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

决策分析中最具挑战性的问题之一是如何在不确定的情况下找到最优决策。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向量或随机过程表示。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积分的封闭形式，然后将其转化为确定性优化问题，也可以使用各种统计和数值积分方法，如情景法，利用蒙特卡罗采样法和正交规则制定一些近似格式，然后用标准线性/非线性规划代码求解。如果决策者没有关于随机变量分布的完整信息，情况就会变得复杂得多。例如，如果决策者除了随机变量的范围之外没有任何信息，那么根据随机参数的最坏情况选择最优决策，以使风险免受不确定性的影响，可能是一种合理的策略。这种决策框架被称为鲁棒优化，它在工程设计中众所周知，其中最优设计必须考虑到极端（尽管罕见）事件。然而，这种健壮的方案并不一定经济，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是棘手的。另一种可能不那么保守的鲁棒优化模型被称为分布鲁棒优化，它是考虑一组具有历史数据、计算机模拟或主观判断的分布，这些分布包含具有一定置信度的真实分布，并且根据最坏分布而不是最坏情况选择最优决策。在这个项目中，我们专注于一类分布鲁棒优化问题，其中分布集是通过随机矩阵的矩来估计的，随机矩阵捕获了一些部分信息，如均值、标准差或随机变量的相关性。利用凸分析中的对偶理论，将分布鲁棒优化问题转化为具有半无限和半确定约束的数学规划。这就产生了两个基本问题：如果力矩是从样本中计算的，那么通过求解MPSISDC获得的最优值和最优解（如果问题是非凸的，则为平稳点）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，但对于鲁棒或分布鲁棒优化的研究很少。2. 如何解决MPSISDC？这是一个包含矩阵变量的半定约束和半无限约束的确定性优化问题。如果底层函数是线性的或二次的，而随机变量的支持是多项式的或半代数的，那么MPSISDC可能被重新塑造为半确定规划问题或凸二次规划问题，但在这里我们没有假设特定的结构，因此没有现成的优化方法可以很容易地应用于解决MPSISDC。本课题以MPSISDC为平台，建立分布鲁棒优化问题的渐近分析理论，并开发求解分布鲁棒优化问题的新数值方法。

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.