variables, namely the robustness evaluation value of the particle.
The invention has the following beneficial effects:
1. the invention is applicable to a wide range of engineering problems;
2. the method takes the evaluation of robustness and the calculation of an original objective function as two optimized sub-objectives simultaneously, so that a designer can select a reasonable compromise solution between a performance index and a variation amplitude generated by an uncertain domain according to actual needs;
3. the method adopts the improved multi-target particle swarm optimization algorithm based on CVT population initialization and dynamic inertia weight, so that the capability of searching a robust optimization solution set is greatly enhanced and the algorithm convergence is faster compared with the multi-target optimization particle swarm optimization algorithm based on random initialization population and fixed inertia weight;
4. the method of the invention provides a novel method for evaluating robustness, and the method takes the average gradient value from a sample point selected in an uncertain domain of an evaluation point to the evaluation point as the robustness evaluation value of the evaluation point;
5. the method adopts uniform design to avoid exponential increase of calculated amount along with the increase of the decision space dimension;
6. the method of the invention uses relatively few and evenly distributed sample points to calculate the relative change amplitude of the target space and the decision space of the evaluation point in the uncertain domain, and fully embodies the concept of robustness.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a flow chart of a method of an embodiment of the present invention;
FIG. 2 is a flow diagram of a solution process of an embodiment of the invention;
FIG. 3 is a schematic diagram of an uncertainty region according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of random population initialization according to an embodiment of the present invention;
FIG. 5 is a schematic diagram of population initialization based on the CVT method according to the embodiment of the present invention;
FIG. 6 is a functional schematic of validation of an embodiment of the present invention;
FIG. 7 is a schematic diagram of an optimization result solution set for validity verification according to an embodiment of the present invention;
FIG. 8 is a schematic diagram of a solution set of a multi-objective particle swarm algorithm employing random initialization and fixed inertia weights according to an embodiment of the present invention;
fig. 9 is a solution set diagram of a multi-target particle swarm algorithm using CVT method initialization and dynamic inertia weights according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The method is suitable for a wide range of engineering problems, and the method is further explained in detail by selecting the field of design of the Mars detection track in the embodiment.
In the design of a mars detection track, how to ensure that a detection task is finished by using as little fuel as possible and the function of a system can be kept when a decision variable has small disturbance is a very challenging and urgent problem to be solved. The initial orbit design from the earth to the mars is realized by a conical curve splicing method, and the method comprises the following specific steps: firstly, determining an emission window and transfer duration, obtaining a solar-centric transfer orbit, further calculating an earth escape orbit and a Mars capture orbit, and finally splicing the obtained three orbits to obtain an initial orbit.
Arranged in the equatorial inertial coordinate system of the earth center, the height of the near point of the escape track is r ep Velocity at near point of magnitude v ep The inclination angle of the track is i e Angular distance of near to ground is omega e The right ascension at the intersection is omega e . In the equatorial inertial coordinate system of the fire center, the height of the near fire point of the capture track is r mp The near-ignition velocity is v mp Inclination of trackIs i m Angular distance of near ignition is omega m The right ascension at the intersection point is omega m . Then according to the above 10 variables, the position and speed of the detector at the near-point and near-fire point can be calculated, and the emission time t is given 1 And arrival time t 2 The escape orbit and capture orbit parameters can be determined, the near-site acceleration magnitude and the near-fire braking magnitude can be obtained through further calculation, and the sum of the near-site acceleration magnitude and the near-fire braking magnitude can reflect the fuel required to be consumed. The optimization model of the problem of the original rail design of the old ground fire is as follows
minf(x)
x＝[r ep ,v ep ,i e ,ω e ,Ω e ,t 1 ,r mp ,v mp ,i m ,ω m ,Ω m ,t 2 ]
General of i e 、r ep 、i m And r mp Determination of scientific goals according to delivery and exploration, t 1 、t 2 Given the range according to engineering requirements, other parameters are defined according to engineering constraints.
The model cannot give an analytical expression, but each group of decision variables (process parameters) corresponds to a unique objective function f value in the value range of the decision variables (process parameters).
Calculating an objective function according to the discrete relation between the objective function model and the process parameters; the method for calculating the objective function is various and can be obtained through experiments in engineering or by establishing a mathematical model for an original system.
