A DUAL-RELAX PENALTY FUNCTION APPROACH FOR SOLVING NONLINEAR BILEVEL PROGRAMMING WITH LINEAR LOWER LEVEL PROBLEM

doi:10.1016/S0252-9602(11)60265-8

数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (2): 652-660.doi: 10.1016/S0252-9602(11)60265-8

A DUAL-RELAX PENALTY FUNCTION APPROACH FOR SOLVING NONLINEAR BILEVEL PROGRAMMING WITH LINEAR LOWER LEVEL PROBLEM

万仲平|王广民|吕一兵

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China； School of Economics and Management, China University of Geosciences, Wuhan 430074, China； Institute of Information and Mathematics, Yangtze University, Jingzhou, Hubei, China

收稿日期:2008-10-28 修回日期:2010-07-26 出版日期:2011-03-20 发布日期:2011-03-20
基金资助:
This research was partially supported by the National Science Foundation of China (70771080) and Social Science Foundation of Ministry of Education (10YJC630233).

A DUAL-RELAX PENALTY FUNCTION APPROACH FOR SOLVING NONLINEAR BILEVEL PROGRAMMING WITH LINEAR LOWER LEVEL PROBLEM

WAN Zhong-Ping, WANG Guang-Min, LV Yi-Bing

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China； School of Economics and Management, China University of Geosciences, Wuhan 430074, China； Institute of Information and Mathematics, Yangtze University, Jingzhou, Hubei, China

Received:2008-10-28 Revised:2010-07-26 Online:2011-03-20 Published:2011-03-20
Supported by:
This research was partially supported by the National Science Foundation of China (70771080) and Social Science Foundation of Ministry of Education (10YJC630233).

摘要/Abstract

摘要：

The penalty function method, presented many years ago, is an important numerical method for the mathematical programming
problems. In this article, we propose a dual-relax penalty function approach, which is significantly different from penalty function
approach existing for solving the bilevel programming, to solve the nonlinear bilevel programming with linear lower level problem. Our
algorithm will redound to the error analysis for computing an approximate solution to the bilevel programming. The error estimate is obtained among the optimal objective function value of the dual-relax penalty problem and of the original bilevel programming problem. An example is illustrated to show the feasibility of the proposed approach.

关键词: Nonlinear bilevel programming, penalty function approach, dual-relax strategy

Abstract:

Key words: Nonlinear bilevel programming, penalty function approach, dual-relax strategy

中图分类号:

90C26

万仲平, 王广民, 吕一兵. A DUAL-RELAX PENALTY FUNCTION APPROACH FOR SOLVING NONLINEAR BILEVEL PROGRAMMING WITH LINEAR LOWER LEVEL PROBLEM[J]. 数学物理学报(英文版), 2011, 31(2): 652-660.

WAN Zhong-Ping, WANG Guang-Min, LV Yi-Bing. A DUAL-RELAX PENALTY FUNCTION APPROACH FOR SOLVING NONLINEAR BILEVEL PROGRAMMING WITH LINEAR LOWER LEVEL PROBLEM[J]. Acta mathematica scientia,Series B, 2011, 31(2): 652-660.

参考文献

[1] Anandalingam G, White D J. A solution for the linear static Stackelberg problem using penalty function. IEEE Transactions Automatic Control, 1990, 35: 1170--1173

[2] Bard J F. Practical Bilevel Optimization:Algorithms and Applications. Dordrecht: Kluwer Academic Publishers, 1998

[3] Campêlo M, Dantas S, Scheimberg S. A note on a penalty function approach for solving bilevel linear programs. Journal of Global Optimization, 2000, 16: 245--255

[4]Colson B, Marcotte P, Savard G. Bilevel programming: a survey. 4OR Quarterly Journal of Operations Research, 2005, (3): 87--107

[5]Dempe S. Foundations of Bilevel Programming. Kluwer Academic Publishers, 2002

[6]Dempe S. Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization, 2003, 52: 333--359

[7]Liu G, Han J, Zhang J. Exact penalty functions for convex bilevel programming problems. Journal of Optimization Theory and Applications, 2001, 110: 621--643

[8]Lv Y, Hu T, Wang G, Wan Z. A penalty function method based on Kuhn-Tucker condition for solving linear bilevel programming. Applied Mathematics and Computation, 2007, 188: 808--813

[9]Meng Z, Dang C, Yang X. On the Smoothing of the Square-Root Exact Penalty Function for Inequality Constrained Optimization. Computational Optimization and Applications, 2006, 35: 375--398

[10]Migadalas A, Paradalos P M, Värbraud P. Multilevel Optimization Algorithms and Applications. Kluwer Academic Publishers, 1998

[11]Vicente L N, Calamai P H. Bilevel and multibilevel programming: a bibliography review. Journal of Global Optimization, 1994, (5): 291--306

[12]Wan Z, Zhou S. The convergence of approach penalty function method for approximate bilevel programming problem. Acta Mathematica Scientia, 2001, 21: 69--76

[13]Wen U P, Hsu S T. Linear bilevel programming problem - a review. Journal of the Operational Research Society, 1991, 42: 125--133

[14] White D J, Anandalingam G. A penalty function approach for solving bi-level linear programs. Journal of Global Optimization, 1993, (3): 397--419

A DUAL-RELAX PENALTY FUNCTION APPROACH FOR SOLVING NONLINEAR BILEVEL PROGRAMMING WITH LINEAR LOWER LEVEL PROBLEM

A DUAL-RELAX PENALTY FUNCTION APPROACH FOR SOLVING NONLINEAR BILEVEL PROGRAMMING WITH LINEAR LOWER LEVEL PROBLEM

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