Optimal solution search processing device and optimal solution search processing method

Abstract

To realize efficient optimum solution search processing for optimization problem such as combinational optimization problem etc., by reducing a calculation amount.SOLUTION: Solutions generated in a previous stage of an optimum solution search are narrowed down. More specifically, an information processor 10 as an optimum solution search processor that executes an optimum solution search for solving an optimization problem has: an optimum solution search part 415a which generates solution candidates for the optimization problem; and an executable solution candidate search part 415b which executes the narrowing-down processing for the solution candidates corresponding to the solution candidates for the optimization problem and then specifies a solution for the optimization problem from among the narrowed-down solution candidates.SELECTED DRAWING: Figure 4

Description

Technical Field

[0001] The present invention relates to an optimal solution search technique for solving combinatorial optimization problems with multiple constraint conditions.

Background Art

[0002] In order to improve the productivity of social systems, it is required to efficiently solve practical combinatorial optimization problems by making full use of digital technologies. For this purpose, techniques for converting combinatorial optimization problems into ground state search problems of interaction models such as Ising models and solving them with physical phenomena or algorithms simulating them have attracted attention.

[0003] In Patent Document 1, the aim is to provide a means for efficiently obtaining a solution to a set partitioning problem. Patent Document 1 discloses an acquisition means for acquiring candidate information which is information on candidates for subsets in a set partitioning problem, and a generation means for generating an expression of an Ising model corresponding to the set partitioning problem based on the candidate information acquired by the acquisition means.

Prior Art Documents

Patent Documents

[0004]

Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0005] As described above, when using an interaction model such as an Ising model, in order to solve an optimization problem with complex constraints, a method of generating solution candidates by logic and solving the combination as a set partitioning (covering) problem is effective. Here, there is a trade-off relationship between the pre-step (generation of solution candidates) and the post-step (combination of solution candidates) of the optimal solution search. In order to obtain a high-quality and more accurate solution in the post-step, it is important to preferentially generate high-quality solution candidates in the pre-step, but this has been difficult with the prior art. [Means for solving the problem]

[0006] To solve the above problems, the present invention narrows down the candidate solutions generated in the pre-process of the optimal solution search. More specifically, in an optimal solution search processing device for solving an optimization problem, the present invention comprises an optimal solution search unit that generates candidate solutions for the optimization problem, and an executable solution candidate search unit that performs a narrowing down process of the candidate solutions to identify a solution for the optimization problem. The feasible solution candidate search unit performs a process to narrow down the solution candidates by executing a linear programming method and a quadratic programming method. Furthermore, the present invention also includes an optimal solution search method using an optimal solution search processing device. [Effects of the Invention]

[0007] According to the present invention, it is possible to reduce the amount of computation and realize an efficient optimal solution search process. [Brief explanation of the drawing]

[0008] [Figure 1] This graph shows the relationship between the variable array and the objective function value in one embodiment of the present invention. [Figure 2] This is a diagram illustrating the Ising model in one embodiment of the present invention. [Figure 3] This figure shows the hardware configuration of an information processing device in one embodiment of the present invention. [Figure 4] This is a functional block diagram of an information processing device in one embodiment of the present invention. [Figure 5] This is a flowchart showing the flow of the optimal solution search process in one embodiment of the present invention. [Figure 6A] This is a diagram (part 1) illustrating an example of applying one embodiment of the present invention to a delivery planning problem. [Figure 6B] This is a diagram (part 2) illustrating an example of applying one embodiment of the present invention to a delivery planning problem. [Figure 7] This figure shows an example of implementation in a cloud system according to one embodiment of the present invention. [Modes for carrying out the invention]

[0009] The following describes in detail, with reference to the drawings, an optimal solution search for a combinatorial optimization problem, which is one embodiment of the present invention. Figure 1 is a graph showing the relationship between the variable array and the objective function value in this embodiment. Generally, an optimization problem is one in which the objective function value is minimized or maximized for a decision target (variable array) under multiple constraints (hereinafter, it will be simply described as minimization). In Figure 1, the objective function value of A is shown to be the minimum value in A, B, and C. In other words, the objective function value corresponding to A is found to be the optimal solution. Now, the variable array shown in Figure 1 will be explained.