And next, taking the objective function and the robustness as two optimization sub-objectives, and performing dual-objective optimization solution.
The dual-target optimization solution method comprises the following steps:
6.1 Set population size, maximum algebra of iteration, maximum inertia weight and minimum inertia weight and random variables R1, R2 between [0,1 ];
6.2 Initializing a population, initializing positions of particles and initializing velocities of the particles with a CVT method;
initializing the position of the particles comprises the steps of:
step 6.2 a), a density function rho (x), a positive integer q and a constant alpha are given 1 ，α 2 ，β 1 And beta 2 Wherein: alpha is alpha 2 >0，β 2 >0，α 1 +α 2 ＝1，β 1 +β 2 ＝1；
Point setDenotes the population to be initialized, where i = 1.., N; for i =1 i The initial values are all 1; selecting an initial point set in decision space by Monte Carlo method
Step 6.2 b), selecting q points in decision space according to the density function rho (x)
Step 6.2 c) for collectionsEach point in (1) will be collectedNeutral z i The nearest point (i.e., point z) i Points in the Voronoi region) into a set w i In, if set w i Is empty, z i Keeping the same; otherwise, calculate w i Average position u of points within the set i And updating z according to the following expression i ；
j i ＝j i +1；
Step 6.2 d) if the updated particle set meets the convergence criterion, the updating is stopped; otherwise, go to step 6.2 b).
6.3 Calculating a fitness function, namely an objective function and robustness of each particle, taking the position of each particle as a decision variable, calculating the objective function according to a discrete relation between an objective function model and a process parameter (decision variable), and calculating the robustness of each particle by using an average gradient value of a sample point in a particle uncertainty domain;
calculating the robustness of the particle comprises the following steps:
6.3 a), setting the dimension of a decision variable of disturbance of the current particle as c, setting the dimension of the uncertain domain of the particle as c, uniformly dividing each dimension of the uncertain domain into t sections, wherein c and t respectively represent a factor number and a level number, selecting a proper uniform design table according to the factor number and the level number to carry out factor level data arrangement, and selecting a sample point set according to the uniform design table in the uncertain domain of the current particle; the particle uncertain domain is preset manually;
6.3 b), calculating the absolute difference between the objective function value of each sample point and the objective function value of the current particle and the Euclidean distance between the decision variable of each sample point and the decision variable of the current particle, and recording the ratio of the absolute difference of the objective function to the Euclidean distance of the decision variable;
6.3 c) calculating the average value of the ratio of the absolute difference of the objective functions of all the sample points to the Euclidean distance of the decision variables, namely the robustness evaluation value of the particle.
6.4 According to Pareto dominance principle), storing the positions of non-dominating particles in the population in an external archive file;
wherein the non-dominant particles are defined as follows: let p and q be two different individuals in the population, when p dominates q, the following two conditions must be met: (1) p is not worse than q for all sub-targets; (2) at least one sub-target exists such that p is better than q; q is called dominant particle, when no other particle in the population dominates p, p is called non-dominant particle;
6.5 Initializing a memory file of each particle, and recording the memory file of the particle to the optimal position of the particle so far;
wherein the optimal position of the particles up to now is defined as follows: if the current evolutionary generation of the particles is matched with the previous generation, the optimal position of the particles is the current generation; if the previous generation of the particles dominates the current generation, the optimal position of the particles is the previous generation; if the current generation and the previous generation of the particles are not dominant, the optimal position of the particles is one of the current generation and the previous generation randomly selected;
6.6 Selecting an optimal location for the population from an external archive file;
the concrete implementation is as follows: dividing a decision space into a plurality of hypercubes, determining the fitness value of each hypercube according to the number of particles contained in the hypercube, firstly selecting the hypercube by a roulette method according to the fitness value, and then randomly determining the optimal individual from the selected hypercube;
6.7 Calculate the velocity of each particle;
the concrete implementation is as follows: respectively calculating the product of the inertia weight and the previous generation speed of the particle, the product of the distance from the optimal position of the particle to the current position and R1, and the product of the distance from the optimal position of the population to the current position of the particle and R2, and setting the sum of the three products as the speed of the particle; the inertia weight linearly decreases from the maximum inertia weight to the minimum inertia weight along with the increase of evolution algebra;
6.8 Update the position of the particle, add the position of the previous generation of the particle to the velocity of the particle, and keep the particle in the search space;
6.9 ) updating the fitness function of the particles, and the calculation method is the same as the step 6.3);
6.10 Updating the external archive file and the particle representations in the divided hypercubes; inserting the current non-dominated solution into the external archive file, the dominated solution to be deleted from the external archive file; when the external archive file is full, preferentially storing the solution of the region with few particles in the target space;
6.11 Update the memory file of the particle;
according to the Pareto domination mechanism, when the current position of the particle is better than the position in the memory archive file, the best position of the particle needs to be updated; when the current position of the particle is different from the memory file, the best position of the particle does not need to be updated; if the two are not dominant, one is randomly selected;
6.12 If the iteration reaches the maximum iteration algebra, stopping and outputting the optimal solution; otherwise, return to step 6.6).
