JP2019153011A - Knee point selection program, knee point selection method, and knee point selection apparatus - Google Patents

Abstract

To select robust knee points from samples including noise.SOLUTION: A knee point selection apparatus 100 receives a plurality of non-inferior solutions as for an optimization problem using a plurality of objective functions. The knee point selection apparatus 100 repeatedly performs a process of assigning height information based on a specific direction set on a multidimensional space having a plurality of objective functions as coordinate axes for each of a plurality of non-inferior solutions while changing weights assigned to the non-inferior solutions, thereby calculating the number of times at which the height information is minimized for each non-inferior solution. The knee point selection apparatus 100 selects a non-inferior solution corresponding to a knee point based on the calculated number of times.SELECTED DRAWING: Figure 9

Description

  The present invention relates to a knee point selection program and the like.

  In the multi-objective optimization problem, each objective function is in a trade-off relationship, and the optimal solution cannot be determined as one, so a non-inferior solution set of each objective function is obtained. A non-inferior solution set is a non-inferior solution set to which superiority or inferiority cannot be assigned, and can be said to be a state in which other objective function values are corrupted when one of the objective function values is improved. The surface formed by the non-inferior solution set is called a Pareto front.

FIG. 17 is a diagram illustrating an example of a Pareto front. 17, the horizontal axis is an axis corresponding to the objective function value of the objective function f _1, increases toward the left, becomes the objective function values for the objective function f ₁ is better. The vertical axis is an axis corresponding to the objective function value of the objective function f _2, increases toward the lower side, becomes the objective function values for the objective function f ₂ is better.

The decision maker selects one final non-inferior solution from the non-inferior sets included in the Pareto front. A preferable non-inferior solution is a knee point P with a well-balanced trade-off ratio. In the vicinity of the knee point P, there is a characteristic that if the objective function f ₁ is slightly improved, the objective function f ₂ is greatly degraded, and if the objective function f ₂ is improved, the objective function f ₁ is largely degraded.

  For example, as a conventional technique for detecting a Pareto front knee point, there is a technique using a normal boundary crossing method. In this prior art, the point farthest in the dominant normal direction from the boundary line C connecting the end points A and B of the Pareto front is detected as a knee point. In the example shown in FIG. 17, P is detected as a knee point.

I. Das. On characterizing the “knee” of the Pareto curve based on normal-boundary intersection. Structural Optimization, 18 (2): 107-115, 1999.

  However, the above-described conventional technique has a problem that a robust knee point cannot be selected from a sample including noise.

  In one aspect, an object of the present invention is to provide a knee point selection program, a knee point selection method, and a knee point selection device capable of selecting a robust knee point from a sample including noise.

  In the first plan, the computer executes the following processing. The computer receives a plurality of non-inferior solution inputs for an optimization problem using a plurality of objective functions. The computer repeatedly executes a process for assigning height information based on a specific direction set in a multidimensional space having a plurality of objective functions as coordinate axes for each of the non-inferior solutions while changing the weight for the non-inferior solution. By doing so, the number of times the height information is minimized for each non-inferior solution is calculated. The computer selects a non-inferior solution that becomes a knee point based on the calculated number of times.

  Robust knee points can be selected from samples containing noise.

FIG. 1 is a diagram illustrating an example of a data structure of non-inferiority set information. FIG. 2 is a diagram for explaining the normalization process. FIG. 3 is a diagram illustrating an example of a data structure of undirected graph information. FIG. 4 is a diagram illustrating an example of a data structure of the directed graph information. FIG. 5 is a diagram (1) for explaining a process of detecting a knee point by the information processing apparatus. FIG. 6 is a diagram (2) for explaining a process of detecting a knee point by the information processing apparatus. FIG. 7 is a diagram for explaining a problem of the reference technique. FIG. 8 is a diagram for explaining processing of the knee point selection device according to the present embodiment. FIG. 9 is a functional block diagram illustrating the configuration of the knee point selection device according to the present embodiment. FIG. 10 is a diagram illustrating an example of the data structure of the non-inferiority set information according to the present embodiment. FIG. 11 is a diagram illustrating an example of the data structure of the weight set information according to the present embodiment. FIG. 12 is a diagram illustrating an example of a data structure of the minimum frequency table according to the present embodiment. FIG. 13 is a diagram illustrating an example of update processing of the local minimum number table. FIG. 14 is a diagram for explaining the processing of the selection unit according to the present embodiment. FIG. 15 is a flowchart illustrating the processing procedure of the knee point selection device according to the present embodiment. FIG. 16 is a diagram illustrating an example of a hardware configuration of a computer that realizes the same function as the knee point selection device. FIG. 17 is a diagram illustrating an example of a Pareto front.

  Embodiments of a knee point selection program, a knee point selection method, and a knee point selection device disclosed in the present application will be described below in detail with reference to the drawings. Note that the present invention is not limited to the embodiments.

