约束集值优化问题的适定性研究及相关分析

This project focuses on some arresting new issues in the avenue of well-posedness for constrained set-valued optimization problems and related problems, and its applications to Nash equilibrium problems. Around these new problems, we will concentrate on the following issues: we investigate sufficient condition of Levitin-Polyak well-posedness for constrained set-valued optimization problems, and optimize the conditions ensuring the conclusions tenable ; taking advantage of subdifferential of set-valued mappings and Ekeland variational principle, we investigate sufficient condition of Hadamard well-posedness for constrained set-valued optimization problems and constrained vector variational inequalities , furthermore, we discuss the equivalence of between the Hadamard well-posedness of the two problems; we study the relationships between well-posedness for constrained set-valued optimization problems and stability(upper semicontinuity, lower semicontinuity and Lipschitz continuity) of their solution mappings(optimal value mappings). As applications, we will investigate sufficient condition of well-posedness for Nash equilibrium problems, and analysis the stability of its predictions. This project will not only enrich and promote theoretical research of well-posedness in optimization field, but also provide new theoretical basis for the stability of Nash equilibrium prediction analysis method.

本项目主要考虑约束集值优化问题及相关问题的适定性中值得研究的新问题，及其在纳什均衡问题适定性分析上的应用。围绕这些新问题，我们的研究内容有：建立研究约束集值优化问题的Levitin-Polyak适定性成立的充分性条件，并优化结论成立的条件；借助集值映射次微分，利用Ekeland变分原理，讨论约束集值优化问题和约束向量变分不等式问题的Hadamard适定性成立的充分性条件，建立二者Hadamard适定性之间的等价关系；分析约束集值优化问题适定性与其解集映射（最优值映射）上、下半连续性及Lipschitz连续性之间的等价关系，建立约束集值优化问题适定性的度量刻画。作为应用，研究纳什均衡问题的适定性成立的充分性条件，判断其预测分析方法的稳定性。本项目的开展不仅可以丰富和促进优化领域适定性的理论研究，还可为纳什均衡预测分析方法的稳定性提供新的理论依据。

本项目研究了具有集值约束条件的集值优化问题及相关问题的Hadamard适定性及Levitin-Polyak 适定性成立的充分性条件，建立其与约束向量变分不等式问题Hadamard适定性之间的等价关系；研究了约束集值优化问题适定性与其解集映射（最优值映射）上、下半连续性及Lipschitz连续性之间的等价关系，建立了约束集值优化问题适定性的度量刻画；研究了纳什均衡问题的适定性成立的充分性条件；研究了参数优化问题的对偶性及灵敏性分析；研究了解约束优化问题及相关问题的算法；研究了解非线性随机变分不等式问题的一种关于正则化间隙函数最小绝对偏差与最小二乘法的凸组合期望法。到目前为止，已在国内外重要学术刊物上发表论文15篇，接受论文1篇，2篇论文正在审稿中，出版专著1部。本项目的研究不仅可以丰富和促进了优化领域的理论研究，还为经济金融及最优控制领域问题的稳定性分析提供了新的理论依据。