WO2017145664A1 - Système, procédé et programme d'optimisation - Google Patents

Système, procédé et programme d'optimisation Download PDF

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WO2017145664A1
WO2017145664A1 PCT/JP2017/003358 JP2017003358W WO2017145664A1 WO 2017145664 A1 WO2017145664 A1 WO 2017145664A1 JP 2017003358 W JP2017003358 W JP 2017003358W WO 2017145664 A1 WO2017145664 A1 WO 2017145664A1
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distribution
optimization
sample
function
sampling
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PCT/JP2017/003358
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Japanese (ja)
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優輔 村岡
遼平 藤巻
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日本電気株式会社
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q10/00—Administration; Management
    • G06Q10/04—Forecasting or optimisation specially adapted for administrative or management purposes, e.g. linear programming or "cutting stock problem"
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q10/00—Administration; Management
    • G06Q10/06—Resources, workflows, human or project management; Enterprise or organisation planning; Enterprise or organisation modelling

Definitions

  • the present invention relates to an optimization system, an optimization method, and an optimization program for optimizing a problem including an uncertain variable.
  • 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.
  • an uncertain set of elliptical ranges is particularly important, which corresponds to a multivariate Gaussian generation process with uncertain inputs.
  • several methods for solving a robust optimization problem using an uncertain set of elliptic ranges are known (for example, see Non-Patent Document 1).
  • 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.
  • Equation 1 ⁇ is an uncertain (data) set and satisfies Equation 2 below.
  • This uncertain set corresponds to the confidence interval of the Gaussian distribution N ( ⁇ , ⁇ ).
  • this optimization problem can be solved as SOCP (second ⁇ order ⁇ cone programming).
  • 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.
  • Equation 3 f is a linear function of x, and g (x, ⁇ ) is a convex function of x.
  • f and g are functional forms that can be solved by any convex optimization solver for an arbitrary ⁇ .
  • this problem can be solved by some convex optimization problem solver for any ⁇ .
  • Equation 4 ⁇ (1) ,..., ⁇ (N) are sampled from ⁇ .
  • ⁇ (1) ,..., ⁇ (N) may be uniformly sampled from the ridges. Then, using this sample, this problem is approximated as shown in Equation 4 below.
  • Non-Patent Document 2 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.
  • 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 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.
  • an optimization means for solving a robust optimization problem including:
  • Another optimization system 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.
  • the robust optimization problem can be efficiently solved even when a correlated uncertain variable according to a non-Gaussian distribution is assumed.
  • 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.
  • 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).
  • 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.
  • the expression format of the input copula function 11 and the peripheral distribution 12 is arbitrary.
  • 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.
  • the distribution of a multidimensional random variable ⁇ is calculated by the copula function C (u 1 ,..., U D ) and the peripheral distribution F 1 ( ⁇ 1 ),..., F D ( ⁇ D ). It is possible to define. Note that ⁇ 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.
  • 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.
  • 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)
  • G for example, a multivariate Gaussian distribution
  • G D for example, a Gaussian distribution marginalized from the multivariate function G
  • a copula as shown in Equation 5 may be used.
  • 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.
  • the allowable risk can be controlled by the sample size.
  • a sample is not directly defined, and this sample in which the sampling means 10 considers the risk is directly used.
  • Sampling means 10 includes copula function C (copula function 11), marginal distribution F 1 ,..., F D (marginal distribution 12), confidence level ⁇ (hereinafter also referred to as “d”) and problem definition f,
  • the target ⁇ 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 ⁇ corresponds to the confidence interval and is determined in advance.
  • Equation 6 d x is the dimension of the original problem, and ⁇ 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 ⁇ (1) ,..., ⁇ (N) from the target ⁇ 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.
  • FIG. 2 is a flowchart illustrating an example of the sampling operation according to the first embodiment.
  • the sampling means 10 samples t (n) from the distribution function G (step S11).
  • G d is the marginal distribution of the d-th variable of G
  • D is the number of marginal distributions.
  • 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, ⁇ )) including uncertain variables, and is defined in advance by a user or the like.
  • 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.
  • 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).
  • 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.
  • the function of the optimization system may be provided in SaaS (Software as Service) format.
  • each of the sampling means 10 and the optimization means 20 may be realized by dedicated hardware.
  • 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.
  • each device 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.
  • 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.
  • 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).
  • the optimization unit 20 solves the robust optimization problem including the uncertain variable using the generated sample (step S24), and outputs the optimal solution.
  • 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.
  • Non-Patent Document 2 a sample that approximates a set is generated after defining a set of certain uncertain variables.
  • 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).
  • Non-Patent Document 2 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 a second embodiment of the optimization system according to the present invention will be described.
  • the optimization system performs sampling from the entire non-Gaussian distribution and performs robust optimization based on the sampling.
  • 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.
  • 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).
  • the confidence interval can be defined as an ellipse.
  • FIG. 4 is a flowchart illustrating an example of sampling operation according to the second embodiment.
  • the sampling means 10 samples t (n) uniformly from the curved surface of the confidence interval of the multivariate distribution function G (step S31).
  • the curved surface of the confidence interval indicates a boundary in a multidimensional space determined based on the G confidence interval.
  • ⁇ (n) [F 1 ⁇ 1 (u 1 (n) ),..., F D ⁇ 1 (u D (n) )] is calculated (step S13).
  • FIG. 5 is a flowchart illustrating another example of the sampling operation according to the second embodiment.
  • the sampling means 10 calculates a confidence level corresponding to G and ⁇ as an ellipse set shown in Expression 7 below (step S41).
  • ⁇ is a quantity representing a risk that can permit a constraint violation, and specifically, is set so that the square of ⁇ becomes a point of ⁇ percent of the chi-square distribution.
  • the sampling means 10 samples t (n) uniformly from the curved surface determined based on the ellipse set (step S42).
  • the processing until ⁇ (n) is calculated is the same as the processing from step S12 to step S13 in FIG.
  • 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.
  • 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.
  • the portfolio optimization problem is an issue where correlation is very important.
  • FIG. 6 is an explanatory diagram illustrating an example of a distribution of past return ratios of two products.
  • the x-axis represents the return ratio of the product 1
  • the y-axis represents the return ratio of the product 2.
  • This distribution is data obtained by observing past data.
  • a range A shown in FIG. 6 is a range including a variable with a probability of 50%.
  • variable 1 log normal distribution of the return ratio of product 1
  • variable 2 log normal distribution of the return ratio of product 2
  • 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.
  • 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.
  • 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.
  • 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.
  • the distribution function may be a function that can define a multidimensional confidence interval.
  • the sampling means 81 may generate a sample from the curved surface of the confidence interval of the defined distribution function.
  • 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.
  • 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.
  • the present invention is suitably applied to, for example, an optimization system that solves an optimization problem using these indices.

