Optimization device, optimization method, and optimization program

Abstract

An optimization device according to an embodiment, which is capable of solving an online matching program with which it is possible to control each node and probabilities of appearance, is provided with: an acquisition unit that acquires input data, including information about nodes, remaining quantities assigned to fixed nodes among said nodes, probabilities of appearance assigned to appearing nodes among said nodes, and a reward assigned to each edge upon matching; a formulation unit that formulates a first optimization problem on the basis of the input data; a determination unit that determines whether all appearing nodes satisfy a prescribed assumption; a transformation unit that, if the prescribed assumption is satisfied, transforms the first optimization problem into a second optimization problem with which it is possible to obtain an approximate solution that consists of a matching strategy and variables for controlling the weight and the probability of appearance of each node of the first optimization problem; a problem solving unit that obtains the approximate solution by solving the second optimization problem; and an output control unit that outputs the approximate solution.

Description

[Technical Field]

[0001] The present invention relates to an optimization device, an optimization method, and an optimization program. [Background technology]

[0002] Online matching is known as an optimization problem that can be applied to various purposes. It is a special matching problem for a bipartite graph G = (U, V, E). This problem involves assigning u∈U to v∈V that appears at each time t, given a fixed node set U that exists in advance and a set of nodes V that may appear in the future. Here, each fixed node u has a remaining capacity r u and cannot be allocated beyond this.

[0003] For example, online matching can be applied to the allocation of internet advertisements (U), which allocates a given advertisement slot to a website visitor (V) without knowing in advance which website the advertisement will appear on.

[0004] Online matching can also be applied to crowdsourcing, which allocates tasks (U) to be solved to workers (V) that appear sequentially via the Internet, or to taxi platforms that allocate available taxis (U) to customers (V) that appear sequentially.

[0005] In this case, the reward for assigning a task to a worker is "monetary value of the task completed - wage", so wage x vt Also, wage x vt Depending on the time, each worker decides whether to participate in the market at time t, which also affects the appearance probability of each worker at time t.

[0006] For example, in Non-Patent Document 1, in online matching with controllable rewards and arrival rates, such a variable x vtA technique for determining a matching strategy is disclosed. [Prior art documents] [Non-patent literature]

[0007] [Non-Patent Document 1] Yuya Hikima, Yasunori Akagi, Naoki Marumo, and Hideaki Kim. Online matching with controllable rewards and arrival probabilities. In International Joint Conference on Artificial Intelligence, 2022. Internet<URL: https: / / www.ijcai.org / proceedings / 2022 / 0254.pdf> [Non-patent document 2] Alaei, Saeed, MohammadTaghi Hajiaghayi, and Vahid Liaghat. "Online prophet inequality matching with applications to allocation ad." Proceedings of the 13th ACM Conference on Electronic Commerce. 2012. [Non-patent document 3] Ahuja, RK; Magnanti, TL; and Orlin, JB 1993. Network Flows: Theory, Algorithms, and Applications. Prentice-Hall Summary of the Invention [Problem to be solved by the invention]

[0008] For example, in Non-Patent Document 1, the approximation rate, which is a theoretical guarantee of the quality of the output solution, is low at 1 / 2, which causes a problem that a good solution may not be output.u When takes a large value, the calculation takes a long time.

[0009] This invention was made with the above in mind, and its purpose is to develop a matching strategy and a variable x that controls the weight and appearance probability of a node. vt The object of the present invention is to provide an approximate solution method that guarantees a better approximation rate than conventional methods for the optimization problem of simultaneously determining the remaining capacity r of each node u. u To provide a technique in which the calculation time does not increase even when takes a large value. [Means for solving the problem]

[0010] In order to solve the above-mentioned problems, one aspect of the present invention is an optimization device capable of solving online matching problems in which it is possible to control the occurrence probability of each node, the optimization device comprising: an acquisition unit that acquires input data including information about the nodes, a residual amount assigned to fixed nodes among the nodes, an occurrence probability assigned to occurrence nodes among the nodes, and a reward assigned to each edge when matching is achieved; a formulation unit that formulates the first optimization problem as a first optimization problem that maximizes the total reward obtained based on the input data; a determination unit that determines whether all of the occurrence nodes satisfy predetermined assumptions; a transformation unit that, if the predetermined assumptions are satisfied, transforms the first optimization problem into a second optimization problem from which it is possible to obtain an approximate solution for variables that control the weights and occurrence probabilities of each node and the matching strategy; a problem solving unit that solves the second optimization problem to obtain the approximate solution; and an output control unit that outputs the approximate solution. [Effects of the Invention]

