WO2017145664A1

WO2017145664A1 - Optimization system, optimization method, and optimization program

Info

Publication number: WO2017145664A1
Application number: PCT/JP2017/003358
Authority: WO
Inventors: 優輔村岡; 遼平藤巻
Original assignee: 日本電気株式会社
Priority date: 2016-02-26
Filing date: 2017-01-31
Publication date: 2017-08-31
Also published as: JP6973370B2; JPWO2017145664A1

Abstract

A sampling means 81 defines a distribution of uncertainty variables according to the non-Gaussian distribution by a copula function and a peripheral distribution and generates samples from the defined distribution. An optimization means 82 uses the generated sample to solve a robust optimization problem including the uncertainty variables.

Description

Optimization system, optimization method and optimization program

The present invention relates to an optimization system, an optimization method, and an optimization program for optimizing a problem including an uncertain variable.

In recent practical operations research, optimization functions defined using many uncertain inputs generated from stochastic processes, such as prediction values generated by machine learning and observation results from sensors including noise Or there is a scene where an optimization problem including constraints is solved. In such a case, there is a known problem that even if the change in the input value is small, the optimum result can change greatly.

Robust optimization is one of the most effective approaches to alleviate such uncertainties, and the worst case scenario for a set of slidable input values (hereinafter sometimes referred to as uncertain sets). This is a method of obtaining a robust solution by optimizing the objective function when assumed.

In fact, an uncertain set of elliptical ranges is particularly important, which corresponds to a multivariate Gaussian generation process with uncertain inputs. Also, several methods for solving a robust optimization problem using an uncertain set of elliptic ranges are known (for example, see Non-Patent Document 1).

Note that Non-Patent Document 2 describes an approximate solution for robust optimization by sampling. Non-Patent Document 3 describes a method of applying robust optimization to a portfolio.

The portfolio optimization described above is a typical problem where risk is very important for obtaining stable results. For example, according to the description of Non-Patent Document 3, if x _d is the investment weight of asset d and ζ _d is the return ratio of asset d, the robust optimization problem is formulated as The

In Equation 1, Ζ is an uncertain (data) set and satisfies Equation 2 below.

This uncertain set corresponds to the confidence interval of the Gaussian distribution N (μ, Σ). Here, by using the method described in Non-Patent Document 1, this optimization problem can be solved as SOCP (second 方法 order 記載 cone programming).

However, since the return ratio is thought to be distributed from a lognormal distribution, it was difficult to identify an appropriate risk when using this Gaussian uncertainty set. For this reason, it is more preferable to solve the uncertain set として as a confidence interval of a distribution whose peripheral distribution is a lognormal distribution.

Also, in the case of data where the distribution model of ζ is unknown and ζ appears not to be a Gaussian distribution, it is also possible to use an empirical distribution that is a non-Gaussian distribution.

As a method for solving a general robust optimization problem, Non-Patent Document 2 describes a method for approximating a robust optimization problem by using samples from the bag. According to the method described in Non-Patent Document 2,

Assuming that, the target robust optimization problem is formulated as shown in Equation 3 below. J represents the number of constraints.

In Equation 3, f is a linear function of x, and g (x, ζ) is a convex function of x. Assume that f and g are functional forms that can be solved by any convex optimization solver for an arbitrary ζ. Assuming a “non-robust” version of this problem (specifically, a problem that excludes max _ζ from the problems described above), this problem can be solved by some convex optimization problem solver for any ζ.

In this method, first, ζ ⁽¹⁾ ,..., Ζ ^(N) are sampled from Ζ. For example, ζ ⁽¹⁾ ,..., Ζ ^(N) may be uniformly sampled from the ridges. Then, using this sample, this problem is approximated as shown in Equation 4 below.

Since all constraints g (x, ζ ^(N) ) have a similar form (eg, linear, quadratic cone, etc.) as a “non-robust” version, this problem solves the “non-robust” version. It is possible to solve using the same solver as.

However, there is no framework for solving robust optimization problems using uncertain sets based on non-Gaussian distributions. It is difficult to use the method described in Non-Patent Document 2.

Therefore, the present invention provides an optimization system, an optimization method, and an optimization program that can efficiently solve a robust optimization problem even when a correlated uncertain variable that follows a non-Gaussian distribution is assumed. Objective.

