JP2018163396A - Piecewise linear approximation function generation apparatus and method - Google Patents

Abstract

A piecewise linear approximation function having a small approximation error and a small number of sections is generated with a small calculation load.
An optimization problem formulation unit 15B formulates an optimization problem for obtaining a slope and an intercept for each of linear approximation functions based on input / output data 14A and node candidates 14B. An error term E _D representing the approximation error of the piecewise linear approximation function with respect to the data 14A is formulated as an objective function related to the optimization problem, and a regularization term E _W representing the slope difference relating to the linear approximation function of the adjacent section among the categories is expressed as follows: Formulation is made as an objective function related to the optimization problem, and the approximate values obtained by the linear approximation functions of adjacent sections coincide with each other as a constraint condition related to the optimization problem.
[Selection] Figure 1

Description

  The present invention relates to a piecewise linear approximation function generation technique for generating a piecewise linear approximation function obtained by dividing a complex nonlinear continuous function into a plurality of sections and approximating with a linear function.

  When creating an operation plan to be used for the operation of multiple air-conditioning heat source facilities, a complex nonlinear continuous function indicating the characteristics of the heat source equipment is approximated by a piecewise linear approximation function, and the resulting function is a mixed integer programming (MIP) (For example, refer to Patent Document 1). As a function generation technique for generating a piecewise linear approximation function, there is a method of generating a piecewise linear approximation function that minimizes an approximation error (see, for example, Patent Document 2). However, this cannot be applied when it cannot be determined in advance that the function to be generated is a convex function or a concave function.

  Conventionally, as such a piecewise linear approximation function generation technique, a technique called Douglas-Peucker algorithm has been proposed (see, for example, Patent Document 3). FIG. 15 is an explanatory diagram showing the Douglas-Peucker algorithm. In this method, Step 1: Connect both end points of the curve to be approximated with a straight line, Step 2: Detect the point on the curve farthest from the straight line, Step 3: Use the detected point as a node to set each segment curve Step 4: The number of sections is increased by repeating Step 2-3 until the number of sections and the error become the desired values.

JP 2010-169328 A JP 2006-099910 A JP 2006-113457 A

  However, in the conventional technique according to Patent Document 3 described above, when generating a linear approximation function of data including an error, if there is a point with a large error, that is, an outlier, the point is likely to become a node, and therefore, the point with a large error. There was a problem that an unnatural linear approximation function different from the intention was generated.

FIG. 16 is an example of generating an unnatural linear approximation function. Here, the input data is arranged at equal intervals of 0.01 in the range of [−2, 4]. Output data is obtained by placing Gaussian noise on a non-linear function and placing an outlier by one point. The total number of input / output data is 601.
According to this generation example, the linear approximation of data including errors is stopped at the time when the number of segments is 3, but an outlier that is far away from the majority of data groups is selected as a node. Therefore, it can be seen that the linear approximation function is greatly distorted compared to the data group.

Generally, when generating an approximate function, it is better that the approximation error is small. Also, instead of focusing on one point with the maximum approximation error, the overall approximation error should be reduced. Thereby, the influence of the outlier which was the subject of patent document 3 can be reduced.
In order to reduce the approximation error of the piecewise linear approximation function, the number of sections may be increased. However, if the number of categories is increased, fine approximation can be performed, but “over-learning” and “over-fitting” (creating a function that lacks versatility specialized in given data) Cause a problem called. Therefore, it is necessary to generate a piecewise linear approximation function with a small approximation error and a small number of sections.

As one of such methods, for example, the boundary between divisions, that is, node candidates and the maximum number of usable nodes are set in advance so that the approximation error due to the generated piecewise linear approximation function is minimized. A method of selecting a suitable node is conceivable.
According to this method, it is possible to generate a function that has a small approximation error and the number of sections does not exceed “the maximum number of usable nodes−1”.

FIG. 17 is an explanatory diagram showing node candidates. FIG. 18 is an example of generating a piecewise linear approximation function.
FIG. 17 shows a function f (x) to be approximated and node candidates x _i (i = 1, 2,..., 11) set on the x axis. In this case, the maximum number of sections is 10.
Assume that a piecewise linear approximation function with the smallest approximation error is obtained by using a maximum of two nodes in addition to the lower limit value x ₁ and the upper limit value x ₁₁ , and as a result, the piecewise linear approximation function shown in FIG. 18 is obtained. .

In FIG. 18, in addition to the lower limit value x ₁ and the upper limit value x ₁₁ , x ₃ and x ₇ are selected as nodes, and the number of sections is three. Therefore, the piecewise linear approximation function h (x) is a combination of three linear functions, and the slope g and intercept b of the corresponding linear approximation functions of the sections S ₁ , S ₂ , S ₃ are expressed as g _[1] , g _{[ 2]} , g _[3] and b _[1] , b _[2] , b _[3] can be expressed as the following equation (19).

Such a piecewise linear approximation function can be generated, for example, by performing the following calculation.
First, without selecting any node other than the lower limit value x ₁ and the upper limit value x ₁₁ , one linear approximation function is obtained as a whole, and an approximation error is calculated. Next, an arbitrary one of the node candidates x _i other than the lower limit value x ₁ and the upper limit value x ₁₁ is selected, divided into two sections, a linear approximation function is obtained for each section, and the approximation error of all sections Is calculated. Repeat this 9 times. Next, any two of the node candidates x _i (i = 2, 3,..., 10) other than the lower limit value x ₁ and the upper limit value x ₁₁ are selected and divided into three sections, and linear approximation is performed for each section. After obtaining the function, the approximation error of all sections is calculated. This is repeated for the number of combinations, ie 36 times. Thereafter, the combination of nodes selected when the approximation error is the smallest is adopted.

  The above method is a “combination optimization problem”. The combination optimization problem has a property that the calculation load increases as the number of combination candidates increases. For example, when there are 500 node candidates other than the upper and lower limits, and a maximum of 5 nodes are employed, the number of combinations exceeds 200 billion.

  The present invention is for solving such problems, and provides a piecewise linear approximation function generation technique capable of generating a piecewise linear approximation function with a small approximation error and a small number of pieces with a small calculation load. It is an object.

