机会锥约束分布鲁棒优化问题的渐近收敛性研究

Chance constrained distributionally robust optimization has broad applications in financial risk optimization, engineering reliability and other fields, which has been successfully used in modelling and analysis of data-driven problems. In the classical chance constrained distributionally robust optimization model, the chance constraint is usually defined by equalities and inequalities, whereas there exist many important problems in practice with the chance constraints being defined by non-polyhedral cones, and the research on this class of distributionally robust problems is still at the exploratory stage. Due to the asymptotic convergence of problems can provide theoretical grounding for both modelling and numerical design, it is of great significance in theory and application to systematically study the asymptotic convergence of the chance constrained distributionally robust optimization problems which are defined by non-polyhedral cones. By using the theories of variational analysis, perturbation analysis and probability measure, this project is devoted to investigating the asymptotic convergence of chance conic constrained distributionally robust optimization problems which are defined by semidefinite cone, complementarity cone and equilibrium cone, respectively. For different types of the set of distributions, the qualitative and quantitative asymptotic convergence of the optimal value functions and optimal solution mappings of the problems under the perturbation of the set of distributions are established by deriving the continuity of robust probability functions, convergence of the feasible set mappings and optimality conditions of the problems. It is expected that this project will make a contribution to the study of chance constrained distributionally robust optimization problems.

机会约束分布鲁棒优化在金融风险优化和工程可靠性等领域有着广泛的应用，被成功地用于数据驱动问题的建模和分析。在经典的机会约束分布鲁棒优化模型中，机会约束通常由等式和不等式定义，而实际中很多重要问题的机会约束是由非多面体锥定义的，这类分布鲁棒优化问题的研究尚处在初步阶段。由于问题的渐近收敛性将为建模和算法设计提供理论保证，因此系统地研究由非多面体锥定义的机会约束分布鲁棒优化问题的渐近收敛性有着重要的理论和应用价值。本项目以变分分析、扰动分析和概率测度理论为工具，研究由半定锥、互补锥、均衡锥定义的机会锥约束分布鲁棒优化问题的渐近收敛性。对于不同形式的分布集合，通过研究鲁棒概率函数的连续性、可行集映射的收敛性和问题的最优性条件，建立问题的最优值函数和最优解映射在分布集合扰动下的定性和定量的渐近收敛性。期望本项目能为机会约束分布鲁棒优化问题的研究做出贡献。

随着科学技术的不断发展，出现在金融和工程技术等领域中的很多数学优化模型越来越复杂，其中参数信息的不完整性是导致模型复杂性增加的一个重要因素，如参数是随机的，参数的概率分布是不确定的，或模型中函数的表达形式是不确定的等。本项目主要建立了机会锥约束分布鲁棒优化问题的渐近收敛性分析，并对相关的锥约束随机规划问题、两阶段分布鲁棒优化问题以及风险鲁棒优化问题的稳定性进行了探讨。首先，对于机会锥约束分布鲁棒优化问题，利用已有的信息构造分布集合，建立分布集合的误差界理论；通过研究鲁棒概率函数的连续性和可行集映射的收敛性，建立问题的最优值函数和最优解映射关于分布集合扰动的渐近收敛性理论。如果把问题中的分布集合看作是参数，这里的渐近收敛性指的是定性或定量地描述参数的变化给最优值函数和最优解映射带来的影响。然后，基于机会锥约束模型的研究，本项目也对二阶锥约束随机规划问题的Karush-Kuhn-Tucker系统、带线性半定锥补偿的两阶段分布鲁棒优化问题以及由二次补偿诱导的k-阶占优约束分布鲁棒优化问题的最优值函数和最优解映射在分布集合扰动下的稳定性理论进行了研究。最后，本项目把分布鲁棒优化的思想应用到探讨带有未知函数的优化模型，包括带有未知效用函数的极大化期望模型和带有未知函数的极小化风险优化模型，研究相应鲁棒优化模型在函数集合扰动下的稳定性理论。本项目建立的结论推广了现有的稳定性理论结果。