The optimization system according to the present invention includes a sampling means for defining a distribution of uncertain variables that follow a non-Gaussian distribution by a copula function and a marginal distribution, and generates a sample from the defined distribution, and an uncertain variable using the generated sample. And an optimization means for solving a robust optimization problem including:

Another optimization system according to the present invention is characterized by comprising optimization means for solving an optimization problem including uncertain variables that follow a non-Gaussian distribution by robust optimization.

The optimization method according to the present invention defines a distribution of uncertain variables according to a non-Gaussian distribution by a copula function and a marginal distribution, generates a sample from the defined distribution, and includes a robust variable including the uncertain variable using the generated sample. It is characterized by solving optimization problems.

The optimization program according to the present invention uses a sampling process for defining a distribution of uncertain variables according to a non-Gaussian distribution by a copula function and a marginal distribution, and generating a sample from the defined distribution, and the generated sample. And performing an optimization process for solving a robust optimization problem including uncertain variables.

According to the present invention, the robust optimization problem can be efficiently solved even when a correlated uncertain variable according to a non-Gaussian distribution is assumed.

It is a block diagram which shows the structural example of 1st Embodiment of the optimization system by this invention. It is a flowchart which shows the operation example of sampling of 1st Embodiment. It is a flowchart which shows the operation example of the optimization system of 1st Embodiment. It is a flowchart which shows the operation example of the sampling of 2nd Embodiment. It is a flowchart which shows the other operation example of the sampling of 2nd Embodiment. It is explanatory drawing which shows the example of distribution of the past return ratio of 2 goods. It is explanatory drawing which shows the example which produced | generated the sample. It is explanatory drawing which shows the other example which produced | generated the sample. It is a block diagram which shows the outline | summary of this invention. It is a block diagram which shows the other outline | summary of this invention.

Hereinafter, embodiments of the present invention will be described with reference to the drawings.

Embodiment 1. FIG.
FIG. 1 is a block diagram showing a configuration example of a first embodiment of an optimization system according to the present invention. The optimization system of this embodiment solves a robust optimization problem using an uncertain variable that follows a non-Gaussian distribution. The optimization system of this embodiment includes a sampling unit 10 and an optimization unit 20.

The sampling means 10 generates a sample used for optimization from an uncertain variable that follows a non-Gaussian distribution. In the present embodiment, the sampling means 10 receives the copula function 11 and the peripheral distribution 12, and generates a sample from the non-Gaussian distribution using the input copula function 11 and the peripheral distribution 12.

The sampling means 10 may read and input the copula function 11 and the peripheral distribution 12 from, for example, a storage unit (not shown) realized by a magnetic disk or the like, or an input device connected via a communication network ( You may receive and input from (not shown). In other words, the sampling means 10 can be said to be an input means (first input means) for inputting the marginal distribution 12 of uncertain variables and a copula function 11 that follow a non-Gaussian distribution.

Note that the expression format of the input copula function 11 and the peripheral distribution 12 is arbitrary. For example, the sampling means 10 may input information indicating the type and parameters of the copula function 11 and the peripheral distribution 12, or may input a function or a mathematical expression itself.

Here, according to Sklar's theorem, the distribution of a multidimensional random variable ζ is calculated by the copula function C (u ₁ ,..., U _D ) and the peripheral distribution F ₁ (ζ ₁ ),..., F _D (ζ _D ). It is possible to define. Note that ζ satisfies the following. Sklar's theorem is described in, for example, Theorem 2.2 in Reference Document 1 below. The contents of the following Reference 1 are incorporated herein as constituting a part of this specification.
<Reference 1>
Elidan, Gal, "Copula bayesian networks.", Advances in neural information processing systems, p.2, 2010.

That is, Copula function C and marginal distribution F ₁ the distribution of the _zeta, ..., can be defined by F _D. Therefore, the sampling unit 10 defines the distribution of the uncertain variable according to the non-Gaussian distribution by the input copula function 11 and the peripheral distribution 12, and generates a sample of the uncertain variable from the defined distribution. For example, the sampling unit 10 may generate a random number based on a distribution defined by the copula function 11 and the marginal distribution 12 to generate an uncertain variable sample.

The sampling means 10 can use any copula function 11 having a method for generating samples from a defined distribution. The sampling means 10 is generated by, for example, a multivariate function G (for example, a multivariate Gaussian distribution) and its peripheral distributions G ₁ ,..., G _D (for example, a Gaussian distribution marginalized from the multivariate function G) A copula as shown in Equation 5 may be used.

Further, the sampling means 10 may use any one-dimensional distribution that can calculate the cumulative distribution F ₁ (x ₁ ),..., F _D (x _D ) and an inverse function. The sampling means 10 may use, for example, a lognormal distribution, an exponential distribution, an empirical distribution, or a combination thereof.