  In order to achieve such an object, the piecewise linear approximation function generation device according to the present invention performs the input based on a plurality of input / output data whose inputs are one-dimensional and a plurality of preset node candidates. A piecewise linear approximation function generating device that generates a piecewise linear approximation function for approximating each piece of output data with a linear approximation function, wherein a slope and a slope for each of the linear approximation functions based on the input / output data and the node candidates An optimization problem formulation unit that formulates an optimization problem for obtaining an intercept, an optimization processing unit that obtains a solution of the optimization problem by executing an optimization operation, and the above-described solution based on the obtained solution A piecewise linear approximation function generation unit that generates each of the linear approximation functions, and the optimization problem formulation unit represents an error representing an approximation error of the piecewise linear approximation function with respect to the input / output data. Is formulated as one or both of an objective function and a constraint condition related to the optimization problem, and a regularization term representing a slope difference related to a linear approximation function of an adjacent section of the sections is expressed as an objective function related to the optimization problem And the restriction condition is formulated as one or both of them, and the fact that the approximation values by the linear approximation function of the adjacent sections coincide at the nodes is formulated as the restriction condition regarding the optimization problem.

  In addition, another piecewise linear approximation function generation device according to the present invention determines each piece of the input / output data based on a plurality of input / output data whose inputs are multidimensional and a plurality of preset vertex candidates. A piecewise linear approximation function generation device for generating a piecewise linear approximation function approximated by a linear approximation function, wherein an optimum for obtaining a slope and an intercept for each of the linear approximation functions based on the input / output data and the vertex candidates An optimization problem formulation unit that formulates the optimization problem, an optimization processing unit that obtains a solution of the optimization problem by executing an optimization operation, and each of the linear approximation functions based on the obtained solution A piecewise linear approximation function generation unit for generating, and the optimization problem formulation unit adds an error term representing an approximation error of the piecewise linear approximation function to the input / output data to the optimization problem. A regularization term that expresses a slope difference for a linear approximation function of an adjacent section among the sections, and is expressed as one of the objective function and the constraint condition for the optimization problem. It is formulated as one or both, and it is formulated as a constraint condition related to the optimization problem that the approximation values by the linear approximation functions of the adjacent sections coincide at the apex.

Further, in a configuration example of the piecewise linear approximation function generation device according to the present invention, the regularization term is composed of an L1 norm related to all the slope differences.
Also, in a configuration example of the piecewise linear approximation function generation device according to the present invention, the regularization term is composed of a sum of L2 norms of the slope difference.

  In addition, a configuration example of the piecewise linear approximation function generation device according to the present invention further includes a standardization unit that standardizes the regularization term using a predetermined coefficient provided for each of the slope differences, Each consists of the corresponding absolute value of the slope difference obtained by solving the optimization problem assuming that the coefficient is 1.

  The piecewise linear approximation function generation method according to the present invention linearly approximates each piece of the input / output data based on a plurality of input / output data having a one-dimensional input and a plurality of preset node candidates. A piecewise linear approximation function generation method for generating a piecewise linear approximation function approximated by a function, wherein the optimization problem formulation unit is configured to determine a slope and a slope for each of the linear approximation functions based on the input / output data and the node candidates. An optimization problem formulation step that formulates an optimization problem for obtaining an intercept; an optimization processing step in which an optimization processing unit performs an optimization operation to obtain a solution of the optimization problem; and a piecewise linear An approximate function generation unit including a piecewise linear approximation function generation step for generating each of the linear approximation functions based on the obtained solution, and the optimization problem formulation step includes: Formulating an error term representing an approximation error of the piecewise linear approximation function with respect to input / output data as one or both of an objective function and a constraint condition related to the optimization problem; Formulating the regularization term representing the slope difference for the approximation function as one or both of the objective function and the constraint condition for the optimization problem, and the approximation value by the linear approximation function of the adjacent segment coincide at the node To formulate as a constraint on the optimization problem.

  Further, another piecewise linear approximation function generation method according to the present invention is configured to determine each piece of the input / output data based on a plurality of input / output data whose inputs are multidimensional and a plurality of preset vertex candidates. A piecewise linear approximation function generation method for generating a piecewise linear approximation function to be approximated by a linear approximation function, wherein an optimization problem formulation unit is configured for each of the linear approximation functions based on the input / output data and the vertex candidates. An optimization problem formulation step that formulates an optimization problem for obtaining a slope and an intercept; an optimization processing step in which an optimization processing unit obtains a solution of the optimization problem by executing an optimization operation; A piecewise linear approximation function generation unit including a piecewise linear approximation function generation step for generating each of the linear approximation functions based on the obtained solution, and the optimization problem formulation step includes: Formulating an error term representing an approximation error of the piecewise linear approximation function with respect to the input / output data as one or both of an objective function and a constraint condition related to the optimization problem; and Formulating a regularization term representing a slope difference with respect to a linear approximation function as one or both of an objective function and a constraint condition with respect to the optimization problem, and an approximation value by the linear approximation function of the adjacent section at a vertex And formulating matching as a constraint on the optimization problem.

  According to the present invention, the optimization problem for obtaining the slope and intercept for each of the linear approximation functions can be formulated as an optimization problem using only continuous variables. Therefore, it is possible to generate a piecewise linear approximation function with as few pieces as possible, and to produce a piecewise linear approximation function with a small approximation error and a small number of pieces with a small calculation load. For this reason, problems called “overlearning” and “overfitting” in the piecewise linear approximation function can be avoided.

  The heat source operation plan also supports situations such as making a plan with a short period of about 30 minutes, or instantly reestablishing a plan in response to changes in site conditions such as equipment failure or sudden changes in heat load. can do. For this reason, applications such as regenerating functions based on actual data and generating a large number of functions within the calculation cycle of the heat source operation plan are possible, and it is possible to obtain extremely significant effects on the industry. Become.

It is a block diagram which shows the structure of the piecewise linear approximation function production | generation apparatus concerning 1st Embodiment. It is a flowchart which shows the piecewise linear approximation function production | generation process concerning 1st Embodiment. It is explanatory drawing which shows input / output data and a node candidate. It is explanatory drawing which shows the inclination and approximate value of a linear approximation function. It is an example of generation of a piecewise linear approximation function. It is an example of generation of a piecewise linear approximation function according to the first embodiment. It is a block diagram which shows the structure of the piecewise linear approximation function production | generation apparatus concerning 2nd Embodiment. It is a flowchart which shows the piecewise linear approximation function production | generation process concerning 2nd Embodiment. It is an example of the production | generation of the piecewise linear approximation function by 2nd Embodiment. It is a block diagram which shows the structure of the piecewise linear approximation function generator concerning 5th Embodiment. It is a flowchart which shows the piecewise linear approximation function production | generation process concerning 5th Embodiment. It is explanatory drawing which shows the division | segmentation in case an input is two-dimensional. It is an example of a piecewise linear approximation function related to input / output data of two inputs and one output. It is an example of the production | generation of the piecewise linear approximation function by 5th Embodiment. It is explanatory drawing which shows Douglas-Peucker algorithm. It is an example of generation of an unnatural linear approximation function. It is explanatory drawing which shows a node candidate. It is an example of generation of a piecewise linear approximation function. It is explanatory drawing which shows the difference of the inclination of the linear approximation function of an adjacent division.

