WO2024029022A1

WO2024029022A1 - Device for accelerating branch-and-bound method, method, and program

Info

Publication number: WO2024029022A1
Application number: PCT/JP2022/029908
Authority: WO
Inventors: まな美小川; 俊介金井; 正崇佐藤; 和陽明石; 麻悠山添
Original assignee: 日本電信電話株式会社
Priority date: 2022-08-04
Filing date: 2022-08-04
Publication date: 2024-02-08

Abstract

This device for accelerating a branch-and-bound method comprises a branching unit, an interval determination unit, a presumed threshold determination unit, and a comparison unit. The branching unit performs a branching operation that divides an input problem into a plurality of problems. The interval determination unit determines a first threshold that defines a first interval where a solution set for each divided problem may exist on the basis of feasible solutions of each divided problem, and determines a second threshold that defines the first interval by solving a relaxation problem of each divided problem. The presumed threshold determination unit uses the first threshold, the second threshold, a probability distribution representing a probability that an optimal solution for each divided problem exists in a second interval narrower than the first interval and with one threshold defining the second interval as the first threshold, and a set probability value to determine a third threshold as the other threshold defining the second interval. The comparison unit executes pruning of the solution sets by comparing the first threshold with the third threshold for each solution set.

Description

Apparatus, method and program for speeding up the branch and bound method

The embodiments relate to an apparatus, method, and program for speeding up the branch and bound method.

An optimization problem is a problem that seeks the optimal solution among the solutions that satisfy the conditions. One of the optimization problems is a combinatorial optimization problem in which optimal solutions of combinations are found. In a combinatorial optimization problem, the number of optimal solution candidates generally increases exponentially with respect to the input. For this reason, it is difficult to enumerate all feasible solutions in a combinatorial optimization problem and find the optimal solution from among the enumerated feasible solutions.

The branch and bound method is one of the methods to avoid enumerating all possible solutions and find the optimal solution. The branch and bound method consists of two operations: a branch operation and a limit operation. A branching operation is an operation in which a problem to be solved is divided into multiple problems by case classification. The limit operation reduces the amount of time required to search for an optimal solution by comparing the solution sets of each problem divided into multiple problems, and as a result of the comparison, discontinuing the search for solutions in solution sets where the optimal solution cannot exist. This is a time saving operation. This kind of aborting of the search for a solution is called pruning. For example, in the case of a minimization problem, the upper and lower limits of the range of solutions within a limited solution set are determined. Then, as a result of comparing the existence ranges of two different solution sets, the solution set with the larger optimal solution cannot contain the optimal solution to the original problem, so the search for a solution in that solution set is discontinued.

In a minimization problem, an appropriate solution from among the feasible solutions is used as the upper limit. On the other hand, the lower limit can be obtained by appropriately setting and solving the relaxation problem. Here, for a solution set where the difference between the upper and lower limits of the range of solutions in a certain solution set is large, that is, the range of solutions that exist is wide, as a result of comparison with different solution sets, it is determined that the optimal There may be cases in which a solution may exist. In this case, the search will not be aborted and the search time for a solution will not be reduced.

The embodiments provide an apparatus, method, and program for speeding up the branch-and-bound method, which completes the search for an optimal solution faster than conventional branch-and-bound methods.

An apparatus for speeding up a branch and bound method according to one embodiment includes a branching section, an interval determining section, a deemed threshold determining section, and a comparing section. The branching unit performs a branching operation to divide the input problem into multiple problems. The interval determination unit determines a first threshold value for determining a possible first interval of the solution set of each divided problem from the feasible solutions of each divided problem, and a first threshold value for determining the first interval. The threshold value of 2 is determined by solving the relaxation problem of each divided problem. The deemed threshold determining unit is configured such that the first threshold, the second threshold, and the optimal solution of each divided problem are narrower than the first interval, and one threshold that determines the interval is the first threshold. A third threshold, which is the other threshold that determines the second interval, is determined based on the probability distribution representing the probability of existing in the second interval, and the set value of the probability. The comparison unit performs pruning of the solution set by comparing the first threshold value and the third threshold value for each solution set.

According to the embodiment, an apparatus, method, and program are provided for speeding up the branch-and-bound method, which completes the search for an optimal solution faster than the conventional branch-and-bound method.

