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### 存零约束优化问题的部分罚函数方法

1. 1. 重庆工商大学数学与统计学院, 重庆 400067;
2. 重庆师范大学数学科学学院, 重庆 401331;
3. 社会经济应用统计重庆市重点实验室, 重庆 400067
• 收稿日期:2021-09-22 修回日期:2022-01-06 出版日期:2022-05-25 发布日期:2022-07-23
• 通讯作者: 李高西,Email:ligaoxicn@126.com.
• 基金资助:
国家自然科学基金(11901068,12171060),重庆市基础研究与前沿探索(cstc2019jcyj-msxmX0456,cstc2021jcyj-msxmX0499),重庆工商大学科研项目(1952034,ZDPTTD201908)资助课题.

ZHANG Tingting, LI Gaoxi, TANG Liping, HUANG Yingquan. Partial Penalty Function Method for Switching Constraints Optimization Problem[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(5): 1234-1245.

### Partial Penalty Function Method for Switching Constraints Optimization Problem

ZHANG Tingting1, LI Gaoxi1,3, TANG Liping2, HUANG Yingquan1

1. 1. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067;
2. School of Mathematics Sciences, Chongqing Normal University, Chongqing 401331;
3. Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing 400067
• Received:2021-09-22 Revised:2022-01-06 Online:2022-05-25 Published:2022-07-23

In this paper, the switching-constrained optimization problem is a new optimization problem proposed in recent years. Due to the existence of switchingconstrained, the commonly used constraint qualification is not satisfied, thus, most of the convergence results of existing algorithms cannot be directly applied to this problem. In this paper, we put the difficult switching constraint on the objective function and propose a partial penalty function method. It is proved that under the linear independent constraint qualification with switching constraint, the convergence of the sequence of stable points of the penalty problem is the weak stationary point of the original problem. At the same time, for any strictly local optimal solution of the original problem, there exists a local optimal solution sequence of the penalty problem which converges to it. Finally, numerical results show that the penalty function method is feasible.

MR(2010)主题分类:

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 [1] Mehlitz P. Stationarity conditions and constraint qualifications for mathematical programs with switching constraints with applications to either-or-constrained programming. Mathematical Programming, 2020, 181:149-186.[2] Clason C, Rund A, Kunisch K. Nonconvex penalization of switching control of partial differential equations. Systems and Control Letters, 2017, 106:1-8.[3] Gugat M. Optimal switching boundary control of a string to rest in finite time. XAMM-Journal of Applied Mathematics and Mechanics, 2010, 88(4):283-305.[4] Hante F M, Sager S. Relaxation methods for mixed-integer optimal control of partial differential equations. Computational Optimization and Applications, 2013, 55(1):197-225.[5] Wang L, Yan Q. Bang-bang property of time optimal controls for some semilinear heat equation. SIAM Journal on Control and Optimization, 2016, 54(6):477-499.[6] Dempe S, Schreier H. Operations Research:Deterministische Modelle und Methoden. Berlin:Springer-Verlag, 2007.[7] Dempe S. Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization, 2003, 52(3):333-359.[8] Luo Z Q, Pang J S, Ralph D, et al. Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints. Mathematical Programming, 1996, 75(1):19-76.[9] Liu G S, Ye J, Zhu J P. Partial exact penalty for mathematical programs with equilibrium constraints. Set-Valued Analysis, 2008, 16(5-6):785-804.[10] Hoheisel T, Kanxow C, Outrata J. Exact penalty results for mathematical programs with vanishing constraints. Nonlinear Analysis, 2010, 72(5):2514-2526.[11] Ixmailov A F, Solodov M V. Mathematical programs with vanishing constraints:Optimality conditions, sensitivity, and a relaxation method. Journal of Optimization Theory and Applications, 2009, 142(3):501-532.[12] Hu X M, Ralph D. Convergence of a penalty method for mathematical programming with complementarity constraints. Journal of Optimization Theory and Applications, 2004, 123(2):365-390.[13] Lian S J, Zhang L S. A simple smooth exact penalty function for smooth optimization problem. Journal of Systems Science&Complexity, 2012, 25(3):521-528.[14] 吕一兵,万仲平.一类半向量二层规划问题的精确罚函数方法.系统科学与数学, 2016, 36(6):800-809.(Lü Y B, Wan Z P. An exact penalty function method for a class of bilevel programming problems with half vectors. Journal of Systems Science and Mathematical Sciences, 2016, 36(6):800-809.)[15] Kanxow C, Mehlitx P, Steck D. Relaxation schemes for mathematical programmes with switching constraints. Optimization Methods and Software, 2019, DOI:10.1080/10556788.2019.1663425.[16] Mehlitx P. On the linear independence constraint qualification in disjunctive programming. Optimization, 2020, 69(10):2241-2277.[17] Shikhman V. Topological approach to mathematical programs with switching constraints. SetValued and Variational Analysis, 2021, DOI:10.1007/s11228-021-00581-5.[18] Liang Y C, Ye J J. Optimality conditions and exact penalty for mathematical programs with switching constraints. Journal of Optimization Theory and Applications, 2021, 190:1-31.
 [1] 方桃 ，朱淑倩 ，孟敏 ，张承慧. 正线性时滞系统的${\bm L}_{\bf 1}({\bm l}_1)$-增益性能分析及正控制器设计[J]. 系统科学与数学, 2014, 34(2): 158-170. [2] 吕一兵;陈忠;万仲平;王广民. 非线性-线性二层规划问题的罚函数方法[J]. 系统科学与数学, 2009, 29(5): 630-636. [3] 薛文娟;沈春根. 一种修改的非单调线搜索SQP算法[J]. 系统科学与数学, 2007, 27(6): 923-934. [4] 杨若黎;顾基发. 一类神经网络模型的稳定性[J]. 系统科学与数学, 1999, 19(3): 309-318. [5] 简金宝;赖炎连. 一族超线性收敛的投影拟牛顿算法[J]. 系统科学与数学, 1996, 16(2): 105-112. [6] 杨波艇;张可村. 用L_1-罚函数作线性搜索函数的一种修正约束变尺度算法[J]. 系统科学与数学, 1996, 16(1): 11-016. [7] 时贞军;王嘉松. 一般线性或非线性约束下的共轭投影变尺度方法[J]. 系统科学与数学, 1995, 15(4): 312-318.