As described in Non-Patent Document 2, since the sample size corresponds to the probability that the solution violates the constraint, the allowable risk can be controlled by the sample size. In this embodiment, a sample is not directly defined, and this sample in which the sampling means 10 considers the risk is directly used.

Hereinafter, a specific procedure in which the sampling unit 10 generates a sample will be described. Sampling means 10 includes copula function C (copula function 11), marginal distribution F ₁ ,..., F _D (marginal distribution 12), confidence level δ (hereinafter also referred to as “d”) and problem definition f, The target ζ distribution is input by g _j . The problem definition f, g _j corresponds to f, g _j used in Equation 3 described above. The confidence level δ corresponds to the confidence interval and is determined in advance.

The sampling means 10 may set the sampling number N based on, for example, an upper limit analysis based on the confidence level δ. Specifically, when it is assumed that g _j (x, δ) <= 0 in the δ confidence interval of the probability distribution, this is expressed as “g _j (x, δ)> 0 is less than δ in a certain probability distribution. In other words.

Here, when sampling is performed with a certain probability distribution, when Expression 6 shown below is satisfied, “the probability ε or less and g _j (x, δ)> 0 should be δ or less” is satisfied. It is known.
N> 1 / ε (log1 / δ + d _x ) (Formula 6)

In Equation 6, d _x is the dimension of the original problem, and ε is an appropriate small number determined according to sampling. Taking advantage of this property, the sampling means 10 may determine N that satisfies the above-described Expression 6.

The sampling means 10 first samples ζ ⁽¹⁾ ,..., Ζ ^(N) from the target ζ distribution. At this time, the sampling means 10 may generate a sample with a particularly high risk based on the problem definitions f and g _j . As a method for generating a sample with a high risk, the sampling means 10 may sample a sample with a high probability density × loss that causes an error by, for example, importance sampling. The importance sampling is described in Reference Document 2 below, for example. The contents of the following Reference 2 are incorporated herein as constituting a part of this specification.
<Reference 2>
Glynn, Peter W and Iglehart, Donald L, "Importance sampling for stochastic simulations", Management Science, INFORMS, vol.35, No. 11, p.1367-1392, 1989
Then, for example, in the case of a copula function based on the distribution function G of the formula 5 above, sampling means 10, by the processing of step S13 from step S11 illustrated below, zeta ^(1), ..., sampling the zeta ^(N) it can. FIG. 2 is a flowchart illustrating an example of the sampling operation according to the first embodiment.

First, the sampling means 10 samples t ⁽ⁿ⁾ from the distribution function G (step S11). Next, the sampling means 10 calculates u _d ⁽ⁿ⁾ = G _d ⁻¹ (t _(n) ) for each d from 1 to D (step S12). Here, G _d is the marginal distribution of the d-th variable of G, and D is the number of marginal distributions. Then, the sampling means 10 calculates ζ ⁽ⁿ⁾ = [F ₁ ⁻¹ (u ₁ ⁽ⁿ⁾ ),..., F _D ⁻¹ (u _D ⁽ⁿ⁾ )] (step S13).

The optimization means 20 receives the uncertain variable sample 21 generated by the sampling means 10 and the optimization problem 22, and uses the input sample 21 to solve the optimization problem 22 by robust optimization. The optimization problem 22 is an optimization problem (f (x, ζ)) including uncertain variables, and is defined in advance by a user or the like. In other words, the optimization unit 20 has a function of solving an optimization problem including an uncertain variable that follows a non-Gaussian distribution by robust optimization. Moreover, since the optimization means 20 inputs the optimization problem 22, it can be said that it also functions as an input means (second input means).

The method by which the optimization means 20 solves the robust optimization problem is arbitrary. The optimization unit 20 may solve the robust optimization problem using, for example, a modification of the problem described in Non-Patent Document 2. For example, following the operation of the sampling illustrated in FIG. 2, the optimization means 20, as shown in Equation 4 above, samples zeta ^(1), ..., it may be converted problems with zeta ^(N). That is, the optimization means 20 may convert the problem into a “non-robust” version of the problem with these samples. The optimization unit 20 can obtain an optimization result by solving the converted problem.

The sampling means 10 and the optimization means 20 are realized by a CPU of a computer that operates according to a program (optimization program). For example, the program may be stored in a storage unit (not shown) included in the optimization system, and the CPU may read the program and operate as the sampling unit 10 and the optimization unit 20 according to the program. Further, the function of the optimization system may be provided in SaaS (Software as Service) format.

