非光滑 Lipschitz 连续函数优化束方法与应用

项目名称： 非光滑 Lipschitz 连续函数优化束方法与应用

项目编号： No.11301246

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 沈洁

作者单位： 辽宁师范大学

项目金额： 22万元

中文摘要： 非光滑Lipschitz连续函数优化问题具有重要理论和应用价值，如最优控制中的大量非光滑问题以及工程领域大量的实际问题（如大坝抗震问题）均是Lipschitz函数优化问题。本课题以凸分析、变分分析为基础，致力于Lipschitz连续优化的高效可执行有效算法的研究。主要内容包括：(1)利用广义微分的UV分解理论，构造Lipschitz函数优化束方法的理论框架，包括算法设计，收敛性分析。(2)对分片线性函数，利用函数非精确信息构建新型近似模型，并结合次梯度删除、度量准则，调整或重新定义线性化误差, 构造相应数值算法，并讨论收敛速度。(3)将取得的上述成果用于解决几个理论与实际问题，包括: 利用束方法求解变分不等式中辅助优化问题；构造求解特殊均衡约束规划问题的可执行双层束方法；利用束方法研究混凝土大坝抗震安全问题。本课题取得的成果将对非光滑最优化理论与数值方法的研究起到促进作用。

中文关键词： 非光滑优化；束方法；Lipschitz 连续函数；线性近似；

英文摘要： Optimization problems with Lipschitz continuous functions are of great importance in both theory and practice, for examples, a large number of nonsmooth problems in optimal control and practical problems arising from engineering fields (such as the anti-sesmic problems) are Lipschitz continuous optimization problems. Based on the foundations of convex analysis and variational analysis,this project aims at construting implementable numerical algorithms with high efficiency for Lipschitz continuous optimization. The main research work includes: (1)Construct theoretical framework of bundle methods for Lipschitz continuous optimization by using UV decomposition theory of general differential, which includes the design of algorithms and the analysis of convergence. (2)For piecewise linear functions, construct new approximation models by utilizing the approximate function values and subgradients, and construct corresponding numerical algorithms by adjusting or redefining linearization errors and by combining subgradient deletion rules with subgradient locality measures. At the same time, we also discuss the rate of convergence. (3)Apply the results of Lipschitz continuous optimization to several theoretical and practical problems, which includes: study safe evaluation technology for concrete dam by applying no

英文关键词： nonsmooth optimization；bundle method；Lipschitz continuous function；linear approximation；