  Before describing the embodiments of the present invention, reference techniques devised by the inventor will be described. This reference technique is not a prior art. When receiving the non-inferiority set information, the information processing apparatus according to the reference technique performs processing to normalize the non-inferiority set information.

  Here, the non-inferior solution set information is non-inferior solution information for a plurality of objective functions in the multi-objective optimization problem.

FIG. 1 is a diagram illustrating an example of a data structure of non-inferiority set information. For example, as shown in FIG. 1, non-inferior solutions for N objective functions are shown as solutions on an N-dimensional space, and the non-inferior solution set information 20 includes a non-inferior solution ID, f ₁ , f ₂ , It associates the f _3. The non-inferior solution ID is information that uniquely identifies the non-inferior solution. f ₁ , f ₂ , and f ₃ are objective function values corresponding to the respective objective functions f ₁ , f ₂ , and f ₃ . For example, the non-inferior solution corresponding to the non-inferior solution ID “1” is “f ₁ = 1.4, f ₂ = −2.4, f ₃ = 7.7”. In the example shown in FIG. 1, f _{1 to} f ₃ are shown, but the non-inferior solution set information may include other objective function values.

A process in which the information processing apparatus according to the reference technology normalizes the non-inferiority set information will be described. FIG. 2 is a diagram for explaining the normalization process. The information processing apparatus reads the non-inferiority set information 20 and searches for the maximum value and the minimum value from the values in each column. For example, in the column f ₁ , the maximum value is “4.4” and the minimum value is “1.4”. In the column f ₂ , the maximum value is “2.6” and the minimum value is “−2.4”. In the column f ₃ , the maximum value is “12.6” and the minimum value is “5.6”.

  The information processing apparatus updates each objective function value by subtracting the minimum value from the value of each column of the non-inferiority set information 20 and dividing the result by (maximum value−minimum value), thereby intermediate data 20a. Is generated.

For example, a case where the objective function value f ₁ “1.4” of the non-inferior solution ID “1” of the non-inferior solution set information 20 is updated will be described. (1.4-1.4) / (4.4- 1.4) = 0. For this reason, the objective function f ₁ of the non-inferiority ID “1” of the intermediate data 20a is “0”. The case of updating the objective function f ₁ “2.6” of the non-inferior solution ID “2” of the non-inferior solution set information 20 will be described. (2.6-1.4) / (4.4-1.4 ) = 0.4. For this reason, the objective function f1 of the non-inferiority ID “2” of the intermediate data 20a is “0.4”. The information processing apparatus updates other objective function values in the same manner.

The information processing apparatus multiplies the value of the intermediate data 20a by a weight corresponding to the type of the objective function, thereby updating each value and generating normalized information 20b. For example, the information processing apparatus multiplies the objective function value of the objective function f ₁ by w ₁ , multiplies the objective function value of the objective function f ₂ by w _2, and multiplies the objective function value of the objective function f ₃ by w ₃ . To do. The weights w ₁ , w ₂ , and w ₃ are set in advance. For example, w ₁ = “1”, w ₂ = “2”, and w ₃ = “3”.

  The information processing device gives “height information” to the non-inferior solution placed in the multi-dimensional space, and the value becomes larger as the non-inferior solution located in the non-dominant direction in the multi-dimensional space. From each non-inferior solution to which information is assigned, a non-inferior solution that is a minimum point is detected as a knee point.

For example, the information processing apparatus, height information for each non-dominated solution ID, and calculated based on the height function h _w. The height function h _w is expressed by the equation (1). The height information given to the non-inferiority ID calculated by the equation (1) corresponds to the sum in the row direction of the normalized information 20b.

  The information processing apparatus performs clustering for classifying non-inferior solutions with a short distance (Eugrid distance) into the same cluster for each non-inferior solution belonging to the same height information. For example, the information processing apparatus classifies each non-inferior solution into clusters by executing clustering using Mapper. For example, the information processing apparatus uses the clustering by Mapper as `` G. Singh, F. Memoli, and G. Carlsson.Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition.In Proceedings of the Symposium on Point Based Graphics, The technology described in “Prague, Czech Republic, 2007” is used.

  The information processing apparatus generates undirected graph information based on the result of clustering by Mapper. Undirected graph information is information in which clusters clustered based on Mapper are connected by overlapping clusters and edges. Here, that two clusters overlap means that there is a non-inferior solution that belongs to both of the two clusters in the non-inferior solution set.

  FIG. 3 is a diagram illustrating an example of a data structure of undirected graph information. In the example shown in FIG. 3, in the undirected graph information 25a, height information is given to each cluster, and overlapping clusters are connected by edges. The height information of each cluster is given based on the height information of each non-inferior solution clustered based on Mapper.

  That is, for non-inferior solutions arranged in a multi-dimensional space, height information whose value increases as the non-inferior solution located in the non-dominant direction in the multi-dimensional space is given, and each cluster c1 to c7 is given. Is given height information based on non-poor height information constituting the cluster. For example, with respect to clusters existing in a multidimensional space, the value of height information increases as the cluster is located in the non-dominant direction. In the example shown in FIG. 3, the height information 1, 3, 2, 1, 2, 3, 2 is given to the clusters c1, c2, c3, c4, c5, c6, c7.