Further, each of the sampling means 10 and the optimization means 20 may be realized by dedicated hardware. Moreover, a part or all of each component of each device may be realized by a general-purpose or dedicated circuit (circuitry), a processor, or a combination thereof. These may be configured by a single chip or may be configured by a plurality of chips connected via a bus. Part or all of each component of each device may be realized by a combination of the above-described circuit and the like and a program.

In addition, when some or all of the constituent elements of each device are realized by a plurality of information processing devices and circuits, the plurality of information processing devices and circuits may be arranged in a concentrated manner or distributedly arranged. May be. For example, the information processing apparatus, the circuit, and the like may be realized as a form in which each is connected via a communication network, such as a client and server system and a cloud computing system.

Next, an operation example of this embodiment will be described. FIG. 3 is a flowchart illustrating an operation example of the optimization system according to the first embodiment. The sampling means 10 inputs the copula function 11 and the marginal distribution 12 (step S21). The sampling means 10 defines a distribution of uncertain variables based on the inputted copula function and the peripheral distribution (step S22), and generates a sample from the defined distribution (step S23). Then, the optimization unit 20 solves the robust optimization problem including the uncertain variable using the generated sample (step S24), and outputs the optimal solution.

As described above, according to this embodiment, the sampling means 10 defines the distribution of uncertain variables according to the non-Gaussian distribution by the copula function 11 and the peripheral distribution 12, and generates a sample from the defined distribution. Then, the optimization unit 20 solves the robust optimization problem including the uncertain variable using the generated sample. That is, the optimization means 20 solves an optimization problem including an uncertain variable that follows a non-Gaussian distribution by robust optimization. Therefore, the robust optimization problem can be efficiently solved even when an uncertain correlated variable that follows a non-Gaussian distribution is assumed.

For example, in the method described in Non-Patent Document 2, a sample that approximates a set is generated after defining a set of certain uncertain variables. However, in the robust optimization problem using uncertain variables that follow a non-Gaussian distribution as described in this embodiment, the figure of the set of corresponding uncertain variables such as an ellipse assumed in the case of a normal distribution is unknown. is there. For this reason, it has been difficult to apply a general sampling method from a non-Gaussian distribution to a method as described in Non-Patent Document 2 (sampling approximation method for robust optimization).

However, in this embodiment, instead of using the method described in Non-Patent Document 2, a sample corresponding to the risk of non-Gaussian distribution is obtained by directly sampling from non-Gaussian distribution. Therefore, even when a correlated uncertain variable that follows a non-Gaussian distribution is assumed, the robust optimization problem can be efficiently solved.

Embodiment 2. FIG.
Next, a second embodiment of the optimization system according to the present invention will be described. In the first embodiment, the optimization system performs sampling from the entire non-Gaussian distribution and performs robust optimization based on the sampling. In the present embodiment, a method for performing robust optimization by controlling the magnitude of the probability that an uncertain variable is included in a non-Gaussian distribution in order to improve the accuracy of the sample will be described.

The configuration of this embodiment is the same as that of the first embodiment. However, in this embodiment, in order to control the probability of the range to be considered by sampling, a copula function 11 defined based on a distribution function that can define a multidimensional confidence interval is assumed. An example of such a copula function is a normal copula (Gaussian copula). For example, when the distribution function is a multidimensional normal distribution, the confidence interval can be defined as an ellipse.

Hereinafter, the operation of the sampling means 10 of the present embodiment will be described by taking as an example the case where a copula based on the multivariate distribution function G of Equation 5 described above is used. FIG. 4 is a flowchart illustrating an example of sampling operation according to the second embodiment.

First, the sampling means 10 samples t ⁽ⁿ⁾ uniformly from the curved surface of the confidence interval of the multivariate distribution function G (step S31). Here, the curved surface of the confidence interval indicates a boundary in a multidimensional space determined based on the G confidence interval.

Thereafter, the sampling means 10 calculates u _d ⁽ⁿ⁾ = G _d ⁻¹ (t _(n) ) for each d from 1 to D, similarly to the processing from step S12 to step S13 shown in FIG. (Step S12). ζ ⁽ⁿ⁾ = [F ₁ ⁻¹ (u ₁ ⁽ⁿ⁾ ),..., F _D ⁻¹ (u _D ⁽ⁿ⁾ )] is calculated (step S13).