FIG. 1 is a block diagram showing the configuration of an apparatus for speeding up the branch and bound method according to the embodiment. FIG. 2 is a graph showing the probability density distribution of equation (4) when α=6, β=1.1, U _i =2.5, and L _i =1.5. FIG. 3 is a block diagram showing a hardware configuration of an example of a device for speeding up the branch and bound method according to the embodiment. FIG. 4 is a flowchart showing the operation of the apparatus for speeding up the branch and bound method according to the embodiment.

Hereinafter, embodiments will be described with reference to the drawings. First, the branch and bound method in the embodiment will be explained. The branch-and-bound method in the embodiment is used to quickly find an approximate solution to the original problem on the premise that the probability distribution of where the optimal solution exists in the solution set limited by the limiting operation is known. Here, the probability distributions are independent for each limiting operation.

When the probability distribution of where the optimal solution exists in the solution set limited by the limiting operation is known, from this probability distribution, the probability that the optimal solution exists in the solution set limited by the limiting operation is p(0<p<1 ) to know the existing interval. In the embodiment, this interval is regarded as the range in which a new optimal solution exists in the solution set limited by the limiting operation. At this time, since the range of existence of the new optimal solution is narrower than the range of existence of the original optimal solution in the limited solution set, it is expected that the number of times the solution search will be aborted will increase. If the number of times the search for a solution is aborted increases, it is expected that the search for the optimal solution to the original problem will end sooner.

Here, if n limiting operations are required until a solution to the original problem is output by the branch and bound method, the probability that the output solution is the optimal solution is p ⁿ . Furthermore, the probability that the difference between the output solution and the optimal solution is within a predetermined value can also be calculated from the probability distribution of where the optimal solution exists in the solution set limited by the limiting operation.

The branch and bound method of the embodiment will be further explained. In the following, an application example for a minimization problem will be explained. First, the solution set of each problem divided by the branching operation is expressed as S _i (i≦m). Here, m represents the number of problems divided by one branching operation. Further, an upper limit value (upper bound) and a lower limit value (lower bound), which are threshold values that determine the interval in which the optimal solution can exist in the solution set S _i , are represented by U _i and L _i , respectively. Here, in the case of a minimization problem, an appropriate solution among the feasible solutions is used as U _i , for example. A feasible solution is a solution that satisfies the conditions set for the problem. The feasible solution may be calculated using any search algorithm such as a greedy method or a local search method. On the other hand, L _i can be obtained, for example, by solving a relaxation problem. A relaxed problem is a problem in which some of the conditions set for the problem are relaxed. Furthermore, a probability density function that expresses where in the closed interval [L _i , U _i ] the optimal solution in the solution set S _i exists as a probability distribution is a function f _i ₍ x, _L _i , U _i ). However, f _i satisfies the relationship of equation (1) below. Here, x is a variable indicating the value of a possible solution to [L _i , U _i ].

If f _i is known for each S _i , a closed interval [l _i , U _i ] in which the optimal solution exists with probability p (0<p<1) can be found. However, p and l _i satisfy the following equation (2).

For different solution sets S _k and S _l (k and l are both elements of i), in the conventional branch and bound method, the optimal solution for the original problem is found in the solution set S _k that satisfies U _l <L _k . is determined not to exist. On the other hand, in the embodiment, it is determined that the optimal solution to the original problem does not exist in the solution set S _k that satisfies U _l <l _k by using the lower limit l _i derived from p and f _i . Since L _i <l _i , it is easier to abort the search for the optimal solution if the limiting operation is performed in the interval [l _i , U _i ] than if the limiting operation is performed in the interval [L _i , U _i ]. The possibility of this occurring increases. Therefore, it is expected that the search for the optimal solution to the original problem will be completed quickly. Hereinafter, l _i will be referred to as a "deemed lower limit value." The deemed lower limit l _i can be calculated from the lower cumulative probability calculated from equation (2). Basically, the smaller the deemed lower limit l _i is, that is, the wider the interval [l _i , U _i ], the closer the value of probability p approaches 1. If the probability p in each limiting operation is a constant value, and if n limiting operations occur until a solution is output, the probability that the output solution will be equal to the actual optimal solution is p ⁿ . Therefore, if the probability p is set to a value close to 1, the probability that the output solution will be the actual optimal solution increases. On the other hand, the narrower the interval [l _i , U _i ], the higher the speed. In this way, there is a trade-off between speedup and solution accuracy.

If the probability distribution of the interval in which the optimal solution can exist is unimodal, and the mode in the probability distribution is close to the upper limit U _i , then the lower limit l _i that gives the probability p close to 1 is The value becomes close to the upper limit U _i . In this case, both high speed and high precision are expected.