  Further, the cluster c2 and the cluster c3 are connected by the side h1. Cluster c3 and cluster c4 are connected by side h2. Cluster c4 and cluster c5 are connected by side h3. Cluster c5 and cluster c6 are connected by side h4. Cluster c6 and cluster c7 are connected by side h5.

  The information processing apparatus converts the undirected graph information 25a into the directed graph information 25b based on the height information of each cluster. The directed graph is information in which each side indicated in the undirected graph information 25a described in FIG. For example, when the height information of the first cluster is larger than the height information of the second cluster, the information processing apparatus sets the direction of the side connecting the first cluster and the second cluster to the first cluster. The direction is set from the second cluster toward the first cluster. When the height information of the second cluster is larger than the height information of the first cluster, the information processing apparatus sets the direction of the side connecting the first cluster and the second cluster to the first information The direction is set from the cluster toward the second cluster.

  FIG. 4 is a diagram illustrating an example of a data structure of the directed graph information. As shown in FIG. 4, in addition to the information described in FIG. 3, direction information is given to each side h1 to h5. The side h2 is a directed side from the cluster c4 toward the cluster c3. The side h1 is a directed side from the cluster c3 toward the cluster c2. The side h3 is a directed side from the cluster c4 toward the cluster c5. The side h4 is a directed side from the cluster c5 toward the cluster c6. The side h5 is a directed side from the cluster c7 toward the cluster c6.

  The information processing apparatus detects a cluster including a non-inferior solution corresponding to the knee point based on the directed graph information 25b. FIG. 5 is a diagram (1) for explaining a process of detecting a knee point by the information processing apparatus. The information processing apparatus detects a minimal cluster in which the input order = 0 and the output order ≧ 1. In the example shown in FIG. 5, the clusters c4 and c7 are minimal clusters in which the input order = 0 and the output order ≧ 1. A minimal cluster is one in which no other cluster lower than the minimal cluster exists around the minimal cluster. The information processing apparatus detects the non-inferiority non-inferiority ID included in the minimal cluster c4 and the non-inferiority non-inferiority ID included in the minimal cluster c7 as knee points.

  For example, the non-inferiority non-inferiority ID included in the minimal cluster c4 is “1, 3”, and the non-inferiority non-inferiority ID included in the minimal cluster c7 is “2, 5”. In this case, the information processing apparatus detects the non-inferiority ID “1, 2, 3, 5” as a knee point.

  By the way, even if the information processing apparatus is a non-inferior solution that belongs to a minimal cluster in which the input order = 0 and the output order ≧ 1, if the following condition is further satisfied, the corresponding non-inferior solution is Suppressing detection as a knee point. Specifically, when the non-inferior solution that belongs to the minimal cluster is also included in other maximal clusters, the information processing apparatus suppresses detection of the corresponding non-inferior solution as a knee point. The condition for the cluster to be a maximum cluster is a cluster in which the input order = 1 and the output order = 0.

FIG. 6 is a diagram (2) for explaining a process of detecting a knee point by the information processing apparatus. In FIG. 6, a graph 26 is a graph in which each non-inferior solution is plotted in two dimensions. For example, the horizontal axis is an axis corresponding to the objective function value of the objective function f _1, increases toward the left, the objective function value is better regarding the objective function f _1. The vertical axis is an axis corresponding to the objective function value of the objective function f _2, increases toward the lower side, becomes the objective function values for the objective function f ₂ is better. In the graph 26, the direction from the upper right to the lower left is the dominant direction. For this reason, the height of each non-inferior solution of the graph 26 increases as it is located at the upper right of the graph 26.

  The information processing apparatus performs clustering by Mapper for each non-inferior solution shown in the graph 26 of FIG. 6, thereby generating the directed graph information 25c. For example, in the graph 26, each non-inferior solution included in the regions 26a to 26h is classified into the minimum clusters 30a to 30h, respectively.

  In section division based on the height of the non-inferior solution set by Mapper, adjacent sections are overlapped and divided, and non-inferior solutions belonging to each section are clustered to generate undirected graph information 25a. To do. For this reason, since the non-inferior solution located in the area where the sections overlap each other belongs to one cluster in each section, it belongs to two or more clusters throughout the entire section. Therefore, even when only a single non-inferior solution exists as in the region 26e, if the non-inferior solution is at a position where the sections overlap, the corresponding non-inferior solution is minimized in the generated undirected graph 25a. It belongs not only to the cluster 30e, but also to the next highest cluster 41a, and further to the maximum cluster 42a. Similarly, the non-inferior solution of the region 26f belongs not only to the minimum cluster 30f but also to the cluster 41b and the maximum cluster 42b.

