A genetic algorithm using a finite search space for solving nonlinear/linear fractional bilevel programming problems

The bilevel programming problem is strongly NP-hard and non-convex, which implies that the problem is very challenging for most canonical optimization approaches using single-point search techniques to find global optima. In the present paper, a class of nonlinear bilevel programming problems are considered where the follower is a linear fractional program. Based on a novel coding scheme, a genetic algorithm with global convergence was developed. First, potential bases of the follower’s problem were taken as individuals, and a genetic algorithm was used to explore these bases. In addition, in order to evaluate each individual, a fitness function was presented by making use of the optimality conditions of linear fractional programs. Also, the fitness evaluation, as a sub-procedure of optimization, can partly improve the leader’s objective. Finally, some computational examples were solved and the results show that the proposed algorithm is efficient and robust.

[1]  Roberto Andreani,et al.  An inexact-restoration method for nonlinear bilevel programming problems , 2009, Comput. Optim. Appl..

[2]  Li He.  An Interpolation Based Genetic Algorithm for Solving Nonlinear Bilevel Programming Problems , 2008 .

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

[4]  S. Dempe.  A simple algorithm for the-linear bilevel programming problem , 1987 .

[5]  David E. Boyce,et al.  A general bilevel linear programming formulation of the network design problem , 1988 .

[6]  K. Deb,et al.  An Evolutionary Approach for Bilevel Multi-objective Problems , 2009 .

[7]  J. Arroyo,et al.  On the solution of the bilevel programming formulation of the terrorist threat problem , 2005, IEEE Transactions on Power Systems.

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

[9]  Jean Bosco Etoa Etoa.  Solving convex quadratic bilevel programming problems using an enumeration sequential quadratic programming algorithm , 2010, J. Glob. Optim..

[10]  Thomas Bäck,et al.  Evolutionary Algorithms in Theory and Practice , 1996 .

[11]  Richard L. Church,et al.  A bilevel mixed-integer program for critical infrastructure protection planning , 2008, Comput. Oper. Res..

[12]  Jonathan F. Bard,et al.  A bilevel programming approach to determining tax credits for biofuel production , 2000, Eur. J. Oper. Res..

[13]  H. Abou-Kandil,et al.  Government-private sector relations as a stackelberg game: A degenerate case , 1987 .

[14]  Yuping Wang,et al.  An evolutionary algorithm for solving nonlinear bilevel programming based on a new constraint-handling scheme , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[15]  Herminia I. Calvete,et al.  Bilevel model for production-distribution planning solved by using ant colony optimization , 2011, Comput. Oper. Res..

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

[17]  Patrice Marcotte,et al.  A Trust-Region Method for Nonlinear Bilevel Programming: Algorithm and Computational Experience , 2005, Comput. Optim. Appl..

[18]  Yu-Ping Wang,et al.  An Interpolation Based Genetic Algorithm for Solving Nonlinear Bilevel Programming Problems: An Interpolation Based Genetic Algorithm for Solving Nonlinear Bilevel Programming Problems , 2009 .

[19]  Yuping Wang,et al.  A New Evolutionary Algorithm for a Class of Nonlinear Bilevel Programming Problems and Its Global Convergence , 2011, INFORMS J. Comput..

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

[21]  Herminia I. Calvete,et al.  A genetic algorithm for solving linear fractional bilevel problems , 2009, Ann. Oper. Res..

[22]  K. Swarup.  Letter to the Editor-Linear Fractional Functionals Programming , 1965 .

[23]  Christodoulos A. Floudas,et al.  Global optimization of mixed-integer bilevel programming problems , 2005, Comput. Manag. Sci..

[24]  Ue-Pyng Wen,et al.  A hybrid neural network approach to bilevel programming problems , 2007, Appl. Math. Lett..

[25]  Herminia I. Calvete,et al.  On the Quasiconcave Bilevel Programming Problem , 1998 .

[26]  Athanasios Migdalas,et al.  Bilevel programming in traffic planning: Models, methods and challenge , 1995, J. Glob. Optim..

[27]  Zhongping Wan,et al.  An approximate programming method based on the simplex method for bilevel programming problem , 2010, Comput. Math. Appl..

[28]  ColsonBenoît,et al.  A Trust-Region Method for Nonlinear Bilevel Programming , 2005 .

[29]  J. G. Ecker,et al.  Solving Bilevel Linear Programs Using Multiple Objective Linear Programming , 2009 .

[30]  Stephan Dempe,et al.  Direct search algorithm for bilevel programming problems , 2011, Comput. Optim. Appl..

[31]  Li Hecheng,et al.  Exponential distribution-based genetic algorithm for solving mixed-integer bilevel programming problems , 2008 .

[32]  Stephan Dempe,et al.  Bilevel road pricing: theoretical analysis and optimality conditions , 2012, Ann. Oper. Res..

[33]  P. Saxena,et al.  Linear Fractional Functional Programming in Complex Space , 1979 .

[34]  Jean Bosco Etoa Etoa.  Solving quadratic convex bilevel programming problems using a smoothing method , 2011, Appl. Math. Comput..

[35]  Kang Li,et al.  New data association technique based on ACO with directional information considered , 2008 .

[36]  Le Dung Muu,et al.  A Global Optimization Method for Solving Convex Quadratic Bilevel Programming Problems , 2003, J. Glob. Optim..

[37]  Herminia I. Calvete,et al.  A new approach for solving linear bilevel problems using genetic algorithms , 2008, Eur. J. Oper. Res..

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

[39]  Yuping Wang,et al.  A real-binary coded genetic algorithm for solving nonlinear bilevel programming with nonconvex objective functions , 2011, 2011 IEEE Congress of Evolutionary Computation (CEC).

[40]  Asgeir Tomasgard,et al.  A branch-and-bound method for discretely-constrained mathematical programs with equilibrium constraints , 2013, Ann. Oper. Res..