Complementarity Active-Set Algorithm for Mathematical Programming Problems with Equilibrium Constraints

Abstract In this paper, an algorithm for solving a mathematical programming problem with complementarity (or equilibrium) constraints (MPEC) is introduced, which uses the active-set methodology while maintaining the complementarity restrictions throughout the procedure. Finite convergence of the algorithm to a strongly stationary point of the MPEC is established under reasonable hypotheses. The algorithm can be easily implemented by adopting any active-set code for nonlinear programming. Computational experience is included to highlight the efficacy of the proposed method in practice.

[1]  Michael A. Saunders,et al.  MINOS 5. 0 user's guide , 1983 .

[2]  Jonathan F. BARD,et al.  Convex two-level optimization , 1988, Math. Program..

[3]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

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

[5]  Joaquim Júdice,et al.  A sequential LCP method for bilevel linear programming , 1992, Ann. Oper. Res..

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

[7]  Christian Kanzow,et al.  Complementarity And Related Problems: A Survey , 1998 .

[8]  S. Dirkse,et al.  Frontiers in Applied General Equilibrium Modeling: Mathematical Programs with Equilibrium Constraints: Automatic Reformulation and Solution via Constrained Optimization , 2002 .

[9]  Hayri Önal,et al.  A modified simplex approach for solving bilevel linear programming problems , 1993 .

[10]  Daniel Ralph,et al.  QPECgen, a MATLAB Generator for Mathematical Programs with Quadratic Objectives and Affine Variational Inequality Constraints , 1999, Comput. Optim. Appl..

[11]  Daniel Ralph,et al.  Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints , 1999, SIAM J. Optim..

[12]  Masao Fukushima,et al.  An Implementable Active-Set Algorithm for Computing a B-Stationary Point of a Mathematical Program with Linear Complementarity Constraints , 2002, SIAM J. Optim..

[13]  MANOEL CAMPÊLO,et al.  A Simplex Approach for Finding Local Solutions of a Linear Bilevel Program by Equilibrium Points , 2005, Ann. Oper. Res..

[14]  R. Fletcher,et al.  Numerical experience with solving MPECs as NLPs , 2002 .

[15]  Michal Kočvara,et al.  Nonsmooth approach to optimization problems with equilibrium constraints : theory, applications, and numerical results , 1998 .

[16]  Ein Verfahren zur Maximierung linearer Funktionen in nichtkonvexen Bereichen , 1962 .

[17]  M. Anitescu.  On Solving Mathematical Programs With Complementarity Constraints As Nonlinear Programs , 2002 .

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

[19]  S Scholtes.  Active set methods for inverse linear complementarity problems , 1999 .

[20]  Joaquim J. Júdice,et al.  On the solution of NP-hard linear complementarity problems , 2002 .

[21]  Steven P. Dirkse,et al.  Mathematical Programs with Equilibrium Constraints : Automatic Reformulation and Solution via Constrained Optimization ∗ , 2002 .

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

[23]  Masao Fukushima,et al.  A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints , 1998, Comput. Optim. Appl..

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

[25]  Paul H. Calamai,et al.  Generating Linear and Linear-Quadratic Bilevel Programming Problems , 1993, SIAM J. Sci. Comput..

[26]  Francisco Facchinei,et al.  A smoothing method for mathematical programs with equilibrium constraints , 1999, Math. Program..

[27]  Joaquim Júdice,et al.  A computational analysis of LCP methods for bilinear and concave quadratic programming , 1991, Comput. Oper. Res..

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

[29]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[30]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[31]  Sven Leyffer.  The penalty interior-point method fails to converge , 2005, Optim. Methods Softw..

[32]  A. Fischer.  A special newton-type optimization method , 1992 .

[33]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[34]  Philip E. Gill,et al.  Practical optimization , 1981 .