求解一类双层规划问题的邻近梯度算法  

A Proximal Gradient Method for Solving a Class of Bilevel Programming Problem

作　　者：李丹[1] 高洪韵 陈爽[1] 王金鹤 LI Dan;GAO Hong-yun;CHEN Shuang;WANG Jin-he(College of Information Engineering, Dalian University, Dalian 116622, China;School of Information and Control Engineering, Qingdao University of Technology, Qingdao 266033, China)

机构地区：[1]大连大学信息工程学院,辽宁大连116622 [2]青岛理工大学信息与控制工程学院,山东青岛266033

出　　处：《大连大学学报》2019年第3期1-5,共5页Journal of Dalian University

基　　金：国家自然科学基金项目(11501074);辽宁省博士启动基金(201501194)

摘　　要：非光滑双层规划问题来源于具有广泛应用的经济学、工程学和自然科学等领域。邻近梯度法是一类求解非光滑优化问题的有效算法。当目标函数具有某些特殊结构,特别是当具有某些解析形式的邻近算子时,邻近梯度法是求解非光滑优化问题的高效算法之一。在本文中,首先以凸分析、变分分析与非光滑优化等理论为基础,其次应用参数极小化技术,将具有某些特殊结构的非光滑双层规划问题转化为单层优化问题,最后应用邻近梯度法逐次求解,并建立算法的收敛性分析。Bilevel programming problems arise in a wide range of real-life implementations such as economics,engineering, military, natural sciences and so on. Proximal gradient method is a class of effective algorithm for minimizing the nonsmooth optimization problems. When the objective function owns special substructure, especially analytical proximal operator, proximal gradient method is high effective. In this paper, the basic theory is convex analysis, variational analysis and nonsmooth optimization. The core technology is to transform the nonsmooth bilevel programming problems with some special structures into one-level optimization problems by parameter minimizing technique. Then construct proximal gradient method to solve the parameterized family of subproblems. Finally we give the convergence analysis of the algorithm.

关 键 词：非线性规划 非光滑优化 邻近梯度法 

分 类 号：O221.2[理学—运筹学与控制论]