The method of the invention respectively uses an original objective function and robustness of a design problem as two sub-objectives of fitness function evaluation, and adopts a multi-objective particle swarm optimization algorithm based on a Pareto principle to simultaneously optimize the two objectives, so as to obtain a Pareto solution set with one-dimensional manifold uniformly distributed. As shown in fig. 7.
The type of the uncertain domain in the invention is interval type. FIG. 3 is an uncertain domain type with a decision variable of 2 dimensions. And taking the evaluation point as a center, and on each dimension of the decision variable, taking the value of the evaluation point on the dimension as a reference, and respectively selecting the interval of the reference point in the positive and negative directions delta. In a high-dimensional decision space (decision space dimension greater than 3), the uncertainty region of the evaluation point is a hypercube.
The invention has the beneficial effects that:
1. evaluation of robustness: and adopting an average value of the ratio of the absolute difference of the target function of the sample point in the uncertainty domain of the evaluation point to the Euclidean distance of the decision variable. Sampling in an uncertain domain of an evaluation point, calculating an absolute difference value of an objective function of a sample point and an objective function of the evaluation point and a Euclidean distance from the sample point to the evaluation point in a decision space, wherein the gradient of the sample point is as follows: and the robustness of the evaluation point is the average value of all sample point gradients in the uncertain domain according to the ratio of the absolute difference of the target function to the Euclidean distance of the decision space.
As shown in fig. 6, the robustness is used as an objective function to perform optimization solution, an uncertain domain is set as x ± 0.1, a solution obtained by using the robustness evaluation method provided by the present invention is 10.0, and a corresponding robustness evaluation value is 0.004728, which reflects that the solution is relatively flat in the neighborhood and has good robustness, and the solution obtained by using the robustness as the objective function in the graph is a point c, which confirms the effectiveness of the method.
2. The invention adopts uniform design sampling, and the selection of sample points in an uncertain domain is as follows: the sampling is designed uniformly. Uniformly and equally dividing each dimension with disturbance in the decision space into equal number of equal segments, recording the number as a horizontal number t, recording the number of dimensions of the decision space as a factor number c, selecting a proper uniform design table according to the factor number and the horizontal number to carry out factor horizontal data arrangement, and selecting corresponding sample points in the decision space according to the uniform design table.
Compared with the existing grid sampling method, the method combines the problem of ground fire initial orbit design, the disturbed decision space is 8-dimensional, each dimension is equally divided into 50 sections in an uncertain domain, and if grid sampling is adopted, the sampling frequency is 50 8 While the uniform design only requires 50 samples.
3. By adopting the idea of multi-objective optimization, the result of the optimization is not a single solution, but a solution set of one-dimensional manifold, as shown in fig. 6.
4. Performing population initialization and inertia weight setting of a multi-target particle swarm optimization algorithm: based on CVT method group initialization and linear decreasing dynamic inertia weight. Population initialization based on CVT method: giving a density function, firstly selecting an initial point set in a decision space according to a certain rule (such as a Monte Carlo method), obtaining voronoi division of the point set, then selecting another point set in the decision space according to the density function, and adjusting the position of the initial point set according to the voronoi area of the initial point set and the point number of the point set selected for the second time until the initial point set is uniformly distributed in the decision space. Fig. 4 shows a randomly selected two-dimensional initial population, and fig. 5 shows a more uniformly distributed initial population obtained by the CVT method. Dynamic inertia weight: the inertia weight value is linearly changed according to the following expression in the population searching process.