[0010] When a combinatorial optimization problem is transformed into a search problem for the ground state of an interaction model (the state in which the energy function takes its minimum value), an interaction model is defined, as shown in Figure 2, in which nodes correspond to each variable and interactions occur between the nodes. Each variable can take continuous or discrete values, but in particular, when all variables are binary values, the interaction model reduces to the Ising model.

[0011] In interaction models, the objective function takes on various values ​​depending on the combination of the array of variables. Furthermore, in real-world combinatorial optimization problems, the number of decision-making targets is large, and therefore the number of corresponding variables is also large. Consequently, the number of candidate solutions in interaction models becomes enormous, making the search for the optimal solution difficult. For this reason, it is possible to treat problems such as set partitioning as a ground state search problem of the Ising model using the procedure described in Patent Document 1, but this becomes difficult as the number of variables increases.

[0012] Next, the configuration of this embodiment for generating candidate solutions in the pre-process of the optimal solution search described above will be explained. Figure 3 is a diagram showing the hardware configuration of the information processing device 10 in this embodiment. The information processing device 10 performs an optimal solution search and can be implemented as a so-called computer. For this reason, as shown in Figure 3, the information processing device 10 has a processor 11, a main memory 12, an auxiliary memory 13, an input device 14, an output device 15, a communication device 16, and arithmetic units 17-1 to 17-n, which are connected to each other via a communication channel 18.

[0013] First, the processor 11 can be implemented as a processing unit such as a CPU (Central Processing Unit), and it performs calculations and other processing according to the program stored in the auxiliary storage device 13, which will be described later. The main memory 12 can be implemented as memory, and it is where the program stored in the auxiliary storage device 13 and the information used for processing by the processor 11 are displayed. The auxiliary storage device 13 can be implemented as so-called storage, and it stores the aforementioned program and information. These will be explained later. The auxiliary storage device 13 may also be implemented as various storage media such as an external HDD (Hard Disk Drive), SSD (Solid State Drive), or memory card. Furthermore, it may be implemented as a separate device from the information processing device 10, such as a database.

[0014] Furthermore, the input device 14 has the function of receiving instructions from the user. For this reason, the input device 14 can be implemented as a so-called input device such as a mouse or keyboard. The output device 15 outputs the processing results from the processor 11 and the calculation results from the arithmetic units 17-1 to 17-n. For this reason, the output device 15 can be implemented as a display device such as a display. In addition, the input device 14 and the output device 15 may be configured as an integrated unit, such as a touch panel. Furthermore, the input device 14 and the output device 15 may be implemented as separate devices from the information processing device 10, such as terminal devices.

[0015] Furthermore, the communication device 16 communicates with other devices via the network. In addition, the arithmetic units 17-1 to 17-n are devices that implement algorithms for solving optimization problems. For this reason, the arithmetic units 17-1 to 17-n can be implemented using dedicated circuits such as FPGAs (Field-Programmable Gate Arrays). Furthermore, the communication channel 18 can be implemented using a bus or the like to connect the above-mentioned devices.

[0016] Next, FIG. 4 is a functional block diagram of the information processing apparatus 10 in the present embodiment. The information processing apparatus 10 is an example of realizing the optimal solution search processing apparatus of the present invention. In FIG. 4, the information processing apparatus 10 includes a model conversion unit 411, a model coefficient setting unit 412, a variable value initialization unit 413, an arithmetic control unit 414, a variable value reading unit 416, a storage unit 400, and an arithmetic device control unit 404.

[0017] Here, the model conversion unit 411 converts the combinatorial optimization problem into an interaction model. The model coefficient setting unit 412 sets the coefficients of the converted interaction model. The variable value initialization unit 413 initializes the variable array when solving the combinatorial optimization problem.