  In FIG. 6, minimum clusters 30a to 30d, 30g, and 30h are minimum clusters to be detected. However, the non-inferior solutions belonging to the minimal clusters 30e and 30f are isolated points included in the regions 26e and 26f and are not suitable as knee points. For this reason, the information processing apparatus determines whether the non-inferior solution belonging to the minimum cluster also belongs to another maximum cluster. The information processing apparatus prevents the non-inferior solution included in the areas 26e and 26f from being detected by suppressing the non-inferior solution satisfying such a condition from being detected as a knee point.

  As described above, in the reference technique, a non-inferior solution included in a minimal cluster is detected as a knee point. In the following description, the non-inferior solution included in the minimal cluster is described as “non-inferior solution corresponding to the minimal point” as appropriate.

  Here, the problems of the reference technology will be described. There is a problem that the knee point detected by the reference technique is likely to change due to noise included in the non-inferior solution set information.

FIG. 7 is a diagram for explaining a problem of the reference technique. In FIG. 7, a graph 50a is a graph in which an ideal non-inferior solution set that does not include noise is plotted on two dimensions. Graphs 50b and 50c are graphs in which a non-inferior solution set including noise is plotted two-dimensionally. In the graph 50 a to 50 c, the horizontal axis is an axis corresponding to the objective function value of the objective function _{f 1.} The vertical axis is an axis corresponding to the objective function value of the objective function f _2.

  As shown in FIG. 7, when the non-inferior solution set includes noise, the scale of the coordinate axis changes. When the scale of the coordinate axis changes, the height function changes, and when the height function changes, the detected knee point changes. The knee point detected by the reference technique includes a non-inferior solution that is not a knee point with a slight change in the height function. For example, even if non-inferior solutions 10A to 10F are detected as knee points in the graph 50a, some non-inferior solutions may not be detected as knee points in the graphs 50b and 50c. In addition, as shown in Formula (1), the height function value of the height function changes as w changes. Changing the scale of the objective function f is equivalent to changing w. That is, it is required to select a robust knee point by removing an unstable non-inferior solution that is not a knee point only by a slight change in the height function.

  Next, processing of the knee point selection device according to the present embodiment will be described. The knee point selection device generates a plurality of weight values used in the height function of Equation (1), and performs a process of specifying a non-inferior solution corresponding to the minimum point using a plurality of height functions having different weights. For each non-inferior solution, count the number of times that the minimum score was reached. The knee point selection device selects the upper non-inferior solution with the most frequent minimum points as the knee point.

FIG. 8 is a diagram for explaining processing of the knee point selection device according to the present embodiment. Graphs 55a, 55b, and 55c shown in FIG. 8 are graphs in which height information is added to a non-inferior solution set based on a height function. For example, the horizontal axis of each graph 55a~55c is an axis corresponding to the objective function value of the objective function _{f 1.} The vertical axis of each graph 55a~55c is an axis corresponding to the objective function value of the objective function _{f 2.}

Here, the weights of the height functions when generating the graphs 55a to 55c are different. For example, the height function used when generating the graph 55 a is “h ₁ = 0.2 × f ₁ + 0.8 × f ₂ ”. The height function used when generating the graph 55b is assumed to be “h ₂ = 0.5 × f ₁ + 0.5 × f ₂ ”. The height function used in generating the graph 55c is assumed to be “h ₃ = 0.8 × f ₁ + 0.2 × f ₂ ”.

  For example, in the graph 55a, the non-inferiority ID of the non-inferior answer (knee point) corresponding to the minimum point is “10B, 10C, 10D, 10F”. In the graph 55b, the non-inferiority non-inferiority ID corresponding to the minimum point is “10C, 10D, 10F”. In the graph 55c, the non-inferiority non-inferiority ID corresponding to the minimum point is “10A, 10C, 10D, 10F”. Then, the number of times selected as the minimum point is “one time” for the non-inferior answer of the non-inferior answer ID “10A”, “one time” for the non-inferior answer of the non-inferior answer ID “10B”, and the non-inferior answer ID “ The non-inferior answer of “10C” is “3 times”. Further, the non-inferior answer of the non-inferior answer ID “10D” is “3 times”, the non-inferior answer of the non-inferior answer ID “10E” is “twice”, and the non-inferior answer of the non-inferior answer ID “10F” is “2”. Times ".

  The knee point selecting device selects a top N non-inferior solution as a knee point from those having a large number of times selected as a minimum point. For example, if N = 4, the knee point selection device selects a non-inferior solution with a non-inferior solution ID “10C, 10D, 10E, 10F” as a knee point.

  In this way, the knee point selection device performs a process of specifying a non-inferior solution corresponding to a minimum point using a plurality of height functions having different weights, and each non-inferior solution has a large number of times that the minimum point is obtained. As the knee point. A non-inferior solution with a large number of minimum points is a robust knee point because it is often a knee point even if the height function changes slightly. For this reason, according to the knee point selection device according to the present embodiment, it is possible to select a robust knee point by eliminating an unstable non-inferior solution that is not a knee point only by a slight change in the height function. .