[0011] According to one aspect of the present invention, an approximate solution method is provided for an optimization problem, which guarantees a better approximation rate than conventional methods, and further, the remaining capacity r of each node u is u It is possible to provide a technique in which the calculation time does not increase even when ρ takes a large value. [Brief explanation of the drawings]

[0012] [Figure 1] FIG. 1 is a block diagram showing an example of a hardware configuration of an optimization device according to an embodiment. [Figure 2] FIG. 2 is a block diagram showing the software configuration of the optimization device according to the embodiment in relation to the hardware configuration shown in FIG. [Figure 3] FIG. 3 illustrates an example of the problem addressed by the present invention. [Figure 4] FIG. 4 shows an example of a minimum convex cost flow problem. [Figure 5] FIG. 5 is a flowchart showing an example of the operation of the optimization device for calculating an approximate solution or an optimal solution to an optimization problem in online matching. [Figure 6] FIG. 6 is a flowchart illustrating step ST103 in more detail. DETAILED DESCRIPTION OF THE INVENTION

[0013] Hereinafter, embodiments of the present invention will be described with reference to the drawings. Hereinafter, elements that are identical or similar to elements already described will be designated by the same or similar reference numerals, and duplicate descriptions will generally be omitted. For example, when there are multiple identical or similar elements, a common reference numeral may be used to describe each element without distinguishing between them, or a subnumber may be used in addition to the common reference numeral to describe each element with distinction between them.

[0014] [Embodiment] (composition) FIG. 1 is a block diagram showing an example of a hardware configuration of an optimization device 1 according to an embodiment. The optimization device 1 is a computer that analyzes input data and generates and outputs output data. For example, the optimization device 1 is installed at an arbitrary location set by an administrator who manages the optimization device 1.

[0015] As shown in Fig. 1, the optimization device 1 includes a control unit 10, a program storage unit 20, a data storage unit 30, a communication interface 40, and an input / output interface 50. The control unit 10, the program storage unit 20, the data storage unit 30, the communication interface 40, and the input / output interface 50 are communicatively connected to one another via a bus. Furthermore, the communication interface 40 may be communicatively connected to an external device via a network. Furthermore, the input / output interface 50 is communicatively connected to an input device 2 and an output device 3.

[0016] The control unit 10 controls the optimization device 1. The control unit 10 includes a hardware processor such as a central processing unit (CPU). For example, the control unit 10 may be an integrated circuit capable of executing various programs.

[0017] The program storage unit 20 may use, as a storage medium, a combination of nonvolatile memory that can be written to and read from at any time, such as an EPROM (Erasable Programmable Read Only Memory), an HDD (Hard Disk Drive), or an SSD (Solid State Drive), and a nonvolatile memory such as a ROM (Read Only Memory). The program storage unit 20 stores programs necessary for executing various processes. That is, the control unit 10 can realize various controls and operations by reading and executing the programs stored in the program storage unit 20.

[0018] The data storage unit 30 is a storage that uses a combination of nonvolatile memory such as a HDD or memory card, which can be written to and read from at any time, and volatile memory such as RAM (Random Access Memory), as a storage medium. The data storage unit 30 is used to store data acquired and generated in the process of the control unit 10 executing programs and performing various processes.

[0019] The communication interface 40 includes one or more wired or wireless communication modules. For example, the communication interface 40 includes a communication module for wired or wireless connection to an external device via a network. The communication interface 40 may also include a wireless communication module for wireless connection to an external device such as a Wi-Fi access point or base station. Furthermore, the communication interface 40 may also include a wireless communication module for wireless connection to an external device using short-range wireless technology. In other words, the communication interface 40 may be any general communication interface as long as it can communicate with an external device and send and receive various information under the control of the control unit 10.

[0020] The input / output interface 50 is connected to the input device 2, the output device 3, etc. The input / output interface 50 is an interface that enables transmission and reception of information between the input device 2 and the output device 3. The input / output interface 50 may be integrated with the communication interface 40. For example, the optimization device 1 and at least one of the input device 2 and the output device 3 may be wirelessly connected using short-range wireless technology or the like, and information may be transmitted and received using the short-range wireless technology.