[0018] Also, the arithmetic control unit 414 executes an arithmetic operation according to an algorithm for solving the optimization problem. For this reason, the arithmetic control unit 414 includes an optimal solution search unit 415a and an executable solution candidate search unit 415b. Here, the optimal solution search unit 415a executes generation of solution candidates and narrowing-down processing of solution candidates. Note that the result of the narrowing-down processing does not have to be a solution candidate that can be actually realized. Also, the executable solution candidate search unit 415b obtains a solution to the set partitioning problem. That is, a solution is specified from the solution candidates.

[0019] Furthermore, the storage unit 400 stores each information of problem data 401, quadratic programming format problem data 402, and domain data 403. The problem data 401 is data indicating the combinatorial optimization problem to be solved. The quadratic programming format problem data 402 is problem data converted into a quadratic programming problem which is a non-linear programming problem. Also, the domain data 403 is data indicating a domain given to the combinatorial optimization problem, for example, a constraint condition.

[0020] Furthermore, the arithmetic device control unit 404 controls the arithmetic devices 17-1 to 17-n shown in FIG. 3. In FIG. 4, the arithmetic device control unit 404 controls the arithmetic operations in the model conversion unit 411 to the variable value reading unit 416. Thus, the arithmetic device control unit 404 realizes a calculation using an algorithm for solving the combinatorial optimization problem.

[0021] Here, we will explain the correspondence between the hardware configuration shown in Figure 3 and the functional blocks shown in Figure 4.

[0022] The processor 11 in Figure 3 implements the functions of the arithmetic unit control unit 404. To do this, the arithmetic unit control program 116, described later, is loaded into the main memory 12, and the processor 11 executes the processing. The main memory 12 and auxiliary memory 13 in Figure 3 correspond to the storage unit 400 in Figure 4. As a result, the auxiliary memory 13 stores information such as problem data 401, quadratic plan format problem data 402, domain data 403, and at least the arithmetic unit control program 116. The auxiliary memory 13 can also store programs other than the arithmetic unit control program 116. This will be explained later using Figure 7.

[0023] Furthermore, the input device 14, output device 15, and communication device 16 are omitted in Figure 4. In addition, the arithmetic units 17-1 to 17-n in Figure 3 correspond to the model conversion unit 411 to the variable value reading unit 416 in Figure 4. However, the functions of the model conversion unit 411 to the variable value reading unit 416 may be implemented by the processor 11 according to the program.

[0024] Next, we will explain the optimal solution search process using the information processing device 10 described above. Here, we will mainly explain the parts of the functional block shown in Figure 4. We will also explain the preprocessing for this optimal solution search process. The model conversion unit 411 converts the combinatorial optimization problem that is the target of the optimal solution search process into an interaction model. The model coefficient setting unit 412 sets the coefficients of the converted interaction model. These coefficients may be set according to the user's specifications. Furthermore, the variable value initialization unit 413 initializes the variables (arrays) used in the optimal search process.

[0025] The optimal solution search process shown in Figure 5 is then executed. The optimal solution search process consists of three main steps: generating candidate solutions and narrowing down the generated candidate solutions (pre-process), and identifying the optimal solution as the final result from the candidate solutions (post-process).

[0026] Figure 5 is a flowchart showing the flow of the optimal solution search process in this embodiment. First, in step S1, the feasible solution candidate search unit 415b of the arithmetic control unit 414 determines the computational amount (time, number of candidates) for narrowing down the options. For this purpose, the user of the information processing device 10 may specify the computational amount. In other words, the information processing device 10 can be controlled to perform processing within a predetermined range of conditions.

[0027] Furthermore, in step S2, each of the optimal solution search units 415a generates a candidate solution. In this step, the target is set cover / partition problems and their derivative problems, and candidate solutions are generated using quadratic programming or linear programming methods. However, the generation of candidate solutions is not limited to these methods. Also, the number of optimal solution search units 415a is illustrative and is not limited to the number shown in Figure 5. In other words, one or more optimal solution search units are sufficient.

