非光滑非凸优化问题的交替线性化算法及其应用

项目名称： 非光滑非凸优化问题的交替线性化算法及其应用

项目编号： No.11501074

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 李丹

作者单位： 大连大学

项目金额： 18万元

中文摘要： 本项目研究一类在自然科学和工程实际中具有广泛应用的极小化两个函数和的非光滑非凸优化问题的交替线性化算法及其应用。本课题组以凸分析、变分分析与非光滑优化等理论为基础，主要内容包括研究求解一类具有某些特殊结构的非光滑非凸复合优化问题的交替线性化算法，构造极小化非光滑非凸优化的交替线性化束方法、非精确交替线性化算法与加速交替线性化算法，并将所构造的算法应用于压缩感知等实际应用领域。期望本项目对非光滑非凸优化的理论与数值方法的研究对该领域的发展起到积极的促进作用。

中文关键词： 非光滑优化；非凸优化；非线性规划；交替线性化算法；邻近点

英文摘要： The project will research alternating linearization algorithms and applications for minimizing the sum of two functions for nonsmooth nonconvex optimization problems which arise in many natural science and applicatons of practical engineering interest. Based on the foundations of convex analysis, variational analysis and nonsmooth optimization, the main research contents are as follows. Firstly the alternating linearization algorithms for minimizing composite functions are studied by exploring the separate substructures. Secondly, we try to extend the alternating linearization bundle algorithms to nonsmooth nonconvex setting. Thirdly, the inexact alternating linearization algorithms are constructed for nonsmooth nonconvex optimization problems. Finally, the accelerate technique for algorithms is considered and all the algorithms in our project will be applied to compressed sensing and other fields. The anticipated results will play an active promoting role for the development of theory and numerical methods of nonsmooth nonconvex optimization.

英文关键词： Nonsmooth optimization;nonconvex optimizatioin;nonlinear programming;alternating linearization algorithms;proximal point