[0021] The input device 2 may include, for example, a keyboard, a pointing device, or the like for the user to input various information to the optimization device 1. The input device 2 may also include a reader for reading data to be stored in the program storage unit 20 or the data storage unit 30 from a memory medium such as a USB memory, or a disk device for reading such data from a disk medium.

[0022] The output device 3 includes a display or the like that displays the results of calculations performed by the control unit 10. The output device 3 also includes a printer or the like that prints out the information displayed on the display.

[0023] FIG. 2 is a block diagram showing the software configuration of the optimization device 1 according to the embodiment in relation to the hardware configuration shown in FIG. The control unit 10 includes an acquisition unit 101 , a formulation unit 102 , an optimization unit 103 , and an output control unit 104 .

[0024] The acquisition unit 101 acquires input data. When the input data is input to the input device 2, the input device 2 stores the input data in the parameter storage unit 301. Details of the input data will be described later.

[0025] The formulation unit 102 formulates an optimization problem. The formulation unit 102 acquires input data stored in the parameter storage unit 301. Then, based on the input data, the formulation unit 102 determines a matching strategy that specifies which fixed node is to be assigned to which appearance node, and formulates an optimization problem (P) that maximizes the total reward obtained. Details of the optimization problem (P) will be described later.

[0026] The optimization unit 103 calculates an optimal solution or an approximate solution to the formulated problem, and includes a determination unit 1031, a problem transformation unit 1032, and a problem solution unit 1033.

[0027] The determination unit 1031 determines whether all of the appearance nodes satisfy a predetermined assumption. The predetermined conditions will be described later. If the predetermined assumption is satisfied, the determination unit 1031 outputs the optimization problem (P) to the problem transformation unit 1032. On the other hand, if the predetermined assumption is not satisfied, the determination unit 1031 outputs the optimization problem (P) to the problem solving unit 1033.

[0028] The problem transformation unit 1032 transforms the optimization problem (P) into a minimum convex cost flow problem (FP). The detailed method for transforming the optimization problem (P) into a minimum convex cost flow problem will be described later. The problem transformation unit outputs the transformed minimum convex cost flow problem to the problem solving unit 1033.

[0029] When the problem solving unit 1033 receives a minimum convex cost flow problem (FP), it solves the minimum convex cost flow problem (FP) using an existing solution method to calculate an optimal solution to the minimum convex cost flow problem. This optimal solution corresponds to the approximation rate of the optimization problem. On the other hand, when the problem solving unit 1033 receives an optimization problem (P), it solves the optimization problem (P) using a general method (e.g., a heuristic solution method, an approximation solution method, etc.).

[0030] The output control unit 104 outputs the variables and the matching strategy to the output device 3. For example, the output control unit 104 controls the output device 3 so that the variables and the matching strategy are displayed on a display thereof.

[0031] The data storage unit 30 includes a parameter storage unit 301. The parameter storage unit 301 is used to store the input data acquired by the acquisition unit 101.

[0032] (operation) First, the problem addressed by the present invention will be described.

[0033] FIG. 3 illustrates an example of the problem addressed by the present invention. The example in Figure 3 shows a special online matching problem for a bipartite graph G = (U, V, E). First, let t∈T:={1, 2, ..., t max} is given. Also, for each edge e∈E, a constant (edge ​​weight) w e For each occurrence node v∈V, the function (occurrence probability) p v is given in advance. Furthermore, each fixed node u has a remaining capacity r u is given in advance.

[0034] (1) in Figure 3 shows the initial state. In (1), for each occurrence node v∈V and time t∈T, the variable x vt Determine.

[0035] (2) and (3) in Figure 3 show the situation that is repeated for each time step. In (2), an occurrence node v has an occurrence probability p v (x vt ) or 1-Σ v p v (x vt ) participant v does not appear. In (3), if a certain appearing node v appears, the reward w is obtained by assigning a node u with remaining capacity to the appearing node v. e +x vt After obtaining the remaining capacity r of node u, u Decrement by 1. Or assign nothing to the node. Repeat steps (2) and (3) for as long as time t is t∈T.