Here, if the upper limit value, lower limit value, and distribution function of the optimal solution for the total S of feasible solutions are represented by U ₀ , L ₀ , and f ₀ (x, L ₀ , U ₀ ), respectively, then the output solution The probability p _out that the difference between the output and the actual optimal solution optval is equal to or less than Δ (Δ≧0) can be expressed by the following equation (3). Here, y is a variable indicating a solution that can exist in the interval [output-Δ, output+Δ].

Hereinafter, a device for speeding up the branch and bound method according to the embodiment will be described. FIG. 1 is a block diagram showing the configuration of an apparatus for speeding up the branch and bound method according to the embodiment. The device 1 includes an input section 10, a branching section 20, a limiting section 30, and an output section 40.

The input unit 10 receives input of the original problem to be solved by the branch and bound method. The original problem may be any combinatorial optimization problem. The original problem may be, for example, an optimal route search problem that searches for an optimal route that minimizes travel time and the like when moving a vehicle. In the case of an optimal route search problem, the input unit 10 receives, for example, information on a graph representing a candidate travel route for a vehicle based on, for example, an input from a user. Alternatively, the input unit 10 obtains the original problem from a storage unit in which information on the original problem is stored in advance.

Further, in the embodiment, the input unit 10 receives input of the probability density function f _i and the parameter p. The probability density function f _i is a probability density function that represents as a probability distribution where the optimal solution in the solution set S _i described above exists in the closed interval [L _i , U _i ]. The parameter p is a set value of the probability that the optimal solution in the solution set S _i exists in the closed interval [l _i , U _i ], which is shown on the left side of equation (2). The input unit 10 receives the probability density function f _i and the parameter p from, for example, a user. Alternatively, the input unit 10 obtains the probability density function f _i and the parameter p from a storage unit in which these pieces of information are stored in advance.

In the embodiment, the probability density function f _i is, for example, a beta distribution of the first kind expressed by the following equation (4).

Here, (x-L _i )/(U _i -L _i ) in equation (4) is a random variable in the beta distribution of equation (4), and is a random variable in the beta distribution of equation (4), using the upper limit value U _i and the lower limit value L _i It corresponds to the value of the variable x normalized to a range of 0 to 1. The value of the random variable is determined by determining U _i and L _i during the limiting operation. Further, B(α, β) in equation (4) is a beta function.

Here, the beta distribution has a characteristic that the distribution shape changes greatly depending on the values of the shape parameters α and β. As mentioned above, the branch and bound _method in the _embodiment can speed up processing and Both high accuracy of the solution can be achieved. Generally, when α>1, β>1, and α>β, the beta distribution is unimodal and gives a distribution in which the highest frequency is near the upper limit value U _i . FIG. 2 is a graph showing the probability density distribution of equation (4) when α=6, β=1.1, U _i =2.5, and L _i =1.5.

The branching unit 20 executes a branching operation to divide the input problem into multiple problems. The algorithm for the branching operation by the branching unit 20 may be set as appropriate depending on the type of problem and the like.

The limiting unit 30 performs a limiting operation on each of the solution sets S _i of the problem divided by the branching unit 20 . The limiting section 30 in the embodiment includes a section determining section 31, a deemed lower limit determining section 32, and a comparing section 33.

The interval determining unit 31 determines an upper limit value U _i and a lower limit value L _i of an interval in which each optimal solution of the solution set S _i can exist. In the case of a minimization problem, an appropriate solution from among the feasible solutions is used for the upper limit U _i . Further, the lower limit value L _i can be obtained by solving a relaxation problem.

The deemed lower limit value determining unit 32 determines the deemed lower limit value _{l i} _for the solution set S i from the upper limit value U _i , the lower limit value L _i , the probability density function f _i and the parameter p. The deemed lower limit value l _i can be calculated, for example, by sequentially calculating the lower cumulative probability of the interval [L _i , U _i ] from the probability density function _fi , and the value of the calculated lower cumulative probability is the value of the parameter p. It can be determined as the value of x when there is a match.

The comparison unit 33 compares the existence range of solutions for pruning the solution set S _i based on the upper limit value U _i and the deemed lower limit value l _i determined for each of the solution set S _i . Then, as a result of the comparison, the comparison unit 33 prunes the solution set S _i in which the optimal solution cannot exist. Specifically, the comparison unit 33 prunes the solution set S _k that satisfies U _l <l _k .