  Next, the configuration of the knee point selection device 100 according to the present embodiment will be described. FIG. 9 is a functional block diagram illustrating the configuration of the knee point selection device according to the present embodiment. As illustrated in FIG. 9, the knee point selection device 100 includes a communication unit 110, an input unit 120, a display unit 130, a storage unit 140, and a control unit 150.

  The communication unit 110 is a processing unit that connects to an external device (not shown) via a network and executes data communication with the external device. The communication unit 110 corresponds to a communication device such as a NIC (Network Interface Card).

  The input unit 120 is an input device for inputting various types of information to the knee point selection device 100. The input unit 120 corresponds to a keyboard, a mouse, a touch panel, or the like. For example, the user operates the input unit 120 to input non-inferiority set information 141 and the like described later.

  The display unit 130 is a display device that displays information output from the control unit 150. The display unit 130 corresponds to a liquid crystal display, a touch panel, or the like.

  The storage unit 140 includes non-inferiority set information 141, weight set information 142, mapper parameters 143, undirected graph information 144, directed graph information 145, and minimum count table 146. The storage unit 140 corresponds to a semiconductor memory element such as a random access memory (RAM), a read only memory (ROM), and a flash memory, and a storage device such as a hard disk drive (HDD).

  The non-inferior solution set information 141 is non-inferior solution information for a plurality of objective functions in the multi-objective optimization problem.

FIG. 10 is a diagram illustrating an example of the data structure of the non-inferiority set information according to the present embodiment. For example, the non-inferior solutions for N objective functions are shown as solutions in an N-dimensional space as shown in FIG. 10, and the non-inferior solution set information 141 includes a non-inferior solution ID, f ₁ , f _2, and so on. Associate. The non-inferior solution ID is information that uniquely identifies the non-inferior solution. f ₁ and f ₂ are objective function values corresponding to the objective functions f ₁ and f ₂ , respectively. In the example shown in FIG. 10, f ₁ and f ₂ are shown, but the non-inferiority set information 141 may include other objective function values.

  The weight set information 142 holds weight information given to each objective function when calculating height information. The weight set information 142 includes a plurality of types of weight sets.

FIG. 11 is a diagram illustrating an example of the data structure of the weight set information according to the present embodiment. As shown in FIG. 11, the weight set information 142 associates weight set numbers with weights (w ₁ , w ₂ ). The weight set number is a number for identifying a set of weights w ₁ and w ₂ given to the height function. For example, when the weight set number “W1001” is selected, w ₁ = “7” and w ₂ = “2” are given to the height function shown in Expression (1). FIG. 11 shows a set of weights w ₁ and w ₂ as an example, but other weights (for example, w ₃ , w ₄ ,...) May be included.

  The Mapper parameter 143 includes parameter information used when clustering non-inferior solutions based on Mapper. For example, the mapper parameter 143 includes the number of mapper sections, the overlap rate of mapper sections, and the number of bins in mapper clustering.

  The undirected graph information 144 is information in which clusters clustered based on Mapper are connected by overlapping clusters with edges. For example, the data structure of the undirected graph information 144 corresponds to the data structure of the undirected graph information 25a described with reference to FIG.

  The directed graph information 145 is information in which each side indicated by the undirected graph information 144 has direction information. For example, the direction of the side connecting the first cluster and the second cluster is such that the height information of the first cluster is greater than the height information of the second cluster. The direction is toward one cluster. The direction of the side connecting the first cluster and the second cluster is such that when the height information of the second cluster is larger than the height information of the first cluster, the direction from the first cluster to the second cluster The direction is toward the cluster. For example, the data structure of the directed graph information 145 corresponds to the data structure of the directed graph information 25b described with reference to FIG.

  The minimum frequency table 146 is a table that holds the number of times of non-inferior solutions corresponding to the minimum points for each non-inferior solution. FIG. 12 is a diagram illustrating an example of a data structure of the minimum frequency table according to the present embodiment. As shown in FIG. 12, this minimum frequency table 146 associates a non-inferior solution ID with a minimum frequency. The non-inferior solution ID is information that uniquely identifies the non-inferior solution. The minimum number of times indicates the number of times of non-inferiority corresponding to the minimum point. The initial value of each minimum number is zero. The process using the minimum number table 146 will be described later.

  Returning to the description of FIG. The control unit 150 includes a reception unit 151, a weight generation unit 152, a calculation unit 153, a selection unit 154, and an output unit 155. The control unit 150 can be realized by a CPU (Central Processing Unit), an MPU (Micro Processing Unit), or the like. It can also be realized by hard wired logic such as ASIC (Application Specific Integrated Circuit) or FPGA (Field Programmable Gate Array).

  The receiving unit 151 is a processing unit that receives various types of information from the communication unit 110 or the input unit 120 and stores the received information in the storage unit 140. For example, the reception unit 151 receives the non-inferiority set information 141 and the mapper parameter 143 and stores them in the storage unit 140. In addition, when receiving the weight generation condition information, the reception unit 151 outputs the weight generation condition information to the weight generation unit 152. The weight generation condition information includes conditions (grid points, distribution, etc.) for generating the weight set information 142.

