A cutting plane method for bilevel linear programming with interval coefficients

This article considers the bilevel linear programming problem with interval coefficients in both objective functions. We propose a cutting plane method to solve such a problem. In order to obtain the best and worst optimal solutions, two types of cutting plane methods are developed based on the fact that the best and worst optimal solutions of this kind of problem occur at extreme points of its constraint region. The main idea of the proposed methods is to solve a sequence of linear programming problems with cutting planes that are successively introduced until the best and worst optimal solutions are found. Finally, we extend the two algorithms proposed to compute the best and worst optimal solutions of the general bilevel linear programming problem with interval coefficients in the objective functions as well as in the constraints.

[1]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[2]  Stephan Dempe,et al.  Foundations of Bilevel Programming , 2002 .

[3]  Milan Hladík,et al.  Generalized linear fractional programming under interval uncertainty , 2010, Eur. J. Oper. Res..

[4]  Mehdi Allahdadi,et al.  The optimal solution set of the interval linear programming problems , 2012, Optimization Letters.

[5]  J. Clegg,et al.  Bilevel programming applied to optimising urban transportation , 2001 .

[6]  Mansour Saraj,et al.  SOLVING BI-LEVEL PROGRAMMING PROBLEMS ON USING GLOBAL CRITERION METHOD WITH AN INTERVAL APPROACH , 2012 .

[7]  Patrice Marcotte,et al.  An overview of bilevel optimization , 2007, Ann. Oper. Res..

[8]  Wu Shiquan.  A CUTTING PLANE METHOD FOR LINEAR BILEVEL PROGRAMS , 1998 .

[9]  Tong Shaocheng,et al.  Interval number and fuzzy number linear programmings , 1994 .

[10]  Charles E. Blair,et al.  Computational Difficulties of Bilevel Linear Programming , 1990, Oper. Res..

[11]  Guohe Huang,et al.  Grey linear programming, its solving approach, and its application , 1993 .

[12]  Samir A. Abass,et al.  An interval number programming approach for bilevel linear programming problem , 2010 .

[13]  John W. Chinneck,et al.  Linear programming with interval coefficients , 2000, J. Oper. Res. Soc..

[14]  Charles Audet,et al.  Disjunctive cuts for continuous linear bilevel programming , 2006, Optim. Lett..

[15]  H. Farhad,et al.  BILEVEL LINEAR PROGRAMMING WITH FUZZY PARAMETERS , 2013 .

[16]  Wei Li,et al.  Fault Detection in Discrete Dynamic Systems with Uncertainty Based on Interval Optimization , 2011 .

[17]  Xiaoning Zhang,et al.  Stackelberg games and multiple equilibrium behaviors on networks , 2007 .

[18]  Ajay Kumar Bhurjee,et al.  Efficient solution of interval optimization problem , 2012, Math. Methods Oper. Res..

[19]  Asoke Kumar Bhunia,et al.  Uncertain constrained optimization by interval-oriented algorithm , 2014, J. Oper. Res. Soc..

[20]  Herminia I. Calvete,et al.  Bilevel multiplicative problems: A penalty approach to optimality and a cutting plane based algorithm , 2008 .

[21]  M. Patriksson,et al.  Stochastic bilevel programming in structural optimization , 2001 .

[22]  Patrice Marcotte,et al.  A CUTTING-PLANE ALGORITHM FOR THE LINEAR BILEVEL PROGRAMMING PROBLEM. , 1993 .

[23]  Gang Du,et al.  Research on the method for interval linear bi-level programming based on a partial order on intervals , 2011, 2011 Eighth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD).

[24]  J. Rohn,et al.  Interval linear programming , 2006 .

[25]  M. Hladík.  Optimal value bounds in nonlinear programming with interval data , 2011 .

[26]  G. Anandalingam,et al.  A penalty function approach for solving bi-level linear programs , 1993, J. Glob. Optim..

[27]  Herminia I. Calvete,et al.  Linear bilevel programming with interval coefficients , 2012, J. Comput. Appl. Math..

[28]  Jonathan F. Bard,et al.  Practical Bilevel Optimization , 1998 .

[29]  Jie Lu,et al.  Fuzzy bilevel programming with multiple objectives and cooperative multiple followers , 2010, J. Glob. Optim..

[30]  S. Dempe.  Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints , 2003 .

[31]  Milan Hladík.  Optimal value range in interval linear programming , 2009, Fuzzy Optim. Decis. Mak..

[32]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications , 1998 .

[33]  Milan Hladı´k.  Weak and strong solvability of interval linear systems of equations and inequalities , 2013 .