[Principle of the Invention]
First, the principle of the present invention will be described.
In general, in the piecewise linear approximation function, there is a need to reduce the number of sections as much as possible in order to avoid problems called “overlearning” and “overfitting”. Moreover, when it uses for the mixed integer plan for a heat source operation plan, it is better that the number of divisions is small also from the viewpoint of calculation load.

In particular, in the heat source operation plan, it is necessary to make a plan with a short cycle of, for example, about 30 minutes, or to immediately make a plan according to changes in the field situation such as equipment failure or a rapid change in heat load. For this reason, it is important to reduce the calculation load and output the calculation result in a short time.
In addition, if a piecewise linear approximation function with a small number of sections can be generated easily and at high speed, the function can be regenerated based on the actual data or many functions can be generated within the calculation cycle of the heat source operation plan. Application is possible, and industrial effects are great.

  As described above, the piecewise linear approximation function is a function in which the linear approximation functions of the respective sections are connected. As a method for reducing the number of sections, a method of combining adjacent sections into one can be considered. According to the present invention, as a specific method for grouping adjacent sections, the linear approximation function of adjacent sections is expressed by one linear approximation function, that is, “the difference in the slopes of the linear approximation functions of adjacent sections”. We focused on making it smaller.

  If the “difference in the slope of the linear approximation function of adjacent sections” becomes smaller and eventually becomes the same, these sections can be expressed by the same linear approximation function. FIG. 19 is an explanatory diagram showing the difference in slope of the linear approximation function between adjacent sections. For example, in FIG. 19, when there is no difference between the slopes g1 and g2 of the linear approximation functions of adjacent sections, these linear approximation functions can be expressed by one linear approximation function. For this reason, since these divisions can be regarded as one, the number of divisions should be reduced as a result.

  Here, the generation of the piecewise linear approximation function can be regarded as a combination optimization problem that optimizes the combination of the linear approximation functions for each piece. However, when the objective function or the constraint condition is to reduce the “difference in the slope of the linear approximation function of adjacent sections”, the generation of the piecewise linear approximation function is not a combinatorial optimization problem. It can be formulated as an optimization problem using only continuous variables.

  From this point of view, the present invention provides an objective function and / or a constraint condition that reduces the “approximation error” and an objective function that reduces the “difference in the slope of the linear approximation function of adjacent segments” and / or Or, in addition to the constraint conditions, solve the optimization problem with the constraint condition that the approximate values of the linear approximation functions of adjacent segments match at the nodes, so that a piecewise linear approximation function with a small number of segments is generated. It is a thing.

  In this case, even if an attempt is made to reduce the “difference in the slope of the linear approximation function of adjacent segments”, there may be many segments with slightly different gradients, and the number of segments may not be reduced well. Therefore, the following measures 1 and 2 can be considered.

Countermeasure 1: The magnitude of “the difference between the slopes of the linear approximation functions of adjacent sections” is expressed by the sum of absolute values (L1 norm).
Countermeasure 2: Before taking the sum of absolute values, standardize by dividing each absolute value by a coefficient. In determining the coefficient, optimization is performed once without using the coefficient (= the coefficient is set to 1), and the absolute value obtained as a result is used as the coefficient.
In the present invention, only such countermeasure 1 is implemented, or both countermeasures 1 and 2 are implemented.

[First Embodiment]
Next, embodiments of the present invention will be described with reference to the drawings.
First, a piecewise linear approximation function generation device 10 according to a first exemplary embodiment of the present invention will be described with reference to FIG. FIG. 1 is a block diagram illustrating a configuration of the piecewise linear approximation function generation device according to the first embodiment.

  This piecewise linear approximation function generation device 10 comprises an information processing apparatus such as a server apparatus as a whole, and linearly approximates each piece of input / output data based on a plurality of input / output data and a plurality of preset node candidates. It is a device that generates a piecewise linear approximation function that approximates with a function.

  As shown in FIG. 1, the piecewise linear approximation function generation device 10 includes a communication I / F unit 11, an operation input unit 12, a screen display unit 13, a storage unit 14, and an arithmetic processing unit 15 as main functional units. Is provided.

The communication I / F unit 11 communicates with an external device (not shown) via a communication line, various data used for generating a piecewise linear approximation function, such as input / output data and node candidates, It has a function to exchange data related to linear approximation functions.
The operation input unit 12 includes an operation input device such as a keyboard, a mouse, and a touch panel, and has a function of detecting an operator operation and outputting the operation to the arithmetic processing unit 15.

  The screen display unit 13 includes a screen display device such as an LCD, and displays various screens output from the arithmetic processing unit 15, such as an operation menu screen, various input / setting screens, and a result screen showing the obtained piecewise linear approximation function. It has a function to do.

The storage unit 14 includes a storage device such as a hard disk or a semiconductor memory, and stores various processing data used for a piecewise linear approximation function generation process executed by the arithmetic processing unit 15 and a program 14P registered in advance from the outside. have.
Main processing data stored in the storage unit 14 includes input / output data 14A and node candidates 14B.

  The input / output data 14A is data in which a plurality of sets of input data (x) and output data (y) are registered. The node candidate 14B is data indicating a boundary of each section set in the input data space, that is, a node of a piecewise linear approximation function.

The arithmetic processing unit 15 includes a microprocessor such as a CPU, and has a function of realizing various processing units that perform a piecewise linear approximation function generation process by reading and executing a program in the storage unit 14.
As main processing units realized by the arithmetic processing unit 15, there are a setting processing unit 15A, an optimization problem formulation unit 15B, an optimization processing unit 15C, and a piecewise linear approximation function generation unit 15D.

  The setting processing unit 15A sets the input / output data 14A acquired via the communication I / F unit 11 in the storage unit 14 and the node candidate 14B input by the operation input unit 12 in the storage unit 14. It has a function.

