锥优化问题的光滑逼近精确罚理论与算法研究

项目名称： 锥优化问题的光滑逼近精确罚理论与算法研究

项目编号： No.11271233

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 赵文玲

作者单位： 山东理工大学

项目金额： 60万元

中文摘要： 精确罚函数在约束优化理论和算法方面起着重要的作用。而由于精确罚函数常常不具有光滑性，所以研究此类函数的光滑逼近尤为必要。鉴于目前尚未见到对锥优化问题这方面的研究成果，因此，本项目拟：1）首先对一般的锥优化问题，给出非线性双参数光滑逼近精确罚(简记NDPF), 研究与其相关的理论，包括扰动函数的稳定性态、零对偶间隙成立的充分必要条件等；2)基于NDPF，设计相应的罚算法, 着重研究算法的各种收敛性质，包括算法的全局收敛性、有限终止性与局部收敛性分析等；3）将上述结果应用到三类具体的非自对偶锥中。针对这三类锥的特征，进一步细致地刻划它们的精确罚理论。上述研究成果将进一步丰富锥优化理论体系。

中文关键词： 锥优化；增广拉格朗日函数；光滑逼近精确罚；算法；有限终止性

英文摘要： Exact penalty function plays a very important role in constrained optimization theory and algorithms. However, exact penalty function often is not smooth, so smooth approximation for the study of such functions is particularly necessary. But for the exact penalty of the general cone optimization problem, there is no research results about its smooth approximation currently. Therefore, this project intends to: 1) firstly, for the general cone optimization problem, give a nonlinear double-parameter smooth approximation exact penalty function (denoted NDPF), then study the relative theory, including the stability of the perturbation function and the necessary and sufficient conditions for zero duality gap established, and so on; 2)base on the NDPF, design the appropriate penalty algorithm, focus on studying a variety of convergence properties of the algorithm, including the global convergence, finite termination and the local convergence analysis, etc; 3) apply the above results to three types of specific non-self-dual cone. For the characteristics of these three types of cones, further characterize their exact penalty theory in detail. The above research results will further enrich the theoretical system of the cone optimization.

英文关键词： cone optimization；augment Lagrange function；smooth approximation exact penalty function；algorithm；finite termination