A Note on Multiobjective Optimization and Complementarity Constraints

We propose a new approach to convex nonlinear multiobjective optimization that captures the geometry of the Pareto set by generating a discrete set of Pareto points optimally. We show that the problem of finding an optimal representation of the Pareto surface can be formulated as a mathematical program with complementarity constraints. The complementarity constraints arise from modeling the set of Pareto points, and the objective maximizes some quality measure of this discrete set. We present encouraging numerical experience on a range of test problem collected from the literature.

[1]  Ching-Lai Hwang,et al.  Methods for Multiple Objective Decision Making , 1979 .

[2]  Klaus Schittkowski,et al.  Test examples for nonlinear programming codes , 1980 .

[3]  F. Clarke Optimization And Nonsmooth Analysis , 1983 .

[4]  W. Stadler Multicriteria Optimization in Engineering and in the Sciences , 1988 .

[5]  Wu Li,et al.  A Newton Method for Convex Regression, Data Smoothing, and Quadratic Programming with Bounded Constraints , 1993, SIAM J. Optim..

[6]  M. Florian,et al.  THE NONLINEAR BILEVEL PROGRAMMING PROBLEM: FORMULATIONS, REGULARITY AND OPTIMALITY CONDITIONS , 1993 .

[7]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[8]  S. Dirkse,et al.  The path solver: a nommonotone stabilization scheme for mixed complementarity problems , 1995 .

[9]  J. Dennis,et al.  A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems , 1997 .

[10]  Theoretical and Computational Comparison of Multiobjective Optimization Methods Nimbus and Rd Theoretical and Computational Comparison of Multiobjective Optimization Methods Nimbus and Rd , 1998 .

[11]  John E. Dennis,et al.  Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems , 1998, SIAM J. Optim..

[12]  B. Rustem Algorithms for nonlinear programming and multiple-objective decisions , 1998, Wiley-Interscience series in discrete mathematics and optimization.

[13]  Kaisa Miettinen,et al.  Proper Pareto Optimality In Nonconvex Problems - Characterization With Tangent And Normal Cones , 1998 .

[14]  John E. Renaud,et al.  Interactive Multiobjective Optimization Procedure , 1999 .

[15]  M. Ferris,et al.  Complementarity problems in GAMS and the PATH solver 1 This material is based on research supported , 2000 .

[16]  Serpil Sayin,et al.  Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming , 2000, Math. Program..

[17]  Kaisa Miettinen,et al.  Interactive multiobjective optimization system WWW-NIMBUS on the Internet , 2000, Comput. Oper. Res..

[18]  J. Outrata On mathematical programs with complementarity constraints , 2000 .

[19]  Stefan Scholtes,et al.  Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity , 2000, Math. Oper. Res..

[20]  Paulo Ferreira,et al.  Bi-objective optimisation with multiple decision-makers: a convex approach to attain majority solutions , 2000, J. Oper. Res. Soc..

[21]  Claus Hillermeier,et al.  Nonlinear Multiobjective Optimization , 2001 .

[22]  ROBERT J. VANDERBEI,et al.  METHODS FOR NONCONVEX NONLINEAR PROGRAMMING : COMPLEMENTARITY CONSTRAINTS , 2002 .

[23]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[24]  Sven Leyffer,et al.  On the Global Convergence of a Filter--SQP Algorithm , 2002, SIAM J. Optim..

[25]  Hande Y. Benson,et al.  INTERIOR-POINT METHODS FOR NONCONVEX NONLINEAR PROGRAMMING: COMPLEMENTARITY CONSTRAINTS , 2002 .

[26]  Sven Leyffer,et al.  Solving mathematical programs with complementarity constraints as nonlinear programs , 2004, Optim. Methods Softw..

[27]  Mihai Anitescu,et al.  Global Convergence of an Elastic Mode Approach for a Class of Mathematical Programs with Complementarity Constraints , 2005, SIAM J. Optim..

[28]  Jorge Nocedal,et al.  Interior Methods for Mathematical Programs with Complementarity Constraints , 2006, SIAM J. Optim..

[29]  S. Leyffer Complementarity constraints as nonlinear equations: Theory and numerical experience , 2006 .

[30]  Sven Leyffer,et al.  Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints , 2006, SIAM J. Optim..

[31]  Georgia Perakis,et al.  A Robust SQP Method for Mathematical Programs with Linear Complementarity Constraints , 2006, Comput. Optim. Appl..