非光滑分析中二阶优化条件及其应用研究

On the basis of second-order optimal conditions of nonsmooth functions in an Euclidean space, this proposed project will study several types of generalized second-order "derivatives" of one nonsmooth lower semicontinuous function defined on a Hilbert space and consider several second-order optimal conditions defined by these second-order "derivatives". Theorems on the relationship among these optimal conditions will be provided. As applications, these second-order optimal conditions will be used to characterize strict minimizer of order two so as to establish its characterization theorems. Theorems being established on relationship among these second-order optimal conditions as well as applications will improve the corresponding results when restricted to the Euclidean space setting.

基于欧氏空间中非光滑函数的二阶优化条件，本项目计划研究希尔伯特空间中非光滑下半连续函数的几类广义二阶“导数”，考虑由这些二阶“导数”所定义的几类二阶优化条件，重点研究建它们之间关系。作为应用，本项目将应用所考虑的二阶优化条件刻划非光滑函数二阶严格极小值点的特征，期望建立二阶严格极小值点的特征定理。拟建立的关于二阶优化条件之间关系以及应用性的定理，当限制到欧氏空间情形将改进相对应的结果。

本项目主要针对非光滑分析中二阶优化条件及应用进行了深入研究，取得了如下主要成果：（1）研究并证明了三类二阶优化条件以及它们之间的等价关系；（2）证明了二阶严格极小值点成立的特征定理，并通过反例说明所研究的三类二阶优化条件仅是二阶严格极小值点成立的必要条件，而非充分发件；（3）建立了求解非光滑凸混合整数非线性规划问题的外逼近算法；（4）建立了求解锥约束混合整数非线性规划问题的广义本德斯算法；（5）证明了任意多个闭凸集簇的正规性质与逼近论中线性正则性的等价关系；（6）建立了凸约束系统具有度量次正则性的原始特征。.这些成果在优化领域SCI期刊发表论文7篇，其中1篇发表于Top期刊SIAM J. Optim.