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带等式约束的光滑优化问题的一类新的精确罚函数

连淑君1,*  唐加会2  杜爱华1   

  1. 1. 曲阜师范大学管理学院, 山东日照 273165; 2. 华东理工大学理学院数学系, 上海 200237
  • 收稿日期:2016-12-12 出版日期:2018-12-15 发布日期:2018-12-15
  • 通讯作者: 连淑君 E-mail: lsjsd2003@126.com
  • 基金资助:

    国家自然科学基金(Nos.71371107, 61373027), 山东省自然科学基金(Nos.ZR2013AM013, ZR2016AM10)

A new class of exact penalty functions for equality constrained smooth optimization

LIAN Shujun1,*  TANG Jiahui2   DU Aihua1   

  1. 1. School  of Management, Qufu Normal University, Rizhao 273165, Shandong, China; 2. Department of Mathematics, School of Science, East China University of Science and  Technology, Shanghai 200237, China
  • Received:2016-12-12 Online:2018-12-15 Published:2018-12-15

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摘要：

罚函数方法是将约束优化问题转化为无约束优化问题的主要方法之一. 不包含目标函数和约束函数梯度信息的罚函数, 称为简单罚函数. 对传统精确罚函数而言, 如果它是简单的就一定是非光滑的; 如果它是光滑的, 就一定不是简单的. 针对等式约束优化问题, 提出一类新的简单罚函数, 该罚函数通过增加一个新的变量来控制罚项. 证明了此罚函数的光滑性和精确性, 并给出了一种解决等式约束优化问题的罚函数算法. 数值结果表明, 该算法对于求解等式约束优化问题是可行的.

关键词: 等式约束最优化, KKT 点, Mangasarian-Fromovitz 条件, 精确光滑罚函数

Abstract:

The penalty function method is one of the main approaches to transform the constrained optimization problems into unconstrained optimization problems. If the gradient of the objective function and constrained functions is not involved in the penalty function, the penalty function is simple. For the traditional exact penalty function, it can not be simple and smooth. For equality constrained optimization problems, a new class of simple penalty functions is constructed by adding a new variable to control the penalty terms. In this paper, the simple penalty functions have been proved smooth and exact. An algorithm based on the class of simple exact functions is proposed. Some numerical examples are given to show the efficiency of the algorithm. 

Key words: equality constrained optimization problem, KKT point, the Mangasarian-Fromovitz condition, exact smooth penalty function