[0036] The problem targeted in this embodiment is the variable x in (1) of FIG. vt (v∈V,t∈T) and the matching strategy that specifies which node u should be assigned to the emerging node v in (3), and the problem is to maximize the total reward obtained.

[0037] In this case, this problem can be formulated as the following optimization problem (P):

[0038]

number

[0039] where ξ∈{v1,v2,...,v n ,⊥} T is a random variable, and ξ t =v k is the appearance node (participant) v k appears at time t, and ξ t =⊥ means that no occurrence node has occurred at time t. D(x) is a function of ξ∈{v1,v2,...,v n ,⊥} tmax The probability distribution of x is Pr(ξ|x)=Π t∈T Pr(ξ t |x), where each v∈{v1, v2, ... v n} for Pr(ξ t =v|x)=p v (x vt ) and Pr(ξ t =⊥|x)=1-Σ v∈V p v (x vt ) The variable π represents the matching strategy in (3) of Figure 1, and Π is the set of all strategies. The function f(π,x,ξ) is the total matching reward obtained when (π,x,ξ) is given.

[0040] By solving the optimization problem (P), it is possible to determine the optimal reward x and matching strategy π. Any optimization method can be used as long as it can derive an optimal solution or an approximate solution to the optimization problem (P). For example, the method disclosed in Non-Patent Document 1 can be applied as a solution to the optimization problem (P). However, as mentioned above, the approximation rate is poor and the remaining amount r u As is increased, the calculation time increases significantly.

[0041] Therefore, the approximate solution method described below can be applied when the following assumption 1 is satisfied.

[0042] Assumption 1: For every occurrence node v∈V, the occurrence probability p v (x) is lim x→∞ p v (x)=0. Or the occurrence probability p v The domain of definition includes the variable x for which (x) = 0. Also, -p´ v (x) / p v (x) is monotonically non-decreasing with respect to x, and the occurrence probability p v (x) is bijective and monotonically decreasing. Here, p´ v (x) is p v Represents the derivative of (x).

[0043] This assumption is true, for example, for the occurrence probability p vThis assumption is satisfied when the complementary cumulative distribution functions of the normal distribution and the Gumbel distribution are used as the . These distributions are commonly used in the field of machine learning, and the above assumption satisfies many distributions used in practical applications.

[0044] Next, the function max π∈Π E ξ~D(x) Consider the approximation of [f(π,x,ξ)]. Function max π∈Π E ξ~D(x) Consider a function that approximates [f(π,x,ξ)]. For any x, we use the matching strategy described in Non-Patent Document 2 as H (x). Also, let f^(x) be the optimal value of the following linear programming problem.

[0045]

number

[0046] Here, δ(α) represents the set of edges that connect to node α.

[0047] In this case, the following inequality holds (see, for example, Non-Patent Document 2).

[0048]

number

[0049] Therefore,

[0050]

number

[0051] Then, (x * ,π * ) is an approximate solution that can achieve 1 / (1-√(3+k)) approximation rate for the optimization problem (P). Here, k=min u r u is.

[0052] From the above, the optimization problem (PA)

[0053]

number

[0054] This optimization problem (PA) can be written as follows:

[0055]

number

[0056] By solving the above optimization problem (PA), an approximate solution that achieves an approximation rate of 1 / (1-√(3+k)) for the optimization problem (P) can be obtained. Therefore, in this embodiment, the optimization problem (PA) is solved quickly to obtain an approximate solution.

[0057] When the above assumption 1 is satisfied, the first constraint of the optimization problem (PA), p v x vt ≧Σ e∈δ(v) z et is an optimal solution (x * ,x * ), the equation always holds. That is, p v x vt =Σ e∈δ(v) z et Therefore, in the optimization problem (PA), x vt :=p v -1 (Σ e∈δ(v) z et ) we can consider the following optimization problem (CP):

[0058]

number

[0059] However, Sv is a function p v -1 is the domain of

[0060] The optimal value z for the above problem * About x vt * :=p v -1 (Σ e∈δ(v) z et * ), then (x * ,z * ) is the optimal solution to the optimization problem (PA). Furthermore, it can be shown from assumption 1 that the objective function of the optimization problem (CP) is a convex function. Therefore, this problem can also be solved using various interior point methods for convex optimization problems. However, in this embodiment, as will be explained below, a method will be described that can quickly solve this problem by reducing it to a minimum convex cost flow problem.