  The optimization problem formulation unit 15B functions to formulate an optimization problem for obtaining a slope and an intercept for each of the linear approximation functions in each section based on the input / output data 14A and the node candidates 14B in the storage unit 14. have.

  More specifically, the optimization problem formulation unit 15B formulates an error term representing the approximation error of the piecewise linear approximation function with respect to the input / output data 14A as an objective function related to the optimization problem, and adjacent to the classification. A function for formulating a regularization term representing an inclination difference relating to a linear approximation function of matching sections, for example, a regularization term consisting of an L1 norm for all of the slope differences as an objective function relating to an optimization problem, and a linear approximation function of adjacent sections It has a function to formulate that the approximate value by is coincident at the nodes as a constraint condition for the optimization problem.

The optimization processing unit 15C has a function of obtaining a solution to an optimization problem by executing an optimization calculation.
The piecewise linear approximation function generation unit 15D has a function of generating a piecewise linear approximation function by generating each of the linear approximation functions of each section based on the solution of the optimization problem obtained by the optimization processing unit 15C. Have.

[Operation of First Embodiment]
Next, the operation of the piecewise linear approximation function generation device 10 according to this exemplary embodiment will be described with reference to FIG. FIG. 2 is a flowchart showing a piecewise linear approximation function generation process according to the first embodiment.
The arithmetic processing unit 15 of the piecewise linear approximation function generation device 10 executes the piecewise linear approximation function generation process of FIG. 2 according to the operator operation detected by the operation input unit 12.

  First, in FIG. 2, the setting processing unit 15A registers the input / output data 14A acquired via the communication I / F unit 11 in the storage unit 14 (step 100), and the node input by the operation input unit 12 The candidate 14B is set in the storage unit 14 (step 101).

Assume that N sets of input data and output data (N is an integer of 2 or more) are provided as the input / output data 14A. Further, it is assumed that n (n is an integer of 3 or more) node candidates x _i (i = 1, 2,..., N) are set. x _i is one that is sorted in ascending order with respect to i, i.e., it is assumed that consists is _{x 1 <x 2 <... <} x n. The node candidates may be the input data itself or may be set by dividing the range from x ₁ to x _n at equal intervals. x ₁ and x _n may be the minimum value and the maximum value of the input data, or arbitrary values may be set within a range including the input data.

The output data corresponding to the input data matching the node candidate is y _i (i = 1, 2,..., N). However, when i = j, there is no output data, that is, there is no data on the node candidate. Let j be the number of _j .
Further, the output data corresponding thereto input data does not match the node candidates, when the segment S _i sandwiched node candidate x _i and the node candidate x _{i + 1} is m _i pieces of input data, these entry The input data of the output data is represented as x _{[i] k} (k = 1, 2,..., M _i ), and the corresponding output data is represented as y _{[i] k} . Therefore, the number N of input / output data is expressed by the following equation (1).

FIG. 3 is an explanatory diagram showing input / output data and node candidates. In this example, four _{x 1, x 2, x 3} , x 4 as a node candidate (n = 4) are set, separated by these nodes candidate _{x 1, x 2, x 3} , x 4 Three sections S ₁ , S ₂ , S ₃ are provided. Of the node candidates x ₁ , x ₂ , x ₃ , and x ₄ , x ₁ is a lower limit value and x ₄ is an upper limit value.
Output data y ₂ exists on x ₂ , but no output data exists on x ₁ , x ₃ , and x ₄ , and j = 1, 3, and 4. In addition, there is no input / output data in the section S ₁ between x ₁ and x ₂ , and m ₁ = 0. On the other hand, the division S ₂ between x ₂ and x ₃ there is one input and output data _{(x [2] 1, y} [2] 1), a m ₂ = 1, again, x ₃ and x ₄ the division S ₃ between the present two input data _{(x [3] 1, y} [3] 1) and a _{(x [3] 2, y} [3] 2) is, m ₃ = 2 It is. Therefore, the number N of input / output data is N = 4−3 + (0 + 1 + 2) = 4.

Next, in FIG. 2, the optimization problem formulation unit 15B formulates an error term E _D representing the approximation error of the piecewise linear approximation function for the input / output data 14A as an objective function related to the optimization problem (step 102). .
The piecewise linear approximation function is h (x), and the approximate value of the output data y _i by h (x) is y ^ _i = h (x _i ), y ^ _{[i] k} = h (x _{[i] k} ) It shall be expressed as
Also, let g _{i be} the slope of the linear approximation function in the interval between node candidate x _i and node candidate x _{i + 1} .

FIG. 4 is an explanatory diagram showing the slope and approximate value of the linear approximation function. In this example, three sections S ₁ , S ₂ , S ₃ divided by node candidates x ₁ , x ₂ , x ₃ , x ₄ are provided, and the slopes of these sections S ₁ , S ₂ , S ₃ are provided. Are g ₁ , g ₂ , and g ₃ . Also, the approximate values of the output data y ₁ , y ₂ , y ₃ , y ₄ corresponding to x ₁ , x ₂ , x ₃ , x ₄ are y ^ ₁ , y ^ ₂ , y ^ ₃ , y ^ ₄ . , The approximate value of the input / output data (x _{[2] 1} , y _{[2] 1} ), (x _{[3] 1} , y _{[3] 1} ), (x _{[3] 2} , y _{[3] 2} ) is y ^ _{[2] 1} , y ^ _{[3] 1} , y ^ _{[3] 2} .

Here, in order to place an objective function and / or a constraint condition such that the approximation error of the piecewise linear approximation function for the input / output data is reduced, the error term E _D is defined by the following equation (2). Equation (2) represents the error sum of squares.

In the case where there are a plurality of output data having the same input data in the input / output data 14A, they may be included in the error calculation of Expression (2). For example, when there are two pieces of input / output data y ₁ and y ′ ₁ having the same input data x ₁ , (x ₁ , y ₁ ) and (x ₁ , y ′ ₁ ), (y ₁ −y ^ ₁ ) Not only ² but also (y ′ ₁ −y ^ ₁ ) ² may be included in the sum of the error terms E _D.

Next, in FIG. 2, the optimization problem formulation unit 15B formulates a regularization term E _W representing a slope difference related to the linear approximation function of adjacent sections among the sections as an objective function related to the optimization problem ( Step 103).
Here, the regularization term E _W is defined by the following equation (3) in order to set an objective function and / or a constraint condition that “the difference in the slopes of the linear approximation functions of adjacent segments” becomes small. Equation (3) represents the sum of absolute values for all of the slope differences, ie, the L1 norm.