[0028] Furthermore, in step S3, the feasible solution candidate search unit 415b narrows down the solution candidates. The details of the narrowing process (step S3) are explained below. In this embodiment, the following equations (Equations 1) to (Equations 3) are used.

[0029]

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[0030]

number

[0031]

number

[0032] First, (Equation 1) represents Problem 1, a combinatorial optimization problem to be solved. Then, candidate solutions are calculated using either (Equation 2), which represents Problem 2, a continuous relaxation problem of Problem 1, or (Equation 3), which represents Problem 3, obtained by eliminating the quadratic term. Here, for (Equation 2), the technique described in International Publication No. 2021 / 220445 can be used to efficiently find the solution, and for (Equation 3), a linear programming solver can be used.

[0033] In other words, the optimal solution search unit 415a solves problems 2 and 3, and the feasible solution candidate search unit 415b determines that the xi∈[0,1] with a large value in those solutions are important (highly likely to be adopted) as solution candidates for problem 1 as well. Note that "high probability" can mean a probability greater than or equal to a predetermined value, or it can mean a high number of values ​​with a high probability.

[0034] Then, the optimal solution search unit 415a removes the already obtained solutions and solves again using (Equation 2) and (Equation 3). As a result, the feasible solution candidate search unit 415b searches for xi with a large value. Here, a large value may be greater than or equal to a predetermined value, or it may be the top of the large values ​​(a predetermined number). The feasible solution candidate search unit 415b then continues processing in Problem 2 and Problem 3 until the number of solution candidates meets a certain condition. In other words, if the number of solutions satisfies the matching condition, the processing is terminated. As a result, the solution candidates are narrowed down. As described above, in this embodiment, the solution candidates generated in step S3 and the solution candidates narrowed down by the narrowing process corresponding to the generation of solution candidates are identified.

[0035] Furthermore, the generation of candidate solutions (step S2) and the narrowing down process (step S3) can be performed as follows: (1) Process in the order of linear programming → quadratic programming, and combine the results; (2) Process in the order of quadratic programming → linear programming, and combine the results. In this way, the above processing can be achieved by executing both linear programming and quadratic programming and combining the results. When performing processing with quadratic programming, the quadratic programming format problem data 402 and domain data 403 are used. Otherwise, the problem data 401 and domain data 403 are used.

[0036] Furthermore, examples of generating candidate solutions will be explained below. First, we will explain the generation process for the set partitioning problem. Variable: σi = 1 (0) (Select subset i (or not)) Constants: Mij = 1 (0) (the subset i contains (does not contain) element j), ci (cost of subset i) Constraint: Every element j is contained in at least one subset ΣiMij σi=1 Objective function: Minimize the sum of costs of the subset Σiciσi.

[0037] Next, we will explain the generation problem in the set cover problem. Variable: σi = 1 (0) (Select subset i (or not)) Constants: Mij = 1 (0) (the subset i contains (does not contain) element j), ci (cost of subset i) Constraint: Every element j is contained in at least one subset ΣiMij σi≧1 Objective function: Minimize the sum of costs of the subset Σiciσi.

[0038] Next, we will explain the generation of derivative problems of the set partitioning problem and the set covering problem. Here, a derivative problem is one in which the variables, constants, and constraints are the same as those of the set partitioning problem and the set covering problem described above. Furthermore, in order to achieve the equalization of evaluation metrics and the minimization of variance (for example, cost equalization of each subset), quadratic equations such as (Equation 4) and (Equation 5) below may be added to the objective function.

[0039]

number

[0040]

number

[0041] In this embodiment, the derivative problem refers to a situation where variables and other elements are common, as described above, and only require some modification.

[0042] Furthermore, in step S4, the feasible solution candidate search unit 415b identifies a solution from the narrowed-down solution candidates. In this embodiment, solutions to set partitioning problems, set covering problems, or derived problems thereof can also be found. In each of the above processes, if variables are used, the values ​​read by the variable value read unit 416 are used. Then, the arithmetic control unit 414, in particular the optimal solution search unit 415a, performs various calculations and processes under the control of the arithmetic unit control unit 404.