The output unit 40 outputs the final solution remaining without pruning as the optimal solution. Output may be performed by any method such as displaying the output solution on a display, outputting it on a printer, or transmitting it to an external device. For example, in the case of an optimal route search problem that searches for an optimal route that minimizes travel time, etc., the output unit 40 outputs information on the optimal route combination that minimizes travel time, etc. For example, the output unit 40 displays the optimal route on a map displayed on the display.

FIG. 3 is a block diagram showing the hardware configuration of an example of the device 1. As shown in FIG. 3, the device 1 includes a processor 101, a ROM 102, a RAM 103, a storage 104, a display 105, an input interface 106, and a communication module 107. The device 1 can be, for example, a terminal device such as a personal computer, a smartphone, or a tablet terminal. However, the device 1 is not limited to a terminal device. For example, when used to search for a travel route for a vehicle, the device 1 may be mounted on the vehicle.

The processor 101 is a processor configured to control the operation of the device 1. Processor 101 is, for example, a CPU. The processor 101 may be an MPU or the like instead of a CPU. Further, the processor 101 may be configured as a hardware logic circuit such as an ASIC. Further, the processor 101 does not need to be configured by one CPU or the like, and may be configured by a plurality of CPUs or the like. The processor 101 can operate as the input section 10, the branching section 20, the limiting section 30, and the output section 40 by executing processing according to the search program 1042 stored in the storage 104.

The ROM 102 is composed of, for example, a nonvolatile semiconductor memory. The ROM 102 stores a startup program for the device 1 and the like. The RAM 103 is composed of, for example, a volatile semiconductor memory. The RAM 103 is used, for example, as a working memory in processing in the processor 101.

The storage 104 is a storage such as an SSD (Solid State Drive) or an HDD (Hard Disk Drive). The storage 104 stores, for example, an OS (Operating System) 1041 and a search program 1042. The OS 1041 is a program for realizing the basic functions of the device 1. Various programs stored in the storage 104 are executed under the control of the OS 1041. The search program 1042 is a program that allows the device 1 to search for an optimal solution to an input problem using a branch and bound method. The search program 1042 can be downloaded to the device 1 as needed. Furthermore, the storage 104 may further store either or both of the probability density function 1043 and the parameters 1044. The probability density function 1043 and the parameter 1044 are the probability density function f _i and the parameter p for calculating the deemed lower limit value l _i . The probability density function f _i and the parameter p do not necessarily need to be stored in the storage 104. The probability density function f _i and the parameter p may be input, for example, by the user's operation of the input interface 106, or may be obtained, for example, by communication from a server external to the device 1.

The display 105 is a liquid crystal display, an organic EL display, or the like. Various screens displayed under the control of the search program 1042 are displayed on the display 105. This screen includes, for example, a problem, a probability density function f _i , an input screen for the parameter p, and an optimal solution display screen.

The input interface 106 is an interface that accepts input from buttons, switches, touch panels, etc. provided in the device 1. The input interface 106 converts signals from buttons, switches, touch panels, etc. so that the processor 101 can identify them.

The communication module 107 is a module for the device 1 to communicate with various external devices. The communication module 107 may be either a wired communication module or a wireless communication module. Further, the communication method of the communication module 107 is not limited to a specific one.

Next, the operation of the device 1 will be explained. FIG. 4 is a flowchart showing the operation of the device 1. The operations in FIG. 4 are controlled by processor 101.

In step S1, the processor 101 receives input of a question. Furthermore, the processor 101 receives input of the probability density function f _i and the parameter p. After these inputs are accepted, the process moves to step S2. The input of the question may be performed, for example, by the user's operation of the input interface 106. Further, the probability density function f _i and the parameter p can be input by acquiring information stored in the storage 104, for example. Here, for the following explanation, it is assumed that the probability density function f _i is a beta distribution of the first kind shown in equation (4). Further, the shape parameters α and β are assumed to be 6 and 1.1, respectively. Further, the parameter p is assumed to be 0.95.

