几类矩阵锥变分不等式的理论与数值算法

Matrix cone variational inequality problems have been an active area of optimization,since they possess important applications in the fields of science and engineering,such as statistics,economics,machine learning,compressed sensing and so on.The project studies the theory and methods for solving the matrix cone variational inequality problems in which the matrix cones are defined by the epigraph of four matrix norms.This problem is very important, because many important matrix cone optimation problems can be classified into special cases of matrix cone variational inequality problems.Based on the classic theory of variational analysis and perturbation analysis,the project plans to study the optimality theory and numerical methods with the help of the differential of matrix cone projection operator and the latest theory of several types of matrix cone programming problems.The main contents of this project are described as follows:we shall study some P-properties of matrix cones;we shall present the B-differential of the projection operator; we shall wish to establish the first order and the second order optimality theory with sigma term,and the stability theory;we shall apply the smoothing method to solve some significant applications of matrix cone variational inequality problems.The aim of the project is to obtain the optimality theory of matrix cone variational inequality problems and discuss the theory and implementation of the smoothing method.We hope to make contribution to the study of theoretical research on conic programing problems.

矩阵锥变分不等式问题是目前优化领域的一个研究热点，它在统计学，经济学，机器学习，压缩传感等科学和工程领域有着重要的应用。本项目研究由四类矩阵范数定义的矩阵锥约束变分不等式问题的理论与求解方法，这类问题非常重要，因为目前很多重要的矩阵优化问题都可纳入到这个框架之下。本项目以变分分析和扰动分析为基础，借助矩阵锥投影算子微分和几类矩阵锥规划的最新理论成果，研究几类矩阵锥变分不等式问题的最优性理论和求解方法。本项目内容包括研究矩阵锥的P性质；计算投影算子的B-微分；建立矩阵锥变分不等式问题的一阶与二阶最优性理论以及稳定性理论，得到相应的Sigma项；研究求解矩阵锥变分不等式问题的光滑方法；并用光滑方法求解几个有重大有实用价值的矩阵锥变分不等式问题。本项目旨在获得矩阵锥变分不等式问题的最优性理论，探讨光滑函数方法的理论与实现，期望对锥规划的理论研究做贡献。

本项目以半光滑分析，变分分析和扰动分析等理论为基础，以特征值和奇异值的方向导数，矩阵锥的投影算子与相关的矩阵分析为工具，建立了几类矩阵锥变分不等式问题的最优性理论，包括几类矩阵锥情况下的函数P性质的定义和含有Sigma项的BD正则性条件。构造了求解二阶锥约束变分不等式问题和正挂限锥互补问题的光滑函数方法以及神经网络方法，利用方程组的正则性理论来刻画光滑函数方法的超线性收敛速度。应用非精确LM 牛顿算法求解三参数S‐N 曲线模型，具有较高的精度和使用价值。