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

在不确定条件下寻找最优决策是决策分析中最具挑战性的问题之一。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现有信息，这些不确定性通常由随机变量向量或随机过程在数学上表示。如果决策者对随机参数的分布有完整的信息，那么他可以得到问题中随机函数的积分的封闭形式，然后将其转化为确定性优化问题，或者替代地使用各种统计和数值积分方法，如情景法、蒙特卡罗抽样法和求积规则来开发一些近似方案，然后使用标准的线性/非线性规划代码进行求解。如果决策者没有关于随机变量分布的完整信息，情况可能会变得复杂得多。例如，如果决策者除了随机变量的范围之外没有任何其他信息，那么基于随机参数的最坏情况选择最优决策可能是合理的策略，以使风险免受不确定性的影响。这种决策框架被称为稳健优化，这在工程设计中是众所周知的，在工程设计中，优化设计必须考虑极端(尽管罕见)事件。然而，这种稳健的计划并不一定经济，因为它为防止罕见事件设置了过多的资源。从数值角度来看，由此产生的优化问题可能很难解决。一种保守程度可能较低的分布稳健优化模型，称为分布稳健优化。该模型考虑一组分布的历史数据、计算机模拟或主观判断，其中包含具有一定置信度的真实分布，并且根据最差分布而不是最差情况来选择最优决策。在本课题中，我们主要研究一类分布稳健优化问题，其中分布的矩通过捕捉随机变量的均值、标准差或相关性等部分信息来估计分布的集合。利用凸分析中的对偶理论，我们将分布稳健优化问题转化为具有半无限半定约束的数学规划问题。出现了两个基本问题：1.如果矩是从样本计算出来的，从求解MPSISDC得到的最优值和最优解(如果问题是非凸的，则为固定点)的可靠性如何？这就需要进行全面的定性和定量的统计分析。这种分析在随机规划中被称为渐近收敛分析或稳定性分析，但对于稳健或分布稳健优化的研究很少。2.如何解决MPSISDC问题？这是一个带有矩阵变量的半定和半无限约束的确定性优化问题。如果基础函数是线性的或二次的，且随机变量的支持度是多项式或半代数的，则MPSISDC可以被重塑为半定规划问题或凸二次规划问题，但这里没有假定特定的结构，因此没有现有的优化方法可以很容易地应用于求解MPSISDC。本项目旨在以MPSISDC为平台，建立一类分布稳健优化问题的渐近分析理论，并发展新的数值方法来求解它们。

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.