[0061] Next, we explain how to solve the optimization problem (CP) through the minimum convex cost flow problem. The solution to the following problem is z * Then, for each t∈T and e∈E, z^ e1 =z^ e2 =···=z^ et max =z e * The resulting z^ is the optimal solution to the optimization problem (CP). This is because the optimization problem (CP) has the same structure at each time t∈T. Therefore, we consider solving the following optimization problem (CP´).

[0062]

number

[0063] At this time, new subscripts s and f are prepared. For all fixed nodes u, z su :=Σ e∈δ(u) z e For all occurrence nodes v, z vf :=Σ e∈δ(v) z e Also, z sfare prepared as slack variables. Then, the optimization problem (CP´) can be rewritten as the minimum convex cost flow problem (FP) as follows:

[0064]

number

[0065] FIG. 4 shows an example of a minimum convex cost flow problem (FP). Here, Figure 4 shows the case where U = {u1, u2} and V = {v1, v2, v3}. As shown in Figure 4, the above-mentioned minimum convex cost flow problem (FP) is to send a flow from node s to node f while filling the capacity of each edge, and to find a flow route that minimizes the total cost for each flow.

[0066] Therefore, the minimum convex cost flow problem (FP) can be efficiently solved using the existing solution method for the minimum convex cost flow problem (FP), such as the capacity scaling method (see, for example, Non-Patent Document 3). By solving the minimum convex cost flow problem (FP) using this solution method, an optimal solution to the minimum convex cost flow problem can be obtained.

[0067] FIG. 5 is a flowchart showing an example of the operation of the optimization device 1 to calculate an approximate solution or an optimal solution to an optimization problem in online matching. The control unit 10 of the optimization device 1 reads out and executes the program stored in the program storage unit 20, thereby realizing the operation of this flowchart.

[0068] This flowchart starts when an administrator (user) of the optimization device 1 inputs input data including various parameters and the like into the input device 2. Note that step ST101, which will be described later, is executed when input data is input from the user, but the optimization device 1 does not have to execute steps ST102 to ST104 immediately. For example, the optimization device 1 may execute these steps at a predetermined time or when further instructions are received from the user.

[0069] In step ST101, the acquisition unit 101 acquires input data. When the input data is input to the input device 2, the input device 2 stores the input data in the parameter storage unit 301. Here, the input data is stored in a fixed node set U={1, 2, ..., u max}, the set of nodes that appear V={1,2,...,v max}, the edge set E, and the residual r given to the fixed node u. u ∀u∈U, edge weight (reward given to each edge when matching) w e ∀e∈E, occurrence probability p of each occurrence node v∈V v The acquisition unit 101 outputs the acquired input data to the formulation unit 102.

[0070] In step ST102, the formulation unit 102 formulates an optimization problem (P). The formulation unit 102 acquires input data stored in the parameter storage unit 301. Then, the formulation unit 102 formulates an optimization problem (P) that maximizes the total amount of rewards obtained, based on the input data. The formulation unit 102 outputs the formulated optimization problem (P) to the optimization unit 103.

[0071] In step ST103, the optimization unit 103 calculates an approximate solution or an optimal solution to the formulated optimization problem (P).

[0072] FIG. 6 is a flowchart illustrating step ST103 in more detail. In step ST201, the determination unit 1031 determines whether all of the appearance nodes v∈V satisfy a predetermined assumption, assumption 1. If it is determined that assumption 1 is satisfied, the process proceeds to step ST202. On the other hand, if it is determined that assumption 1 is not satisfied, the process proceeds to step ST204.

[0073] In step ST202, the problem transformation unit 1032 transforms the optimization problem (P) into the above-mentioned minimum convex cost flow problem (FP). Specifically, the problem transformation unit 1032 transforms the optimization problem (P) into an optimization problem (PA) that can obtain the approximation rate of the optimization problem. At this time, by solving (PA), an approximate solution with an approximation rate of (1-√(k+3)) can be obtained. Furthermore, the problem transformation unit 1032 transforms the optimization problem (PA) into an optimization problem (CP) whose objective function is a convex function according to Assumption 1. Then, the problem transformation unit 1032 transforms the optimization problem (CP) into a minimum convex cost flow problem (FP) based on the fact that the optimization problem (CP) has the same structure at each time point. Then, the problem transformation unit 1032 outputs the minimum convex cost flow problem (FP) to the problem solving unit 1033. Furthermore, the problem transformation unit 1032 may output the optimization problem (PA) and the optimization problem (CP) to the problem solving unit 1033.