Next, the operation of the sampling means 10 of this embodiment will be further described by taking as an example the case where the multivariate distribution function G is a multivariate Gaussian distribution (G = N (μ, Σ)). FIG. 5 is a flowchart illustrating another example of the sampling operation according to the second embodiment.

First, the sampling means 10 calculates a confidence level corresponding to G and δ as an ellipse set shown in Expression 7 below (step S41). Here, ε is a quantity representing a risk that can permit a constraint violation, and specifically, is set so that the square of ε becomes a point of δ percent of the chi-square distribution.

Next, the sampling means 10 samples t ⁽ⁿ⁾ uniformly from the curved surface determined based on the ellipse set (step S42). Hereinafter, the processing until ζ ⁽ⁿ⁾ is calculated is the same as the processing from step S12 to step S13 in FIG.

As described above, according to the present embodiment, the sampling unit 10 generates a sample from the curved surface of the confidence interval of the defined distribution function. The distribution function is a function that can define a multidimensional confidence interval. Therefore, in addition to the effect of the first embodiment, the probability of the range to be considered by sampling can be controlled, so that the accuracy of the optimization result can be further improved.

Specifically, the sampling means 10 may define a confidence interval of a normal copula and generate a sample from the curved surface of the confidence interval of the defined distribution function. The sampling means 10 may generate samples uniformly from the curved surface of the confidence interval.

Hereinafter, the operation of the optimization system of this embodiment will be described using a specific example. Hereinafter, a method for solving a portfolio optimization problem using the optimization system of the present embodiment will be described in detail. The portfolio optimization problem is an issue where correlation is very important.

◇ Return ratio (= future price / current price) is an index considered for investment. Here, considering the investment amount and return ratio of two products (product 1 and product 2), it can be said that the portfolio optimization problem solves the problem shown in the following Expression 8. This problem corresponds to the optimization problem 22 of the first embodiment.

min_ {Investment amount for product 1, investment amount for product 2}
-(Return ratio of product 1) x (Investment amount in product 1)
-(Return ratio of product 2) x (Investment amount in product 2) (Formula 8)

However, (current price of product 1) x (amount of investment in product 1)
+ (Current price of product 2) x (investment amount in product 2) <= (budget)

Since the return ratio is an uncertain index calculated from the future price, it is preferable to solve this problem as a robust optimization problem. On the other hand, it is known that the return ratio follows a lognormal distribution. FIG. 6 is an explanatory diagram illustrating an example of a distribution of past return ratios of two products. In the graph illustrated in FIG. 6 (and FIG. 7 and FIG. 8 described later), the x-axis represents the return ratio of the product 1, and the y-axis represents the return ratio of the product 2. This distribution is data obtained by observing past data. In this specific example, it is assumed that the past return ratio of two products has a distribution illustrated in FIG. Further, it is assumed that a range A shown in FIG. 6 is a range including a variable with a probability of 50%.

For example, a Gaussian copula parameter θ = −0.79. This corresponds to the copula function 11 of the first embodiment. The log normal distribution of the return ratio of product 1 (hereinafter referred to as variable 1) and the log normal distribution of the return ratio of product 2 (hereinafter referred to as variable 2) are estimated as follows, for example. Let's say. This corresponds to the peripheral distribution 12 of the first embodiment.
Lognormal distribution of variable 1 (loc, scale, shape) = (-0.18,0.97,0.84)
Lognormal distribution of variable 2 (loc, scale, shape) = (-0.52,0.56,0.43)

Sampling means 10 inputs such information and generates a sample. FIG. 7 is an explanatory diagram showing an example in which 100 samples are generated from FIG. This sample corresponds to the sample 21 of the first embodiment.

Further, as shown in the second embodiment, the sampling unit 10 may generate a sample based on a Gaussian confidence interval of a Gaussian copula. FIG. 8 is an explanatory diagram illustrating another example in which 100 samples are generated from the samples illustrated in FIG. 6 based on the 50% confidence interval. Compared with the generation of the sample illustrated in FIG. 7, it can be seen that the sample illustrated in FIG. 8 is uniformly generated along the curved surface of the confidence interval.

Optimizer 20 inputs the above-described sample and the problem shown in Equation 7 and solves this optimization problem. For example, according to the method described in Non-Patent Document 2, the optimization unit 20 creates an optimization problem in which an expression obtained by substituting the value of the sample generated for the return ratio is added as a constraint. And the optimization means 20 solves a problem using a linear programming solver. Thereby, for example, an optimization result that the investment amount of the product 1 = 12 and the investment amount of the product 2 = 88 is obtained.