Next, in FIG. 2, the optimization problem formulation unit 15B formulates that the approximate values by the linear approximation functions of adjacent sections match at the nodes, that is, the approximate value matching constraint as a constraint condition for the optimization problem. (Step 104).
This approximate value match can be represented by the first line of the following equation (4). The expression in the second row represents that the approximate value inside the section matches the approximate straight line (gets on the straight line).

Next, in FIG. 2, the optimization problem formulation unit 15B integrates the individual objective functions and constraint conditions formulated in this way, and obtains the slope and intercept for each of the linear approximation functions in each section. The optimization problem is formulated as the following equation (5) (step 105).

The objective function to be minimized is a weighted linear sum of the error term E _D and the regularization term E _W. λ> 0 is a weight for the regularization term E _W. If the weight is increased, it can be expected that the number of sections of the piecewise linear approximation function is reduced.
In FIG. 2, the optimization processing unit 15C sets a weight λ for the regularization term E _W in accordance with the operator operation detected by the operation input unit 12 (step 106), and an expression defined by this weight λ ( The optimization problem is solved by executing the optimization operation based on 5) (step 107).

At this time, if the intermediate variable v _i (i = 1, 2,..., N−2) is introduced and the regularization term is newly defined using the intermediate variable v _i , the above equation (5) is expressed by the following equation: It can be expressed as (6). The regularization term Ew in equation (6) is the sum of the intermediate variables v _i . This equation (6) is a so-called “secondary programming problem”. Therefore, this problem can be solved easily and at high speed as compared with the combinatorial optimization problem.

Thereafter, in FIG. 2, the piecewise linear approximation function generation unit 15 </ b> D generates each piece of linear approximation function of each section based on the solution of the optimization problem obtained by the optimization processing unit 15 </ b> C, thereby producing piecewise linear approximation functions. An approximate function is generated (step 108).
Here, as a result of solving the optimization problem of equation (6), the result of arranging the gradients g _i (i = 1, 2,..., N−1) is as shown in the following equation (7). .

Equation (7) means that the first s ₁ slopes, ie, g _i to g _s1 have the same slope. It also means that the next s ₂ slopes, that is, g _s1 to g _s2 have the same slope again. This means that the obtained slope is divided into p pieces. Here, a minute error may be regarded as zero. Even if the inclination is the same, if they are not adjacent, they are considered different.
From the above, the nodes to be adopted are x ₁ , x _{s1 + 1} , x _{s1 + s2 + 1} ,..., X _{s1 +... + Sp + 1} , and the number of nodes to be adopted is p + 1. However, x ₁ is the lower _{limit, x s1 + ... + sp +} 1 is the upper limit value, the number of division is a p number.

Here, when the p slopes are represented by g _[1] , g _[2] ,..., G _[p] , the piecewise linear approximation function h (x) can be expressed by the following equation (8).

The intercepts b _[1] , b _[2] ,..., B _[p] may be obtained from the following equation (9) using the approximate values obtained as a result of solving the optimization problem.

FIG. 5 is an example of generating a piecewise linear approximation function. Here, the division S ₁ from x ₁ to x _3, the gradient g _[1] is derived from m ₁ = 2 single slope g ₁ and g _2, the segment S ₂ from x ₃ to x _7, m ₂ = Slope g _[2] is derived from ₄ slopes g ₃ , g ₄ , g ₅ , g _6, and in section S ₂ from x ₇ to x ₁₁ , m ₃ = 4 slopes g ₇ , g _The slope g _[3] is derived from ₈ , g ₉ and g ₁₀ .

Here, it is confirmed whether the obtained piecewise linear approximation function h (x) is in a desired range (step 109). The obtained piecewise linear approximation function h (x) has a large approximation error or a large number of pieces. If the desired piecewise linear approximation function h (x) is not obtained (step 109: NO), the process returns to step 106 to change the magnitude of the weight λ with respect to the regularization term E _W. If λ is reduced, the approximation error is reduced, and if λ is increased, the number of sections can be expected to be reduced.

  On the other hand, when the obtained piecewise linear approximation function h (x) is within a desired range (step 109: YES), the obtained piecewise linear approximation function h (x) is displayed on the screen display unit 13; Or it outputs to the exterior from the communication I / F part 11, and the production | generation process of a series of piecewise linear approximation functions is complete | finished.

  FIG. 6 is an example of generating a piecewise linear approximation function according to the first embodiment. Here, similarly to FIG. 16 described above, the input data is arranged at equal intervals of 0.01 in the range of [−2, 4]. Output data is obtained by placing Gaussian noise on a non-linear function and placing an outlier by one point. The total number of input / output data is 601.

In this piecewise linear approximation function generation process, all input data are set as node candidates. Therefore, there are 599 node candidates excluding the upper and lower limit values. The weight λ for the regularization term is set to an appropriate value.
As a result, a piecewise linear approximation function having 10 pieces is obtained, and it is understood that there is no influence of outliers. In addition, a general-purpose quadratic programming solver was used for the optimization process, but the calculation was possible in a very short time.

[Effect of the first embodiment]
As described above, according to the present embodiment, the optimization problem formulation unit 15B formulates an optimization problem for obtaining a slope and an intercept for each of the linear approximation functions based on the input / output data 14A and the node candidates 14B. At this time, the error term E _D representing the approximation error of the piecewise linear approximation function for the input / output data 14A is formulated as an objective function relating to the optimization problem, and the regularity representing the slope difference relating to the linear approximation function of the adjacent section among the sections. The formulation term E _W is formulated as an objective function related to the optimization problem, and the approximation value by the linear approximation function of the adjacent sections coincides at the nodes as a constraint condition related to the optimization problem.

  Thereby, the optimization problem for obtaining the slope and intercept for each of the linear approximation functions can be formulated as an optimization problem using only continuous variables. Therefore, it is possible to generate a piecewise linear approximation function with as few pieces as possible, and to produce a piecewise linear approximation function with a small approximation error and a small number of pieces with a small calculation load. For this reason, problems called “overlearning” and “overfitting” in the piecewise linear approximation function can be avoided.