In step S2, processor 101 performs a branching operation. For example, the processor 101 divides the original problem into multiple problems by case classification. For example, in the case of an optimal route search problem, the possible combinations of routes between the starting point and the ending point can be a problem. For the sake of explanation, it is assumed below that the solution to the problem is divided into five solution sets S ₁ , S ₂ , S ₃ , S ₄ , and S ₅ by a branching operation. For example, if the input problem is an optimal route search problem that minimizes the cost such as travel time required for moving the route, the values of the solution sets S ₁ , S ₂ , S ₃ , S ₄ , and S ₅ are the cost can be the value of

In step S3, the processor 101 determines an upper limit value U _i and a lower limit value L _i . As described above, the upper limit value U _i is determined to be an appropriate value from among the feasible solutions. On the other hand, the lower limit value L _i is determined by solving a relaxation problem corresponding to each solution set S _i . In the following, for example, U ₁ =2.5, L ₁ =1.5, U ₂ =3.1, L ₂ =1.9, U ₃ =4.5, L ₃ =0.7, U ₄ =3. .5, L ₄ =2.1, U ₅ =4, and L ₅ =1. Since the solution to the relaxed problem is a solution to the problem whose conditions are more relaxed than the original problem, each optimal solution of the solution set S _i is necessarily within the interval defined by the upper limit value U _i and the lower limit value L _i . can exist inside.

In step S4, the processor 101 determines a deemed lower limit value l _i for each solution set S _i from the upper limit value U _i , the lower limit value L _i , the probability density function f _i and the parameter p. As mentioned above, the deemed lower limit l _i is determined by sequentially calculating the lower cumulative probabilities of the interval [L _i , U _i ] from the probability density function _fi , and It can be determined as the value of x when it matches the value of parameter p. In the following, it is assumed that l ₁ =2.09, l ₂ =2.6, l ₃ =2.94, l ₄ =2.87, and l ₅ =2.77, for example. Each optimal solution of the solution set S _i may exist with probability p in the interval defined by the upper limit value U _i and the assumed lower limit value l _i . Further, the deemed lower limit value l _i is always larger than the lower limit value L _i .

In step S5, the processor 101 determines whether there is S _k such that U _l <l _k . Here, the subscripts l and k are both elements of i. If it is determined that S _k such that U _l <l _k exists, the process moves to step S6. If it is determined that there is no S _k satisfying U _l <l _k , the process moves to step S7. In the example, U ₁ <l ₂ , U ₁ <l ₃ , U ₁ <l ₄ , U ₁ <l ₅ . Therefore, S _k satisfying U _l <l _k is determined to be S ₂ , S ₃ , S ₄ , and S ₅ . Here, in the conventional branch and bound method, it is determined whether or not S _k such that U _l <L _k exists. In this case, U ₁ >L ₂ , U ₁ >L ₃ , U ₁ >L ₄ , and U ₁ >L ₅ . The same goes for other combinations, and as a result, there is no solution set to be pruned. In this case, another branch operation is required.

In step S6, the processor 101 prunes the solution set S _k . That is, the processor 101 terminates the search for the optimal solution for the solution set S _k . In the example, processor 101 prunes solution sets S ₂ , S ₃ , S ₄ , and S ₅ . After that, the process moves to step S7.

In step S7, the processor 101 determines whether to end the search. If the predetermined search termination conditions are met, such as one solution set remaining without pruning and the optimal solution in that solution set being a feasible solution, it is determined that the search is to be terminated. Ru. In step S7, if it is determined that the search is not to end, the process returns to step S2. In this case, the branching and limiting operations are performed again on the solution set S _i that remains unpruned. If it is determined in step S7 to end the search, the process moves to step S8.

In step S8, the processor 101 outputs the searched solution as an output solution. After that, the process in FIG. 4 ends. For example, in the case of an optimal route search problem, the processor 101 displays a route corresponding to the output solution on the display 105, for example.

As explained above, according to this embodiment, in the limiting operation of the branch and bound method, instead of the lower limit determined by solving the relaxation problem, the lower limit of the interval in which the optimal solution can exist with probability p is used. The existence range of the optimal solution is compared using a certain assumed lower limit value. Since the deemed lower limit value is larger than the lower limit value, the possibility that the number of times of pruning will increase is increased by comparing the existence range of the optimal solution based on the deemed lower limit value. By increasing the number of pruning operations, it is expected that the search for the optimal solution will become faster. Also, excessive learning for branching and limiting operations is not required.

Here, in the embodiment, the existence range of the optimal solution is compared based on the deemed lower limit value. Therefore, there is a possibility that the output solution will not be the optimal solution. Since the probability density distribution of where the optimal solution exists in the interval [l _i , U _i ] is known, the probability that the difference y between the output solution and the actual optimal solution is less than or equal to Δ (Δ≧0) can also be determined. can be calculated.