[0043] Furthermore, in step S5, the arithmetic unit control unit 404 receives a response regarding whether the solution identified in step S4 meets expectations. In other words, it receives an evaluation result from the user of the information processing device 10 regarding the identified solution. If the result is that it meets expectations (YES), the process terminates. Otherwise (NO), the process proceeds to step S1.

[0044] Then, if the process transitions to step S1, the following processes are executed targeting one of the following: (1) Perform the following processing on solution candidates other than the solution identified in step S4. In this case, all solution candidates may be included, or the solution candidates extracted according to any rule may be included. (2) A different candidate solution may be calculated and used as the target solution.

[0045] This concludes the explanation of the flowchart in Figure 5. Next, we will describe an example of applying this embodiment to a delivery planning problem such as the transportation of goods. Figure 6A is a diagram (part 1) illustrating an example of applying this embodiment to a delivery planning problem. In this diagram, we assume that goods are delivered from a central base to points A, B, C, D, and E (see bottom of Figure 6A). In this case, candidate solutions such as i=1 to i=3 are generated. Candidates i=1 and i=3 are advantageous in terms of values ​​that can be evaluated using objective functions such as "completion time" and "travel distance". As a result, these candidate solutions are narrowed down as shown in Figure 6B.

[0046] Next, a modified implementation of this embodiment will be described. The information processing device 10 of this embodiment can be implemented as a so-called cloud system. This configuration will be described below. Figure 7 shows an example of implementation in a cloud system according to this embodiment.

[0047] In Figure 7, the cloud system is configured such that the information processing device 10 is connected to terminal devices 20 and production equipment 30 via a network 40. The production equipment 30 is responsible for production in the production system targeted by the optimization problem, and receives the solution identified by the information processing device 10 and performs tasks such as formulating a calculation plan accordingly.

[0048] Furthermore, the terminal device 20 is a terminal used by users performing production tasks, and can be implemented as a computer such as a PC or tablet. In other words, the input device 14 and output device 15 in Figure 4 will be implemented in the terminal device 20.

[0049] Furthermore, the information processing device 10 can be implemented using a computer known as a server. The information processing device 10 includes a processor 11, main memory 12, auxiliary memory 13, communication device 16, and arithmetic units 17-1 to 17-n, which are connected to each other via a communication channel 18. These are the same as those shown in Figure 4.

[0050] Here, the auxiliary storage device 13 stores the model conversion program 111, the model coefficient setting program 112, the variable value initialization program 113, the arithmetic control program 114, the variable value reading program 115, the arithmetic unit control program 116, and various information 120. Here, each program corresponds to the respective parts of Figure 5 as follows. Model conversion program 111: Model conversion unit 411 Model coefficient setting program 112: Model coefficient setting unit 412 Variable value initialization program 113: Variable value initialization unit 413 Variable value reading program 115: Variable value reading unit 416 Arithmetic unit control program 116: Arithmetic unit control unit 404 Furthermore, the arithmetic control program 114 has the functions of the arithmetic control unit 414, which corresponds to the arithmetic units 17-1 to 17-n. Therefore, in Figure 8, either the arithmetic control program 114 or the arithmetic units 17-1 to 17-n can be omitted.

[0051] Furthermore, the arithmetic control program 114 includes a first optimal solution search module, a second optimal solution search module, and an arithmetic device control module, which correspond to the optimal solution search unit 415a and the arithmetic device control unit 404, respectively.

[0052] This concludes the description of this embodiment. This embodiment is applicable to various fields. It can be applied to transportation systems such as railways, logistics services such as home delivery (generation of delivery routes and driver allocation), and optimization of warehouse and production systems. In this case, it becomes possible to formulate operation plans, production plans, personnel plans, etc. [Explanation of Symbols]

[0053] 10...Information processing device, 12...Main memory, 13...Auxiliary memory, 14...Input device, 15...Output device, communication device, 17-1~17-n...Arithmetic unit, 18...Communication channel, 20...Terminal device, 30...Production device, 40...Network, 400...Storage unit, 401...Problem data, 402...Secondary programming format problem data, 403...Domain data, 404...Arithmetic unit control unit, 411...Model conversion unit, 412...Model coefficient setting unit, 413...Variable value initialization unit, 414...Arithmetic control unit, 416...Variable value reading unit