Next, the outline of the present invention will be described. FIG. 9 is a block diagram showing an outline of the present invention. The optimization system according to the present invention defines a distribution of uncertain variables according to a non-Gaussian distribution by a copula function and a marginal distribution, and a sampling means 81 (for example, sampling means 10) that generates a sample from the defined distribution. Optimization means 82 (for example, optimization means 20) for solving a robust optimization problem including uncertain variables using the obtained samples.

Such a configuration can efficiently solve the robust optimization problem even when an uncertain correlated variable that follows a non-Gaussian distribution is assumed.

Also, the distribution function may be a function that can define a multidimensional confidence interval. At this time, the sampling means 81 may generate a sample from the curved surface of the confidence interval of the defined distribution function. With such a configuration, the probability of the range to be considered by sampling can be controlled, so that the accuracy of the optimization result can be further improved.

Specifically, the sampling unit 81 may define a confidence interval of a normal copula and generate a sample from the curved surface of the confidence interval of the defined distribution function. The sampling unit 81 may generate samples uniformly from the curved surface of the confidence interval.

FIG. 10 is a block diagram showing another outline of the present invention. Another optimization system according to the present invention includes optimization means 91 (for example, optimization means 20) that solves an optimization problem including uncertain variables that follow a non-Gaussian distribution by robust optimization. Even with such a configuration, it is possible to efficiently solve the robust optimization problem even when an uncertain correlated variable according to a non-Gaussian distribution is assumed.

Specifically, the optimization system inputs first input means (for example, sampling means 10) for inputting a marginal distribution of uncertain variables and a copula function according to a non-Gaussian distribution, and an optimization problem including the uncertain variables. The second input means (for example, the optimization means 20 may be provided. The optimization means 91 solves the optimization problem by the robust optimization using the samples generated from the copula function and the marginal distribution. Also good.

As mentioned above, although this invention was demonstrated with reference to embodiment and an Example, this invention is not limited to the said embodiment and Example. 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.

This application claims priority based on Japanese Patent Application 2016-035527 filed on February 26, 2016, the entire disclosure of which is incorporated herein.

Since it is considered that indices handled in the financial relationship follow a non-Gaussian distribution, the present invention is suitably applied to, for example, an optimization system that solves an optimization problem using these indices.

10 Sampling means 20 Optimization means

Claims

A sampling means for defining a distribution of uncertain variables following a non-Gaussian distribution by a copula function and a marginal distribution, and generating a sample from the defined distribution;
An optimization system comprising: an optimization unit that solves a robust optimization problem including the uncertain variable using the generated sample.
Distribution function is a function that can define a multidimensional confidence interval,
The optimization system according to claim 1, wherein the sampling unit generates a sample from the curved surface of the confidence interval of the defined distribution function.
The optimization system according to claim 1, wherein the sampling unit defines a confidence interval of a normal copula and generates a sample from a curved surface of a confidence interval of a defined distribution function.
The optimization system according to claim 2, wherein the sampling unit uniformly generates a sample from the curved surface of the confidence interval.
An optimization system characterized by having an optimization means for solving optimization problems including uncertain variables that follow a non-Gaussian distribution by robust optimization.
A first input means for inputting a marginal distribution of uncertain variables following a non-Gaussian distribution and a copula function;
Second input means for inputting an optimization problem including the uncertain variable,
The optimization system according to claim 5, wherein the optimization unit solves the optimization problem by robust optimization using the sample generated from the copula function and the marginal distribution.
Define a distribution of uncertain variables that follow a non-Gaussian distribution with a copula function and a marginal distribution, generate a sample from the defined distribution,
An optimization method characterized by solving a robust optimization problem including the uncertain variable using a generated sample.
The optimization method according to claim 7, wherein the distribution function is a function capable of defining a multidimensional confidence interval, and a sample is generated from a curved surface of the confidence interval of the distribution function to be defined.
On the computer,
A sampling process that defines a distribution of uncertain variables that follow a non-Gaussian distribution with a copula function and a marginal distribution, and generates a sample from the defined distribution; and
The optimization program for performing the optimization process which solves the robust optimization problem containing the said uncertainty variable using the produced | generated sample.
Distribution function is a function that can define a multidimensional confidence interval,
On the computer,
The optimization program according to claim 9, wherein a sample is generated from a curved surface of a confidence interval of a distribution function defined by sampling processing.