Inertia weight = maximum inertia weight- (maximum inertia weight-minimum inertia weight) x iteration algebra ÷ maximum iteration algebra
In the initial stage of iteration, the inertia weight is large, and the algorithm has strong global optimization capability; with the increase of the iteration number, the population is gradually converged, the inertia weight is reduced, and the algorithm has stronger local optimization capability.
And comparing the random initialization and the fixed inertia weight with the performance of a multi-target particle swarm algorithm based on the CVT method initialization and the dynamic inertia weight. Fig. 8 is a solution set of a multi-target particle swarm algorithm using random initialization and fixed inertia weight, fig. 9 is a solution set of a multi-target particle swarm algorithm using CVT method initialization and dynamic inertia weight, both the running generations are 500 generations, it can be seen that the algorithm optimization solution sets using CVT method initialization and dynamic inertia weight are more evenly distributed, more optimized solutions are found, and meanwhile, the algorithm convergence using CVT method initialization and dynamic inertia weight is faster under the same iteration generations.
It will be understood that modifications and variations can be made by persons skilled in the art in light of the above teachings and all such modifications and variations are intended to be included within the scope of the invention as defined in the appended claims.
Claims (5)
1. A robust optimization method based on average gradient values and improved multi-objective particle swarm optimization is characterized by comprising the following steps:
1) In the design of a Mars detection orbit, firstly determining a transmitting window and transfer duration according to actual conditions, obtaining a solar-centric transfer orbit, then further calculating an earth escape orbit and a Mars capture orbit, and finally splicing the obtained three orbits to obtain an initial orbit; determining an optimization model of the ground fire initial rail design problem and process parameters to be optimized in the optimization model according to the obtained initial rail;
the optimization model for determining the design problem of the initial orbit of the ground fire according to the obtained initial orbit specifically comprises the following steps:
arranged in the equatorial inertial coordinate system of the earth center, the height of the near point of the escape track is r ep Velocity at near point of magnitude v ep The inclination angle of the track is i e Angular distance of near place is omega e The right ascension at the intersection is omega e (ii) a In the equatorial inertial coordinate system of the fire center, the height of the near fire point of the capture track is r mp The near-ignition velocity is v mp The inclination angle of the track is i m Angular distance of near ignition is omega m The right ascension at the intersection point is omega m (ii) a Then according to the above 10 variables, the position and speed of the detector at the near-point and near-fire point are calculated, and then the position and speed are givenEmission time t 1 And arrival time t 2 The escape orbit and the capture orbit parameters can be determined, the near-site acceleration magnitude and the near-fire braking magnitude are obtained through further calculation, and the sum of the near-site acceleration magnitude and the near-fire braking magnitude can reflect the fuel required to be consumed; the optimization model of the primary orbit design problem of the old ground fire is as follows
minf(x)
x＝[r ep ,v ep ,i e ,ω e ,Ω e ,t 1 ,r mp ,v mp ,i m ,ω m ,Ω m ,t 2 ]
Wherein i e 、r ep 、i m And r mp Determination of the parameter v from the scientific goals of delivery and detection ep ,ω e ,Ω e ,r mp ,v mp ,ω m ,Ω m Given a range according to engineering constraints, the parameter to be optimized is t 1 、t 2 ，t 1 、t 2 The range is given according to engineering requirements;
2) Determining constraint conditions of process parameters to be optimized according to the actual engineering conditions;
3) Establishing an optimization target and an objective function model of a process parameter to be optimized;
4) Obtaining a discrete relation between an optimization target and welding process parameters through a limited number of tests; calculating an objective function according to the discrete relation between the objective function model and the process parameters;
5) Setting an uncertain domain, and calculating the robustness by using the average gradient value of the uncertain domain samples;
6) And (4) taking the objective function and the robustness as two optimization sub-objectives to carry out double-objective optimization solution.
