Solving Nonlinear Multiobjective Bilevel Optimization Problems with Coupled Upper Level Constraints

For bilevel optimization problems with a multiobjective optimization problem on each level we consider a very general formulation with the upper-level constraint being dependent on the lower-level variable. Extending results of an earlier work we give connections between the solution sets of several multiobjective optimization problems and the induced set of the multiobjective bilevel problem. Using these results together with a refinement strategy involving new sensitivity results for the parameter-dependent ε-constraint scalarization we develop an algorithm for solving such problems. We also demonstrate the applicability of our algorithm on two test problems.

[1]  Xinping Shi,et al.  Model and interactive algorithm of bi-level multi-objective decision-making with multiple interconnected decision makers , 2001 .

[2]  K. Jittorntrum.  Solution point differentiability without strict complementarity in nonlinear programming , 1984 .

[3]  Jane J. Ye,et al.  Optimality conditions for bilevel programming problems , 1995 .

[4]  W. I. Davisson.  Public Investment Criteria , 1964 .

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

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

[7]  Johannes Jahn,et al.  Multiobjective Search Algorithm with Subdivision Technique , 2006, Comput. Optim. Appl..

[8]  J. Morgan,et al.  Semivectorial Bilevel Optimization Problem: Penalty Approach , 2006 .

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

[10]  L. N. Vicente,et al.  Multicriteria Approach to Bilevel Optimization , 2006 .

[11]  Xiaoqi Yang,et al.  Mathematical Programs with Vector Optimization Constraints , 2005 .

[12]  椹木 義一,et al.  Theory of multiobjective optimization , 1985 .

[13]  Gabriele Eichfelder,et al.  An Adaptive Scalarization Method in Multiobjective Optimization , 2008, SIAM J. Optim..

[14]  Johannes Jahn,et al.  Vector optimization - theory, applications, and extensions , 2004 .

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

[16]  P. Serafini,et al.  Scalarizing vector optimization problems , 1984 .

[17]  Mahmoud A. Abo-Sinna,et al.  A Bi-Level Non-Linear Multi-Objective Decision Making under Fuzziness , 2001 .

[18]  Teng Chun.  A Class of Genetic Algorithms on Bilevel Multi-objective Decision Making Problem , 2000 .

[19]  Gabriele Eichfelder,et al.  Scalarizations for adaptively solving multi-objective optimization problems , 2009, Comput. Optim. Appl..

[20]  Paul H. Calamai,et al.  Bilevel and multilevel programming: A bibliography review , 1994, J. Glob. Optim..

[21]  Heinrich von Stackelberg,et al.  Stackelberg (Heinrich von) - The Theory of the Market Economy, translated from the German and with an introduction by Alan T. PEACOCK. , 1953 .

[22]  Mahmoud A. Abo-Sinna,et al.  A multi-level non-linear multi-objective decision-making under fuzziness , 2004, Appl. Math. Comput..

[23]  Martine Labbé,et al.  On a class of bilevel programs , 2000 .

[24]  Yafeng Yin,et al.  Multiobjective bilevel optimization for transportation planning and management problems , 2002 .

[25]  X Shi,et al.  Interactive bilevel multi-objective decision making , 1997 .

[26]  L. Lasdon,et al.  On a bicriterion formation of the problems of integrated system identification and system optimization , 1971 .

[27]  M. Sakawa,et al.  Stackelberg Solutions to Multiobjective Two-Level Linear Programming Problems , 1999 .

[28]  A. Messac,et al.  Concept Selection Using s-Pareto Frontiers , 2003 .

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

[30]  Paul H. Calamai,et al.  Generating quadratic bilevel programming test problems , 1994, TOMS.

[31]  Patrice Marcotte,et al.  Bilevel programming: A survey , 2005, 4OR.

[32]  A. Messac,et al.  Smart Pareto filter: obtaining a minimal representation of multiobjective design space , 2004 .