[0074] In step ST203, the problem solving unit 1033 solves the minimum convex cost flow problem (FP) using an existing solution method (for example, the Capacity scaling method) for the minimum convex cost flow problem to calculate an optimal solution to the minimum convex cost flow problem. The calculated optimal solution, i.e., the variables that control the weights and appearance probabilities of each node of the optimization problem (P) and the matching strategy (x * ,π * ) is calculated. Here, the problem solving unit 1033 may solve the optimization problem (PA) or the optimization problem (CP). The problem solving unit 1033 calculates the calculated variables and the matching strategy (x * ,π * ) to the output control section 104. That is, the process proceeds to step ST104.

[0075] In step ST204, the problem solving unit 1033 solves the optimization problem (P). On the other hand, if assumption 1 is not satisfied in step ST201, the optimization problem (P) cannot be transformed into a minimum convex cost flow problem (FP). In this case, the determination unit 1031 outputs the formulated optimization problem (P) to the problem solving unit 1033. Then, the problem solving unit 1033 solves the optimization problem (P) using a general method (for example, a heuristic solution method, an approximate solution method, etc.). Then, the problem solving unit 1033 outputs the solved solution to the output control unit 104. That is, the process proceeds to step ST104.

[0076] In step ST104, the output control unit 104 calculates the variable x * and matching strategy π * to the output device 3. The output control unit 104 outputs the variable x * and matching strategy π * is controlled to be displayed on the display of the output device 3.

[0077] For example, in applications such as crowdsourcing, k in the approximation rate 1 / (1-√(3+k)) often becomes large. In this case, it is possible to achieve a high approximation rate. For example, in annotation tasks, the amount of each task is often 100 or more. Therefore, k≧100. In this case, the approximation rate becomes 1-1 / (√(3+k))>1-1 / √103>0.9, which is much better than the conventional approximation rate of 1 / 2. In addition, the above solution method uses the remaining amount r of each node u to u Even if takes a large value, the calculation time does not increase significantly.

[0078] (Effects of the embodiment) According to this embodiment, the optimization device 1 can provide an approximate solution to an optimization problem that guarantees a better approximation rate than conventional solutions, provided that certain assumptions are met. u It is possible to provide a technique in which the calculation time does not increase even when ρ takes a large value.

[0079] [Other embodiments] In the above embodiment, an example has been described in which the optimization problem (P) is transformed into the minimum convex cost flow problem (FP) on the condition that Assumption 1 is satisfied. However, Assumption 1 may be any assumption as long as it allows the optimization problem (P) to be transformed into the minimum convex cost flow problem (FP).

[0080] The solution to the optimization problem described in this embodiment is a solution to a general optimization problem, and therefore can be applied not only to the above-mentioned problems but also to various problems that can be reduced to a formulated optimization problem.

[0081] The techniques described in the above embodiments can be stored as a program (software means) that can be executed by a computer on a storage medium such as a magnetic disk (e.g., a floppy disk, a hard disk, etc.), an optical disk (e.g., a CD-ROM, a DVD, an MO, etc.), or a semiconductor memory (e.g., a ROM, a RAM, a flash memory, etc.), and can also be distributed by transmitting the program via a communication medium. The program stored on the medium also includes a configuration program that configures the software means (including not only executable programs but also tables and data structures) that the computer executes. The computer that implements this device loads the program stored on the storage medium and, in some cases, configures the software means using the configuration program, and executes the above-described processing by controlling the operation of the software means. The term "storage medium" as used herein is not limited to storage media for distribution, but also includes storage media such as magnetic disks and semiconductor memories installed inside the computer or in devices connected via a network.