  In addition, in the heat source operation plan, in the heat source operation plan, for example, a plan is made in a short cycle of about 30 minutes, or a plan is immediately reestablished in response to a change in the site situation such as equipment failure or a rapid change in heat load. It is possible to cope with any situation. For this reason, applications such as regenerating functions based on actual data and generating a large number of functions within the calculation cycle of the heat source operation plan are possible, and it is possible to obtain extremely significant effects on the industry. Become.

[Second Embodiment]
Next, the piecewise linear approximation function generation device 10 according to the second exemplary embodiment of the present invention will be described with reference to FIG. FIG. 7 is a block diagram showing a configuration of a piecewise linear approximation function generation device according to the second embodiment, and the same or equivalent parts as those in FIG. 1 are denoted by the same reference numerals.

According to the first embodiment described above, there is a possibility that the number of sections may not be reduced well by the input / output data 14A. In the present embodiment, as shown in FIG. 7, in the configuration of the first embodiment, a standardization processing unit 15E that standardizes the regularization term E _W described above is added to the arithmetic processing unit 15. It is.

The reference processing unit 15E, using the predetermined coefficient provided for each slope difference related to the linear approximation function sections adjacent of each section, and has a function of scaling the regularization term E _W.
That is, in this embodiment, instead of using the above-described equation (3) as the regularization term E _W , “standardization” is performed using a positive coefficient d _i as in the following equation (10). .

Here, a method for setting the coefficient d _i will be described.
In the initial standardization, d _i = 1 is set as an initial value. This means that equation (10) and equation (3) are equivalent. Then, the optimization problem is solved and the obtained gradient is set as g * _i . Using the obtained gradient g * _i , d _i is updated according to the following equation (11).

At this time, ε is a minute positive number for preventing zero division and may be set appropriately in advance. max (a, b) means a function that takes a large value of a and b. d _i may be updated each time optimization is performed.

FIG. 8 is a flowchart showing the piecewise linear approximation function generation processing according to the second embodiment, and the same or equivalent parts as in FIG.
Compared with FIG. 2 described above, between step 106 and step 107, the standardization processing unit 15E performs processing (step 200) for standardizing the regularization term E _W according to the equations (10) and (11). Have been added.
As a result, it is possible to generate a piecewise linear approximation function with a small number of sections even for the input / output data 14A in which the number of sections does not decrease well.

If the desired piecewise linear approximation function h (x) is not obtained (step 109: NO), the process returns to step 106, and the magnitude of the weight λ for the regularization term E _W may be changed. Returning to 200, the regularization term E _W may be rescaled. If λ is reduced, the approximation error is reduced, and if λ is increased, the number of sections can be expected to be reduced.

  FIG. 9 is an example of generating a piecewise linear approximation function according to the second embodiment. Here, similarly to FIG. 16 described above, the input data is arranged at equal intervals of 0.01 in the range of [−2, 4]. Output data is obtained by placing Gaussian noise on a non-linear function and placing an outlier by one point. The total number of input / output data is 601.

In this piecewise linear approximation function generation process, all input data are set as node candidates. Therefore, there are 599 node candidates excluding the upper and lower limit values. The weight λ for the regularization term was set to the same value as in FIG.
As a result, four piecewise linear approximation functions having the number of pieces less than 10 in FIG. 6 are obtained, and it is understood that a more natural approximation is obtained without being influenced by the outlier.

[Third Embodiment]
Next, a piecewise linear approximation function generation device 10 according to a third exemplary embodiment of the present invention will be described.
In the first embodiment, the error term E _D representing the approximation error of the piecewise linear approximation function with respect to the input / output data 14A is formulated as an objective function related to the optimization problem, and the slope relating to the linear approximation function of the adjacent section among the sections. Although the case where the regularization term E _W representing the difference is formulated as an objective function related to the optimization problem has been described as an example, one or both of the error term E _D and the regularization term E _W are determined as the optimization problem. You may formulate it as a constraint condition about.

In order to include the error term E _D and the regularization term E _W in the constraints, the upper limit r of the error term E _{D and} the upper limit l of the regularization term E _W are set, and the following equation (12) is described above. What is necessary is just to include in the constraint condition of Formula (6). Since the error term E _D has a quadratic form, an optimization problem including the error term E _D as a constraint condition is a “secondary constraint problem”. This problem can also be solved easily and faster than the combination optimization problem.

When the error term E _D is included in the constraint condition, it is not necessary to include it in the objective function. Similarly, when the regularization term E _W is included in the constraint condition, it may not be included in the objective function.
If at least one of the error term E _D and the regularization term is not included in the objective function, the weight λ for the regularization term E _W is not necessary and need not be set.
If neither the error term E _D nor the regularization term E _W is included in the objective function, the optimization problem is an “executability problem” in which the objective function does not exist and only the constraints exist. Can be solved.

[Fourth Embodiment]
Next, a piecewise linear approximation function generation device 10 according to a fourth exemplary embodiment of the present invention will be described.
The optimization problem can be a linear programming problem instead of a quadratic programming problem. In that case, the error term E _D may be the sum of absolute values of errors as shown in the following equation (13), not the sum of squares of the error represented by the above equation (2). Note that the conversion from the absolute value expression to the linear expression may be performed using intermediate variables in the same way as the conversion of the regularization term performed in Expression (5) to Expression (6).

[Fifth Embodiment]
Next, a piecewise linear approximation function generation device 10 according to a fifth exemplary embodiment of the present invention will be described with reference to FIG. FIG. 10 is a block diagram illustrating a configuration of a piecewise linear approximation function generation device according to the fifth embodiment, and the same or equivalent parts as those in FIG. 1 are denoted by the same reference numerals.

  In the above-described first to fourth embodiments, the case where the input is a one-dimensional linear approximation function, that is, a so-called broken line is described as an example. In this embodiment, a case where the input is multidimensional will be described. The configuration of the piecewise linear approximation function generation device 10 according to the present embodiment is the same as that shown in FIG. 1, and a detailed description thereof is omitted here.

  When the input is multidimensional, two linear approximation functions are not connected at the nodes, but three or more linear approximation functions are connected at the vertices. It is only necessary to replace “node” with “vertex”. Therefore, the vertex candidate 14C is registered in the storage unit 14 instead of the node candidate 14B.

  That is, according to the present invention, an objective function and / or a constraint condition in which the “approximation error” is small and an objective function and / or constraint condition in which the “difference in the slope of the linear approximation function of adjacent segments” is small. In addition, a piecewise linear approximation function with a small number of sections is generated by solving an optimization problem with a constraint condition that the approximation values of adjacent section linear approximation functions match at the "vertex" It is.