  The weight generation unit 152 is a processing unit that generates the weight set information 142 based on the weight generation condition information. The weight generation unit 152 stores the weight set information 142 in the storage unit 140. For example, the weight generation unit 152 selects weights in a grid pattern from a range Δ that can be weighted. The range Δ is defined as in equation (2). Alternatively, the weight generation unit 152 may select weights with a uniform distribution over the range Δ, or may select weights with a normal distribution around a specified value on the range Δ.

  In this embodiment, as an example, the case where the weight generation unit 152 generates the weight set information 142 has been described. However, the knee point selection device 100 may receive the weight set information 142 from the communication unit 110 or the input unit 120. Good.

  The calculation unit 153 selects one weight set from among the plurality of weight sets in the weight set information 142 and calculates height information for each non-inferior solution included in the non-inferior solution set information 141. The calculation unit 153 identifies the non-inferior solution corresponding to the minimum point based on the height information of each non-inferior solution. The calculation unit 153 refers to the minimum number table 146 and adds 1 to the minimum number corresponding to the non-inferiority non-inferiority ID that is the minimum point. The calculation unit 153 repeatedly executes the above processing while changing each weight set included in the weight set information 142.

An example of processing in which the calculation unit 153 calculates the non-inferior solution height information included in the non-inferior solution set information 141 will be described. The calculation unit 153 selects one weight set from the weight set information 142, sets the weights w ₁ and w ₂ of the selected weight set to the height function shown in Expression (1), and sets the height function to Based on this, non-inferiority height information is calculated. The processing for calculating non-inferiority height information is the same as that of the information processing apparatus of the reference technique described above, except that one weight set is selected from the weight set information 142 and used for the height function.

  Next, an example of a process in which the calculation unit 153 identifies the non-inferior solution corresponding to the minimum point based on the non-inferior solution height information will be described. The calculation unit 153 performs clustering for classifying non-inferior solutions with a short distance (Eugrid distance) into the same cluster for each non-inferior solution belonging to the same height information. The calculation unit 153 classifies each non-inferior solution into clusters by executing mapper clustering using the mapper parameter 143.

  The calculation unit 153 generates undirected graph information 144 based on the result of clustering by Mapper. Also, the calculation unit 153 converts the undirected graph information 144 into the directed graph information 145 based on the height information of each cluster. The process in which the calculation unit 153 generates the undirected graph information 144 and the directed graph information 145 is the same as the information processing apparatus of the reference technique described above.

  Based on the directed graph information 145, the calculation unit 153 detects a cluster (minimal cluster) including a non-inferior solution corresponding to the knee point. The calculation unit 153 detects a non-inferior solution included in the minimal cluster as a knee point (a non-inferior solution corresponding to the minimal point). The process of calculating the knee point by the calculation unit 153 is the same as that of the information processing apparatus of the reference technique described above.

  When the calculation unit 153 detects the non-inferior solution corresponding to the minimum point by executing the above processing, the calculation unit 153 updates the minimum number table 146. FIG. 13 is a diagram illustrating an example of update processing of the local minimum number table. The minimum count table 146a is the minimum count table 146 before update. For example, the non-inferiority non-inferiority ID detected as a knee point by the above processing is set to “10A, 10C, 10D”. In this case, the calculation unit 153 generates the updated minimum count table 146b by incrementing the non-inferiority ID by 1 to the minimum count corresponding to “10A, 10C, 10D”.

  The selection unit 154 is a processing unit that selects the top N non-inferior solutions having a large number of minimums based on the minimum number table 146. N is a predetermined natural number and is set in advance by the user. The non-inferior solution selected by the selection unit 154 is a knee point. The selection unit 154 outputs information on the knee point corresponding to the selected minimum point to the output unit 155.

  FIG. 14 is a diagram for explaining the processing of the selection unit according to the present embodiment. For example, the minimum frequency table 146 obtained as a result of the calculation unit 153 repeatedly executing the above processing while changing each weight set included in the weight set information 142 is referred to as a minimum frequency table 146b. When the selection unit 154 sorts the records included in the minimum count table 146b in descending order of the minimum count, as shown in the minimum count table 146c, the order of the non-inferiority IDs “10D, 10B, 10C, 10A, 10E” Become. When N is set to 3, the selection unit 154 selects the non-inferior solution ID “10D, 10B, 10C” as the non-inferior solution (non-inferior solution ID) corresponding to the minimum point.

  The output unit 155 is a processing unit that causes the display unit 130 to display information on the knee point selected by the selection unit 154. Further, the output unit 155 may notify the detection result to an external device via a network. For example, the output unit 155 acquires the record of the non-inferior solution ID selected as the knee point from the non-inferior solution set information 141, and outputs the acquired record.

  Next, an example of a processing procedure of the knee point selection device according to the present embodiment will be described. FIG. 15 is a flowchart illustrating the processing procedure of the knee point selection device according to the present embodiment. As illustrated in FIG. 15, the reception unit 151 of the knee point selection device 100 receives non-inferiority set information 141, mapper parameters 143, and weight generation condition information (step S101).