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Abstract

Un moyen d'échantillonnage (81) définit une distribution de variables d'incertitude selon la distribution non gaussienne au moyen d'une fonction de copule et d'une distribution périphérique et génère des échantillons à partir de la distribution définie. Un moyen d'optimisation (82) utilise l'échantillon généré pour résoudre un problème d'optimisation robuste comprenant les variables d'incertitude.
PCT/JP2017/003358 2016-02-26 2017-01-31 Système, procédé et programme d'optimisation WO2017145664A1 (fr)

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Cited By (3)

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CN107749638A (zh) * 2017-10-19 2018-03-02 东南大学 多微电网组合的虚拟发电厂分布式随机非重叠抽样的无中心优化方法
CN110097263A (zh) * 2019-04-18 2019-08-06 新奥数能科技有限公司 综合能源系统的设备调控方法及装置
CN111652445A (zh) * 2020-06-11 2020-09-11 广东科创工程技术有限公司 基于高斯分布的污水设备优化运行控制方法

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JP2003108753A (ja) * 2001-09-28 2003-04-11 Tokai Bank Ltd 金融機関のリスク管理システム及びそれを用いた処理方法
WO2006035507A1 (fr) * 2004-09-29 2006-04-06 National Institute Of Information And Communications Technology Systeme d’aide au courtage de titres
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US8170941B1 (en) * 2008-10-16 2012-05-01 Finanalytica, Inc. System and method for generating random vectors for estimating portfolio risk
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JP2002183111A (ja) * 2000-12-13 2002-06-28 Yamatake Corp 曲面モデルの同定方法及びプログラム
JP2003108753A (ja) * 2001-09-28 2003-04-11 Tokai Bank Ltd 金融機関のリスク管理システム及びそれを用いた処理方法
WO2006035507A1 (fr) * 2004-09-29 2006-04-06 National Institute Of Information And Communications Technology Systeme d’aide au courtage de titres
JP2006268558A (ja) * 2005-03-24 2006-10-05 Yamatake Corp データ処理方法及びプログラム
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Publication number Priority date Publication date Assignee Title
CN107749638A (zh) * 2017-10-19 2018-03-02 东南大学 多微电网组合的虚拟发电厂分布式随机非重叠抽样的无中心优化方法
CN107749638B (zh) * 2017-10-19 2021-02-02 东南大学 多微电网组合的虚拟发电厂分布式随机非重叠抽样的无中心优化方法
CN110097263A (zh) * 2019-04-18 2019-08-06 新奥数能科技有限公司 综合能源系统的设备调控方法及装置
CN111652445A (zh) * 2020-06-11 2020-09-11 广东科创工程技术有限公司 基于高斯分布的污水设备优化运行控制方法
CN111652445B (zh) * 2020-06-11 2024-03-22 广东科创智水科技有限公司 基于高斯分布的污水设备优化运行控制方法

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