[Operation of Fifth Embodiment]
Next, with reference to FIG. 11, the operation of the segmented linear approximate function generation device 10 according to the present exemplary embodiment will be described. FIG. 11 is a flowchart showing a piecewise linear approximation function generation process according to the fifth embodiment, and the same or equivalent parts as those in FIG. 2 are denoted by the same reference numerals.
The arithmetic processing unit 15 of the piecewise linear approximation function generation device 10 executes the piecewise linear approximation function generation process of FIG. 11 according to the operator operation detected by the operation input unit 12.

  First, in FIG. 11, the setting processing unit 15A registers the input / output data 14A acquired through the communication I / F unit 11 in the storage unit 14 (step 100), and the vertex input by the operation input from the operation input unit 12 The candidate 14C is set in the storage unit 14 (step 101).

If the number of input dimensions is p, the partition is a simplex having p + 1 vertices. For example, if the input is two-dimensional, the simplex is a triangle.
FIG. 12 is an explanatory diagram showing classification when the input is two-dimensional. For example, when the input is two-dimensional x1 and x2 and the output y is one-dimensional, each section (triangle) can be defined as shown in FIG. In this example, the total number of sections is 224, and the number of vertices is 135.

  FIG. 13 shows an example of a piecewise linear approximation function related to input / output data having two inputs and one output. If each vertex has an output value y, each section is represented by a plane, that is, each section is represented by a separate linear approximation function, as shown in FIG. A piecewise linear approximation function can be generated.

Next, in FIG. 11, the optimization problem formulation unit 15B formulates an error term E _D representing the approximation error of the piecewise linear approximation function for the input / output data 14A as an objective function related to the optimization problem (step 103). .
The multidimensional error term E _D can be expressed by the following equation (14) as in the equation (2).

Here, y _i , i ≠ j is an output value of the vertex i, and n is the total number of the vertex i. When i = j, the vertex does not have an output value.
Further, y _{[I] k,} when a certain segment was S _I, is (but other than vertex) output values included in the classification S _I, and the number is m _i. N is the total number of sections. Even if the output value is located on the boundary between the sections, it is not necessary to add the approximation errors in duplicate.

Next, in FIG. 11, the optimization problem formulation unit 15B formulates the regularization term E _W representing the gradient difference related to the linear approximation function of the adjacent section among the sections as an objective function related to the optimization problem ( Step 103).

  As in the case of the one-dimensional case, if the “difference in the slope of the linear approximation function of adjacent segments” is set to zero, they can be regarded as the same segment, so the number of segments can be reduced. In the two-dimensional case, “adjacent section” means two triangles sharing “sides” (line segments). In the case of the p dimension, the adjacent section means two simplexes sharing the p-1 dimensional hyperplane.

Two adjacent sections are denoted by S _I and S _J , respectively, and S _I and S _J each have a slope in the p-dimensional direction. These _{slopes, g I = [g [I} ] 1, g [I] 2, ..., g [I] p] T and _{g J = [g [J]} 1, g [J] 2, ..., g _{[J] p} ] ^T T is a transpose symbol.
Taking the difference in slope of each dimension, the calculation of the L2 norm is defined as the following equation (15).

Further, the multidimensional regularization term E _W corresponding to the equation (3) can be defined as the sum of the L2 norms as in the following equation (16). However, “I, JεA” means “all combinations of adjacent sections”.

  Next, in FIG. 11, the optimization problem formulation unit 15B formulates that the approximate values by the linear approximation functions of adjacent sections match at the apex, that is, the approximate value matching constraint as a constraint condition for the optimization problem. (Step 104).

１行目は、区分Ｓ_Iの辺ｐ＋１個のうち、ｐ個について置けばよい。これは全ての辺について置くと冗長となるからである。ａ，ｂは各辺の両端の頂点を意味しており、ｘ_aとｘ_bは頂点の位置を意味する。ｇ_Iは区分Ｓ_Iにおける線形近似関数の傾きを表すベクトルを表す。
２行目は、区分Ｓ_Iに含まれる区分内部の近似値が、近似直線と一致する（超平面上に乗る）ことを表している。ｙ＾_aは、Ｓ_Iの任意の頂点上の近似値である。 The multidimensional constraint corresponding to Equation (4) is as shown in Equation (17) below.

The first line of the side p + 1 pieces of partition S _I, it places the p number. This is because putting all sides becomes redundant. a and b mean the vertices at both ends of each side, and x _a and x _b mean the positions of the vertices. g _I represents a vector representing the slope of the linear approximation function in the section S _I.
The second line, the approximate value of the segment internal included in segment S _I is represents that it matches the approximate line (ride on the hyperplane). y ^ _a is an approximate value on an arbitrary vertex of S _I.

Next, in FIG. 11, the optimization problem formulation unit 15B integrates the individual objective functions and constraint conditions formulated in this way, and obtains the slope and intercept for each of the linear approximation functions in each section. The optimization problem is formulated as shown in the following equation (18) (step 105). This equation (18) is a “secondary programming problem” and can be easily and quickly solved as in the case of the one-dimensional case.

Thereafter, the optimization processing unit 15C sets a weight λ for the regularization term E _W according to the operator operation detected by the operation input unit 12 (step 106), and an expression (5) defined by the weight λ. The optimization problem is solved by executing an optimization operation based on (step 107).
Subsequently, the piecewise linear approximation function generation unit 15D generates a piecewise linear approximation function by generating each of the linear approximation functions of each section based on the solution of the optimization problem obtained by the optimization processing unit 15C. (Step 108).

Here, it is confirmed whether the obtained piecewise linear approximation function h (x) is in a desired range (step 109). The obtained piecewise linear approximation function h (x) has a large approximation error or a large number of pieces. If the desired piecewise linear approximation function h (x) is not obtained (step 109: NO), the process returns to step 106 to change the magnitude of the weight λ with respect to the regularization term E _W. If λ is reduced, the approximation error is reduced, and if λ is increased, the number of sections can be expected to be reduced.

  On the other hand, when the obtained piecewise linear approximation function h (x) is within a desired range (step 109: YES), the obtained piecewise linear approximation function h (x) is displayed on the screen display unit 13; Or it outputs to the exterior from the communication I / F part 11, and the production | generation process of a series of piecewise linear approximation functions is complete | finished.