  The weight generation unit 152 of the knee point selection device 100 generates the weight set information 142 based on the weight generation condition information (step S102). The calculation unit 153 of the knee point selection device 100 selects an unselected weight set included in the weight set information 142 (step S103).

  The calculation unit 153 detects a non-inferior solution corresponding to the minimum point (step S104). The calculation unit 153 updates the minimum count table 146 based on the non-inferior solution detection result corresponding to the minimum point (step S105).

  The calculation unit 153 determines whether or not all weight sets included in the weight set information 142 have been selected (step S106). If all the weight sets have not been selected (No at Step S106), the calculation unit 153 proceeds to Step S103. When the calculation unit 153 selects all the weight sets (step S106, Yes), the calculation unit 153 proceeds to step S107.

  The selection unit 154 of the knee point selection device 100 sorts the records of the minimum count table 146 based on the minimum count (step S107). The selection unit 154 selects a non-inferior answer corresponding to the top N record among the sorted records as a knee point (step S108). The output unit 155 of the knee point selection apparatus 100 outputs and displays knee point information on the display unit 130 (step S109).

  Next, the effect of the knee point selection apparatus 100 according to the present embodiment will be described. The knee point selection apparatus 100 performs a process of specifying a non-inferior solution corresponding to a minimum point using a plurality of height functions having different weights, and for each non-inferior solution, a point having a large number of minimum points is converted to a knee point. Choose as. A non-inferior solution with a large number of minimum points is a robust knee point because it is often a knee point even if the height function changes slightly. For this reason, according to the knee point selection apparatus 100 according to the present embodiment, it is possible to select a robust knee point by excluding an unstable non-inferior solution that is not a knee point only by a slight change in the height function. it can.

  The knee point selection apparatus 100 registers the minimum number of times for each non-inferior solution in the minimum number table 146, and sorts the non-inferior solutions in the minimum number table 146 based on the minimum number of times. The knee point selection apparatus 100 selects a non-inferior solution with the highest number of minimum N as a knee point. Thereby, a plurality of robust knee points can be selected.

  The knee point selection apparatus 100 calculates the height information for the non-inferior solution by adding the values obtained by multiplying the values of the plurality of objective functions included in the non-inferior solution by the weight. Thereby, the non-inferior solution corresponding to the minimum point can be identified efficiently.

  Next, an example of a hardware configuration of a computer that realizes the same function as the knee point selection apparatus 100 shown in the above embodiment will be described. FIG. 16 is a diagram illustrating an example of a hardware configuration of a computer that realizes the same function as the knee point selection device.

  As illustrated in FIG. 16, the computer 200 includes a CPU 201 that executes various arithmetic processes, an input device 202 that receives input of data from a user, and a display 203. The computer 200 also includes a reading device 204 that reads a program or the like from a storage medium, and an interface device 205 that exchanges data with a recording device or the like via a wired or wireless network. The computer 200 also includes a RAM 206 that temporarily stores various information and a hard disk device 207. The devices 201 to 207 are connected to the bus 208.

  The hard disk device 207 includes a reception program 207a, a weight generation program 207b, a calculation program 207c, a selection program 207d, and an output program 207e. The CPU 201 reads out each program 207 a to 207 e and develops it in the RAM 206.

  The reception program 207a functions as a reception process 206a. The weight generation program 207b functions as a weight generation process 206b. The calculation program 207c functions as a calculation process 206c. The selection program 207d functions as a selection process 206d. The output program 207e functions as the output process 206e.

  The process of the reception process 206a corresponds to the process of the reception unit 151. The processing of the weight generation process 206b corresponds to the processing of the weight generation unit 152. The process of the calculation process 206c corresponds to the process of the calculation unit 153. The process of the selection process 206d corresponds to the process of the selection unit 154. The process of the output process 206e corresponds to the process of the output unit 155.

  Note that the programs 207a to 207e are not necessarily stored in the hard disk device 207 from the beginning. For example, each program is stored in a “portable physical medium” such as a flexible disk (FD), a CD-ROM, a DVD, a magneto-optical disk, and an IC card inserted into the computer 200. Then, the computer 200 may read and execute each of the programs 207a to 207e.

  The following supplementary notes are further disclosed with respect to the embodiments including the above examples.

(Appendix 1) Accepting multiple non-inferiority inputs for an optimization problem using multiple objective functions,
For each of the plurality of non-inferior solutions, a process of giving height information based on a specific direction set on a multidimensional space having the plurality of objective functions as coordinate axes is repeatedly executed while changing the weight for the non-inferior solution By calculating the number of times the height information is minimized for each non-inferior solution,
A knee point selection program that causes a computer to execute a process of selecting a non-inferior solution that becomes a knee point based on the calculated number of times.