[Modified example]
Modifications of the embodiment will be described below. In the example described above, the deemed lower limit value l _i is calculated based on the probability density function _fi . This is because the problem is assumed to be a minimization problem. If a maximization problem is assumed as the problem, the upper limit value is determined by the relaxation problem. Therefore, in the case of a maximization problem, the deemed upper limit u _i is calculated based on the probability density function f _i . The deemed upper limit value u _i is determined by, for example, sequentially calculating the upper cumulative probability of the interval [L _i , U _i ] from the probability density function f _i , and when the value of the calculated upper cumulative probability matches the value of the parameter p. can be determined as the value of x at Then, in the comparison in step S5, it is determined whether or not S _k such that L _l >u _k exists, and a solution set S _k that satisfies this condition is pruned.

Further, in the embodiment, a beta distribution of the first kind is used as the probability density function _fi . However, the beta distribution of the first kind does not necessarily have to be used as the probability density function _fi . The probability density function f _i may be any function that can express where in the interval [L _i , U _i ] the optimal solution exists as a probability distribution.

Further, each process according to the embodiment described above can also be stored as a program that can be executed by the processor 101, which is a computer. In addition, it can be stored and distributed in a storage medium of an external storage device such as a magnetic disk, an optical disk, or a semiconductor memory. The processor 101 reads a program stored in the storage medium of this external storage device, and its operations are controlled by the read program, thereby being able to execute the above-described processes.

Note that the present invention is not limited to the above-described embodiments, and can be variously modified at the implementation stage without departing from the gist thereof. Moreover, each embodiment may be implemented in combination as appropriate, and in that case, the combined effect can be obtained. Furthermore, the embodiments described above include various inventions, and various inventions can be extracted by combinations selected from the plurality of constituent features disclosed. For example, if a problem can be solved and an effect can be obtained even if some constituent features are deleted from all the constituent features shown in the embodiment, the configuration from which these constituent features are deleted can be extracted as an invention.

1... Device 10... Input section 20... Branching section 30... Limiting section 31... Section determining section 32... Deemed lower limit determining section 33... Comparing section 40... Output section 101... Processor 102... ROM
103...RAM
104...Storage 105...Display 106...Input interface 107...Communication module 1041...OS
1042... Search program 1043... Probability density function 1044... Parameter

Claims

a branching unit that performs a branching operation to divide the input problem into multiple problems;
A first threshold, which is one threshold value that determines a possible first interval of the solution set of each divided problem, is determined from the feasible solutions of each divided problem, and the first interval is determined. an interval determining unit that determines a second threshold, which is the other threshold, by solving a relaxation problem for each divided problem;
The first threshold value, the second threshold value, and the optimal solution of each divided problem are narrower than the first interval, and one threshold that determines the interval is the first threshold value. a deemed threshold determining unit that determines a third threshold, which is the other threshold that determines the second interval, based on a probability distribution representing the probability of existing in the second interval and the set value of the probability;
a comparison unit that performs pruning of the solution set by comparing the first threshold value and the third threshold value for each solution set;
A device for speeding up the branch and bound method, comprising:
The problem is a minimization problem,
The first threshold is an upper limit value in the first section and the second section,
The second threshold is a lower limit value in the first section,
The third threshold is a lower limit value in the second section that is larger than the second threshold;
An apparatus for speeding up the branch and bound method according to claim 1.
Performing a branching operation to divide the input problem into multiple problems;
A first threshold, which is one threshold value that determines a possible first interval of the solution set of each divided problem, is determined from the feasible solutions of each divided problem, and the first interval is determined. determining a second threshold, which is the other threshold, by solving a relaxation problem for each of the divided problems;
The first threshold value, the second threshold value, and the optimal solution of each divided problem are narrower than the first interval, and one threshold that determines the interval is the first threshold value. Determining a third threshold, which is the other threshold that determines the second interval, based on a probability distribution representing the probability of existing in the second interval and the set value of the probability;
pruning the solution set by comparing the first threshold and the third threshold for each solution set;
A method for speeding up branch-and-bound methods.
Performing a branching operation to divide the input problem into multiple problems;
A first threshold, which is one threshold value that determines a possible first interval of the solution set of each divided problem, is determined from the feasible solutions of each divided problem, and the first interval is determined. determining a second threshold, which is the other threshold, by solving a relaxation problem for each of the divided problems;
The first threshold value, the second threshold value, and the optimal solution of each divided problem are narrower than the first interval, and one threshold that determines the interval is the first threshold value. Determining a third threshold, which is the other threshold that determines the second interval, based on a probability distribution representing the probability of existing in the second interval and the set value of the probability;
pruning the solution set by comparing the first threshold and the third threshold for each solution set;
A program for accelerating the branch and bound method on a computer.