  FIG. 14 is an example of generating a piecewise linear approximation function according to the fifth embodiment. FIG. 14A is a plan view and FIG. 14B is a perspective view. Here, a piecewise linear approximation function generated from the input / output data of FIG. 13 described above is shown. Although the original number of sections was 224, it can be seen that the number of sections is reduced to 12 by this method.

[Extended embodiment]
The present invention has been described above with reference to the embodiments, but the present invention is not limited to the above embodiments. Various changes that can be understood by those skilled in the art can be made to the configuration and details of the present invention within the scope of the present invention. In addition, each embodiment can be implemented in any combination within a consistent range.

  For example, the second, third, and fourth embodiments may be applied to the fifth embodiment. In order to apply the second embodiment to the fifth embodiment, the right side D (I, J) of equation (16) may be divided by the normalization coefficient. If the larger one is adopted as the standardization coefficient, the value obtained by solving the optimization problem with the initial value 1 and substituting the obtained slope into the equation (15) and the small positive number ε will be adopted. Good.

  DESCRIPTION OF SYMBOLS 10 ... Piecewise linear approximation function generator, 11 ... Communication I / F part, 12 ... Operation input part, 13 ... Screen display part, 14 ... Memory | storage part, 14A ... Input-output data, 14B ... Node candidate, 14C ... Vertex candidate, DESCRIPTION OF SYMBOLS 15 ... Operation processing part, 15A ... Setting processing part, 15B ... Optimization problem formulation part, 15C ... Optimization processing part, 15D ... Piecewise linear approximation function generation part.

Claims (7)

Piecewise linear approximation that generates a piecewise linear approximation function that approximates each section of the input / output data with a linear approximation function based on a plurality of input / output data with one-dimensional input and a plurality of preset node candidates A function generator,
An optimization problem formulation unit that formulates an optimization problem for obtaining a slope and an intercept for each of the linear approximation functions based on the input / output data and the node candidates;
An optimization processing unit for obtaining a solution to the optimization problem by executing an optimization operation;
A piecewise linear approximation function generator that generates each of the linear approximation functions based on the obtained solution;
The optimization problem formulation unit is:
An error term representing an approximation error of the piecewise linear approximation function for the input / output data is formulated as one or both of an objective function and a constraint condition related to the optimization problem,
A regularization term representing a slope difference related to a linear approximation function of an adjacent section of the sections is formulated as one or both of an objective function and a constraint condition regarding the optimization problem,
A piecewise linear approximation function generation device characterized in that an approximation value by a linear approximation function of the adjacent pieces coincides at a node as a constraint condition related to the optimization problem. Piecewise linear approximation that generates a piecewise linear approximation function that approximates each piece of the input / output data with a linear approximation function based on a plurality of input / output data that is multidimensional and a plurality of preset vertex candidates A function generator,
An optimization problem formulation unit that formulates an optimization problem for obtaining a slope and an intercept for each of the linear approximation functions based on the input / output data and the vertex candidates;
An optimization processing unit for obtaining a solution to the optimization problem by executing an optimization operation;
A piecewise linear approximation function generator that generates each of the linear approximation functions based on the obtained solution;
The optimization problem formulation unit is:
An error term representing an approximation error of the piecewise linear approximation function for the input / output data is formulated as one or both of an objective function and a constraint condition related to the optimization problem,
A regularization term representing a slope difference related to a linear approximation function of an adjacent section of the sections is formulated as one or both of an objective function and a constraint condition regarding the optimization problem,
The piecewise linear approximation function generation device characterized in that the approximation value by the linear approximation function of the adjacent sections coincides at a vertex as a constraint condition regarding the optimization problem. The piecewise linear approximation function generation device according to claim 1,
The piecewise linear approximation function generation device, wherein the regularization term includes an L1 norm related to all of the slope differences. The piecewise linear approximation function generation device according to claim 2,
2. The piecewise linear approximation function generation device according to claim 1, wherein the regularization term includes a sum of L2 norms of the slope difference. In the piecewise linear approximation function generation device according to claim 3 or 4,
A standardization unit that standardizes the regularization term using a predetermined coefficient provided for each of the slope differences;
Each of the coefficients includes a corresponding absolute value of the slope difference obtained by solving the optimization problem on the assumption that the coefficient is 1. The piecewise linear approximation function generation device, wherein: Piecewise linear approximation that generates a piecewise linear approximation function that approximates each section of the input / output data with a linear approximation function based on a plurality of input / output data with one-dimensional input and a plurality of preset node candidates A function generation method,
An optimization problem formulation step for formulating an optimization problem for obtaining a slope and an intercept for each of the linear approximation functions based on the input / output data and the node candidates;
An optimization processing step for obtaining a solution of the optimization problem by executing an optimization operation;
A piecewise linear approximation function generation unit including a piecewise linear approximation function generation step of generating each of the linear approximation functions based on the obtained solution;
The optimization problem formulation step includes:
Formulating an error term representing an approximation error of the piecewise linear approximation function for the input / output data as one or both of an objective function and a constraint on the optimization problem;
Formulating a regularization term representing a slope difference related to a linear approximation function of an adjacent section of the sections as one or both of an objective function and a constraint condition related to the optimization problem;
A method of generating a piecewise linear approximation function, comprising: formulating, as a constraint condition on the optimization problem, that approximate values obtained by linear approximation functions of the adjacent pieces coincide at nodes. Piecewise linear approximation that generates a piecewise linear approximation function that approximates each piece of the input / output data with a linear approximation function based on a plurality of input / output data that is multidimensional and a plurality of preset vertex candidates A function generation method,
An optimization problem formulation unit that formulates an optimization problem for obtaining a slope and an intercept for each of the linear approximation functions based on the input / output data and the vertex candidates; and
An optimization processing step for obtaining a solution of the optimization problem by executing an optimization operation;
A piecewise linear approximation function generation unit including a piecewise linear approximation function generation step of generating each of the linear approximation functions based on the obtained solution;
The optimization problem formulation step includes:
Formulating an error term representing an approximation error of the piecewise linear approximation function for the input / output data as one or both of an objective function and a constraint on the optimization problem;
Formulating a regularization term representing a slope difference related to a linear approximation function of an adjacent section of the sections as one or both of an objective function and a constraint condition related to the optimization problem;
A method of generating a piecewise linear approximation function, comprising: formulating, as a constraint condition on the optimization problem, that approximate values obtained by linear approximation functions of the adjacent pieces coincide at a vertex.

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