2. The robust optimization method according to claim 1, wherein the dual target optimization solution in step 6) comprises the following steps:
6.1 Set population size, maximum algebra of iteration, maximum inertia weight and minimum inertia weight and random variables R1, R2 between [0,1 ];
6.2 Initializing a population, initializing positions of particles and initializing velocities of the particles with a CVT method;
6.3 Calculating a fitness function of each particle, and taking the position of each particle as a decision variable, wherein the decision variable is used for representing a process parameter; calculating an objective function according to the discrete relation between the objective function model and the decision variables; calculating the robustness of the particles by using the average gradient value of the sample points in the particle uncertainty domain;
6.4 According to Pareto domination principle, storing the position of non-dominating particles in the population in an external archive file;
6.5 Initializing a memory file of each particle, and recording the memory file of the particle to the optimal position of the particle so far;
6.6 Selecting an optimal location of the population from an external archive file;
6.7 Calculate the velocity of each particle;
6.8 Update the position of the particle, add the position of the previous generation of the particle to the velocity of the particle, and keep the particle in the search space;
6.9 Update a fitness function of the particle;
6.10 Update the particle representations in the external archive file and the partitioned hypercubes; inserting the current non-dominated solution into the external archive file, the dominated solution to be deleted from the external archive file; when the external archive file is full, preferentially storing the solution of the region with few particles in the target space;
6.11 Update the memory file of the particle;
6.12 If the iteration reaches the iteration maximum algebra, stopping and outputting an optimal solution; otherwise, return to step 6.6).
3. The robust optimization method of claim 2, wherein initializing the positions of the particles in step 6.2) comprises the steps of:
step 6.2 a), a density function rho (x), a positive integer q and a constant alpha are given 1 ，α 2 ，β 1 And beta 2 Wherein: alpha is alpha 2 ＞0，β 2 ＞0，α 1 +α 2 ＝1，β 1 +β 2 ＝1；
Point setDenotes the population to be initialized, where i = 1.., N; for i =1, say, N, set j i The initial values are all 1; selecting an initial point set in decision space by Monte Carlo method
Step 6.2 b), selecting q points in decision space according to the density function rho (x)
Step 6.2 c) for collectionsEach point in (1) will be collectedNeutral z i The nearest points are summarized in the set w i In, if set w i Is empty, z i Keeping the same; otherwise, calculate w i Average position u of points within the set i And updating z according to the following expression i ；
Step 6.2 d) if the updated particle set meets the convergence criterion, the updating is stopped; otherwise, go to step 6.2 b).
4. The robust optimization method according to claim 2, wherein in step 6.3), calculating the robustness of the particles comprises the steps of:
6.3 a), setting the dimension of a decision variable of disturbance of the current particle as c, setting the dimension of an uncertain domain of the particle as c, uniformly dividing each dimension of the uncertain domain into t sections, respectively representing a factor number and a level number by c and t, selecting a proper uniform design table according to the factor number and the level number to carry out factor level data arrangement, and selecting a sample point set according to the uniform design table in the uncertain domain of the current particle; the particle uncertain domain is preset manually;
6.3 b), calculating the absolute difference between the objective function value of each sample point and the objective function value of the current particle and the Euclidean distance between the decision variable of each sample point and the decision variable of the current particle, and recording the ratio of the absolute difference of the objective function to the Euclidean distance of the decision variable;
6.3 c) calculating the average value of the ratio of the absolute difference of the objective functions of all the sample points to the Euclidean distance of the decision variables, namely the robustness evaluation value of the particle.
5. The robust optimization method of claim 2, wherein the step 6.9) of updating the fitness function of the particle, and calculating the robustness of the particle comprises the steps of:
6.9 a), setting the dimension of a decision variable of disturbance of the current particle as c, setting the dimension of an uncertain domain of the particle as c, uniformly dividing each dimension of the uncertain domain into t sections, respectively representing a factor number and a level number by c and t, selecting a proper uniform design table according to the factor number and the level number to carry out factor level data arrangement, and selecting a sample point set according to the uniform design table in the uncertain domain of the current particle; the particle uncertain domain is preset manually;
6.9 b), calculating the absolute difference between the objective function value of each sample point and the objective function value of the current particle and the Euclidean distance between the decision variable of each sample point and the decision variable of the current particle, and recording the ratio of the absolute difference of the objective function to the Euclidean distance of the decision variable;
6.9 c), calculating the average value of the ratio of the absolute difference of the objective functions of all the sample points to the Euclidean distance of the decision variables, namely the robustness evaluation value of the particle.
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