[0082] In short, this invention is not limited to the above-described embodiments, and various modifications can be made in the implementation stage without departing from the spirit of the invention. Furthermore, the embodiments may be implemented in combination as appropriate as possible, and in such cases, the combined effects can be obtained. Furthermore, the above-described embodiments include inventions at various stages, and various inventions can be extracted by appropriately combining the disclosed multiple constituent elements. [Explanation of symbols]

[0083] 1...Optimization device 2...Input device 3...Output device 10...Control unit 101…Acquisition Department 102…Formulation part 103...Optimization section 1031...Judgment section 1032...Problem transformation section 1033…Problem solving department 104...Output control unit 20...Program memory section 30...Data storage unit 301...parameter storage unit 40...Communication interface 50...Input / output interface

Claims

1. An optimization device capable of solving an online matching problem in which each node and its occurrence probability can be controlled, an acquisition unit that acquires input data including information about the nodes, a remaining amount assigned to a fixed node among the nodes, the occurrence probability assigned to an occurrence node among the nodes, and a reward assigned to each edge when matching is performed; a formulation unit that formulates the problem as a first optimization problem that maximizes the total reward obtained based on the input data; a determination unit that determines whether all of the appearance nodes satisfy a predetermined assumption; a transformation unit that transforms the first optimization problem into a second optimization problem that can obtain an approximate solution for a matching strategy and a variable that controls the weight of each node and the occurrence probability of the first optimization problem when the predetermined assumption is satisfied; a problem solver that solves the second optimization problem to obtain the approximate solution; an output control unit that outputs the approximate solution; An optimization device comprising:

2. the transformation unit transforms the second optimization problem into a third optimization problem in which the objective function is a convex function under Assumption 1; The optimization device according to claim 1 , wherein the problem solving unit obtains the approximate solution by solving the third optimization problem.

3. the transformation unit transforms the second optimization problem into a third optimization problem in which the objective function is a convex function under Assumption 1, and transforms the third optimization problem into a minimum convex cost flow problem based on the fact that the third optimization problem has the same structure at each time; The optimization device according to claim 1 , wherein the problem solver obtains the approximate solution by solving the minimum convex cost flow problem.

4. The predetermined assumption is that for all occurrence nodes v∈ a set V of occurrence nodes that may appear in the future, the occurrence probability p ｖ (x) is lim ｘ→∞ p ｖ (x)=0, or the variable is p ｖ The domain of the variable x where (x)=0 is included, and -p' ｖ（ｘ） / p ｖ (x) is monotonically non-decreasing with respect to the variable x, and p ｖ 2. The optimization apparatus according to claim 1, wherein (x) is assumed to be bijective and monotonically decreasing.

5. The variables and the matching strategy are 1 / (1-√(3+k)) approximation rates of the first optimization problem, where k=min ｕ r ｕ where u is the fixed node and r ｕ The optimization device according to claim 1 , wherein ∑ i = ∑ j ...

6. An optimization method executed by a processor of an optimization device capable of solving an online matching problem in which each node and its occurrence probability can be controlled, comprising: Obtaining input data including information about the nodes, a remaining amount to be assigned to a fixed node among the nodes, the occurrence probability to be assigned to an occurrence node among the nodes, and a reward to be assigned to each edge when matching is performed; Formulating a first optimization problem that maximizes a sum of rewards based on the input data; determining whether all of the occurrence nodes satisfy a predetermined assumption; When the predetermined assumption is satisfied, transforming the first optimization problem into a second optimization problem for which an approximate solution can be obtained for variables that control the weights of each node and the occurrence probability of the first optimization problem, and a matching strategy; solving the second optimization problem to obtain the approximate solution; outputting the approximate solution; An optimization method comprising:

7. 1. An optimization program comprising instructions to be executed by a processor of an optimization device capable of solving an online matching problem in which each node and its occurrence probability can be controlled, the instructions comprising: Obtaining input data including information about the nodes, a remaining amount to be assigned to a fixed node among the nodes, the occurrence probability to be assigned to an occurrence node among the nodes, and a reward to be assigned to each edge when matching is performed; Formulating a first optimization problem that maximizes a sum of rewards based on the input data; determining whether all of the occurrence nodes satisfy a predetermined assumption; When the predetermined assumption is satisfied, transforming the first optimization problem into a second optimization problem for which an approximate solution can be obtained for variables that control the weights of each node and the occurrence probability of the first optimization problem, and a matching strategy; solving the second optimization problem to obtain the approximate solution; outputting the approximate solution; An optimization program comprising:

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