Claims

1. In an optimal solution search processing unit that performs optimal solution search for an optimization problem, An optimal solution search unit that generates candidate solutions to the aforementioned optimization problem, The system includes an executable solution candidate search unit that performs a process to narrow down the candidate solutions according to the candidate solutions for the optimization problem, and identifies a solution for the optimization problem from the narrowed-down candidate solutions. The aforementioned feasible solution candidate search unit is an optimal solution search processing device that performs a process to narrow down the solution candidates by executing a linear programming method and a quadratic programming method.

2. In the optimal solution search processing device according to claim 1, The feasible solution candidate search unit is an optimal solution search processing device that executes the quadratic programming method after the linear programming method, or the linear programming method after the quadratic programming method.

3. In the optimal solution search processing device according to claim 1, Furthermore, the identified solution is output to an output device, and the system has a control unit that receives evaluation results from the user of the optimal solution search processing device regarding whether the identified solution meets expectations. If the received evaluation result does not meet expectations, the optimal solution search processing device generates candidate solutions again, and the feasible solution candidate search unit identifies the solution.

4. In the optimal solution search processing device according to claim 3, If the received evaluation result does not meet expectations, the optimal solution search processing device generates the solution from among solution candidates other than the identified solution, and the feasible solution candidate search unit identifies the solution.

5. In the optimal solution search processing device according to claim 4, If the received evaluation result does not meet expectations, the optimal solution search processing device generates the solution using an optimal solution search unit to select a solution candidate extracted from other solution candidates according to an arbitrary rule, and the feasible solution candidate search unit identifies the solution.

6. In the optimal solution search processing device according to claim 3, If the received evaluation result does not meet expectations, the optimal solution search processing device generates the solution using a different candidate solution than the identified solution, and the feasible solution candidate search unit identifies the solution.

7. In an optimal solution search method that uses an optimal solution search processing device to perform an optimal solution search for an optimization problem, The aforementioned optimal solution search processing device comprises an optimal solution search unit and a feasible solution candidate search unit. The aforementioned optimal solution search unit generates candidate solutions to the optimization problem, The executable solution candidate search unit performs a process to narrow down the solution candidates according to the solution candidates for the optimization problem, and identifies the solution for the optimization problem from the narrowed-down solution candidates. An optimal solution search method in which the feasible solution candidate search unit performs a process to narrow down the solution candidates by executing a linear programming method and a quadratic programming method.

8. In the optimal solution search processing method described in claim 7, An optimal solution search method wherein the feasible solution candidate search unit executes the quadratic programming method after the linear programming method, or executes the linear programming method after the quadratic programming method.

9. In the optimal solution search processing method described in claim 7, Furthermore, the optimal solution search processing device has a arithmetic unit control unit, The arithmetic unit control unit outputs the identified solution to the output device, and receives an evaluation result from the user of the optimal solution search processing device regarding whether the identified solution meets expectations. If the received evaluation result does not meet expectations, the optimal solution search unit generates candidate solutions again and identifies the solution in an optimal solution search processing method.

10. In the optimal solution search processing method described in claim 9, An optimal solution search method in which, if the received evaluation result does not meet expectations, the optimal solution search unit generates the solution from among candidate solutions other than the identified solution and identifies the solution.

11. In the optimal solution search processing method according to claim 10, If the received evaluation result does not meet expectations, the optimal solution search method involves the optimal solution search unit generating the solution and identifying the solution from among the candidate solutions other than the identified solution, using an arbitrary rule to select the candidate solution.

12. In the optimal solution search processing method described in claim 9, An optimal solution search method in which, if the received evaluation result does not meet expectations, the optimal solution search unit generates the solution using a different candidate solution than the identified solution, and the feasible solution candidate search unit identifies the solution.

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