(Additional remark 2) The process which selects the non-inferiority used as the said knee point sorts a plurality of non-inferior solutions based on the frequency | count that the said height information becomes the minimum, Among the several non-inferior solutions sorted The knee point selection program according to appendix 1, wherein a non-inferior solution having the highest number of times of minimization is selected as the knee point.

(Additional remark 3) The said process to calculate calculates the height information with respect to the said non-inferior solution by adding the value which multiplied the value of the several objective function contained in a non-inferior solution with the weight, It is characterized by the above-mentioned. The knee point selection program according to appendix 1 or 2.

(Appendix 4) A knee point selection method executed by a computer,
Accepts multiple non-inferior inputs for an optimization problem with multiple objective functions,
For each of the plurality of non-inferior solutions, a process of giving height information based on a specific direction set on a multidimensional space having the plurality of objective functions as coordinate axes is repeatedly executed while changing the weight for the non-inferior solution By calculating the number of times the height information is minimized for each non-inferior solution,
A knee point selection method comprising: executing a process of selecting a non-inferior solution that becomes a knee point based on the calculated number of times.

(Additional remark 5) The process which selects the non-inferior solution used as the said knee point sorts a plurality of non-inferior solutions based on the frequency | count that the said height information becomes the minimum, Among the several non-inferior solutions sorted The knee point selection method according to appendix 4, wherein a non-inferior solution having the highest number of times of minimization is selected as the knee point.

(Additional remark 6) The said process to calculate calculates the height information with respect to the said non-inferior solution by adding the value which multiplied the value of the some objective function contained in a non-inferior solution with the weight, It is characterized by the above-mentioned. The knee point selection method according to appendix 4 or 5.

(Appendix 7) A reception unit that receives a plurality of non-inferiority inputs for an optimization problem using a plurality of objective functions;
For each of the plurality of non-inferior solutions, a process of giving height information based on a specific direction set on a multidimensional space having the plurality of objective functions as coordinate axes is repeatedly executed while changing the weight for the non-inferior solution A calculation unit that calculates the number of times the height information is minimized for each non-inferior solution;
A knee point selection apparatus comprising: a selection unit that selects a non-inferior solution that becomes a knee point based on the calculated number of times.

(Additional remark 8) The said calculation part sorts several non-inferior solutions based on the frequency | count that the said height information becomes the minimum, and the frequency | count that becomes the minimum among the plurality of sorted non-inferior solutions is high. The knee point selection device according to appendix 7, wherein a poor answer is selected as the knee point.

(Additional remark 9) The said calculation part calculates the height information with respect to the said non-inferior solution by totaling the value which multiplied the weight to the value of the some objective function contained in a non-inferior solution. The knee point selection device according to 7 or 8.

DESCRIPTION OF SYMBOLS 100 Knee point selection apparatus 110 Communication part 120 Input part 130 Display part 140 Storage part 141 Non-inferiority set information 142 Weight set information 143 Mapper parameter 144 Undirected graph information 145 Directed graph information 146 Minimal number of times table 150 Control part 151 Reception part 152 Weight Generation unit 153 Calculation unit 154 Selection unit 155 Output unit

Claims (5)

Accepts multiple non-inferior inputs for an optimization problem with multiple objective functions,
For each of the plurality of non-inferior solutions, a process of giving height information based on a specific direction set on a multidimensional space having the plurality of objective functions as coordinate axes is repeatedly executed while changing the weight for the non-inferior solution By calculating the number of times the height information is minimized for each non-inferior solution,
A knee point selection program that causes a computer to execute a process of selecting a non-inferior solution that becomes a knee point based on the calculated number of times.   The process of selecting a non-inferior solution that becomes the knee point sorts a plurality of non-inferior solutions based on the number of times that the height information is minimized, and becomes a minimum among the plurality of sorted non-inferior solutions. The knee point selection program according to claim 1, wherein a non-inferior answer having the highest number of times is selected as the knee point.   2. The height information for the non-inferior solution is calculated by summing values obtained by multiplying values of a plurality of objective functions included in the non-inferior solution by weights. The knee point selection program according to 2. A knee point selection method executed by a computer,
Accepts multiple non-inferior inputs for an optimization problem with multiple objective functions,
For each of the plurality of non-inferior solutions, a process of giving height information based on a specific direction set on a multidimensional space having the plurality of objective functions as coordinate axes is repeatedly executed while changing the weight for the non-inferior solution By calculating the number of times the height information is minimized for each non-inferior solution,
A knee point selection method comprising: executing a process of selecting a non-inferior solution that becomes a knee point based on the calculated number of times. A reception unit that accepts a plurality of non-inferiority inputs for an optimization problem using a plurality of objective functions;
For each of the plurality of non-inferior solutions, a process of giving height information based on a specific direction set on a multidimensional space having the plurality of objective functions as coordinate axes is repeatedly executed while changing the weight for the non-inferior solution A calculation unit that calculates the number of times the height information is minimized for each non-inferior solution;
A knee point selection apparatus comprising: a selection unit that selects a non-inferior solution that becomes a knee point based on the calculated number of times.

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