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Distributionally Robust Optimisation With Matrix Moment Constraints: A Semi-Infinite and Semi-Definite Programming Approach

具有矩阵矩约束的分布鲁棒优化：半无限半定规划方法

基本信息

批准号：
EP/M003191/1
负责人：
Huifu Xu
金额：
$ 29.39万
依托单位：
City, University of London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2014
资助国家：
英国
起止时间：
2014 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FM003191%2F1
关键词：
Distributionally Robust Optimisation Matrix Moment

项目摘要

One of the most challenging issues in decision analysis is to find an optimal decision under uncertainty. The solvability of such a decision problem and the quality of the optimal decision rely heavily on available information on the underlying uncertainties which are often mathematically represented by a vector of random variables or a random process. If a decision maker has complete information on the distribution of the random parameters, then he can either obtain a closed form of the integral of the random functions in the problem and then convert it into a deterministic optimisation problem, or alternatively use various statistical and numerical integration approaches such as scenario method, Monte Carlo sampling method and quadrature rules to develop some approximation schemes and then solve this using standard linear/nonlinear programming codes. The situation can become far more complex if the decision maker does not have complete information on the distribution of the random variables. For instance, if the decision maker does not have any information other than the range of the random variables, then it might be a reasonable strategy to choose an optimal decision on the basis of the worst scenario of the random parameters in order to immunize the risks from the uncertainty. This kind of decision making framework is known as robust optimisation and it is well known in engineering design where an optimal design must take into account of the extreme (albeit rare) event. However, this kind of robust scheme is not necessarily economical in that it sets out excessive resources for preventing a rare event. From numerical perspective, the resulting optimization problem could be intractable. A alternative and possibly less conservative robust optimisation model, which is known as distributionally robust optimisation, is to consider a set of distributions with historical data, computer simulation or subjective judgements which contain the true distribution with certain confidence and the optimal decision is chosen on the basis of the worst distribution rather than the worst scenario.In this project, we concentrate on a class of distributionally robust optimization problems where the set of distributions is estimated through moment of random matrices which capture some partial information such as the mean value, the standard deviation or the correlation of the random variables.Through some duality theory in convex analysis, we transform the distributional robust optimization into mathematical programs with semi-infinite and semi-definite constraints (MPSISDC). Two fundamental questions arise: 1. If the moments are calculated from samples, how reliable are the optimal value and the optimal solution (or stationary points if the problem is nonconvex) obtained from solving the MPSISDC? This requires one to carry out comprehensive qualitative and quantitative statistical analysis. This kind of analysis is known as asymptotic convergence analysis or stability analysis in stochastic programming but little has been done for robust or distributionally robust optimization. 2. How do we solve the MPSISDC? This is a deterministic optimization problem which involves semi-definite and semi-infinite constraints with matrix variables. If the underlying function is linear or quadratic and the support of the random variables are polynomial or semi-algebraic, then the MPSISDC may be recast as a semi-definite programming problem or a convex conic programming problem, but here we do not assume the specific structure and hence there is no existing optimization method which can be readily applied to solve MPSISDC. This project is to use MPSISDC as a platform to establish the theory of asymptotic analysis for the class of distributionally robust optimization problems and develop novel numerical methods for solving them.

决策分析中最具挑战性的问题之一是在不确定性条件下寻求最优决策。这种决策问题的可解性和最优决策的质量在很大程度上依赖于有关潜在不确定性的可用信息，这些不确定性通常在数学上由随机变量或随机过程的向量表示。如果决策者对随机参数的分布有完整的信息，那么他可以获得问题中随机函数的积分的封闭形式，然后将其转换为确定性优化问题，或者使用各种统计和数值积分方法，例如情景方法，蒙特卡罗抽样方法和求积规则来开发一些近似方案，然后使用标准的线性/非线性规划代码来解决这个问题。如果决策者没有关于随机变量分布的完整信息，情况会变得更加复杂。例如，如果决策者除了随机变量的范围之外没有任何信息，那么根据随机参数的最坏情况选择最优决策可能是一种合理的策略，以便使风险免受不确定性的影响。这种决策框架被称为鲁棒优化，并且在工程设计中众所周知，最优设计必须考虑极端（尽管罕见）事件。然而，这种稳健的方案不一定经济，因为它设置了过多的资源来防止罕见事件。从数值的角度来看，由此产生的优化问题可能是棘手的。另一种可能不太保守的稳健优化模型，称为分布稳健优化，是考虑一组包含历史数据、计算机模拟或主观判断的分布，其中包含具有一定置信度的真实分布，并根据最坏分布而不是最坏情况选择最佳决策。在本项目中，本文研究了一类分布鲁棒优化问题，其中分布的估计是通过随机矩阵的矩来实现的，这些矩捕捉了随机变量的均值、标准差或相关性等部分信息，利用凸分析中的对偶理论，将分布鲁棒优化问题转化为半无限半定约束数学规划问题。出现了两个基本问题：1。如果矩是从样本中计算出来的，那么通过求解MPSISDC得到的最优值和最优解（或者如果问题是非凸的，则是稳定点）的可靠性如何？这就需要进行全面的定性和定量统计分析。这类分析被称为随机规划中的渐近收敛分析或稳定性分析，但很少有人做了鲁棒或分布鲁棒优化。2.如何解决MPSISDC？这是一个确定性的优化问题，它涉及半定和半无限的约束与矩阵变量。如果底层函数是线性或二次的，并且随机变量的支持是多项式或半代数的，则MPSISDC可以被改写为半定规划问题或凸锥规划问题，但是这里我们不假设特定的结构，因此没有现有的优化方法可以容易地应用于求解MPSISDC。本计画以MPSISDC为平台，建立分布鲁棒最佳化问题的渐近分析理论，并发展新的数值方法来求解。