Complementarity constraints as nonlinear equations: Theory and numerical experience

Recently, it has been shown that mathematical programs with complementarity constraints (MPCCs) can be solved efficiently and reliably as nonlinear programs. This paper examines various nonlinear formulations of the complementarity constraints. Several nonlinear complementarity functions are considered for use in MPCC. Unlike standard smoothing techniques, however, the reformulations do not require the control of a smoothing parameter. Thus they have the advantage that the smoothing is exact in the sense that Karush-Kuhn-Tucker points of the reformulation correspond to strongly stationary points of the MPCC. A new exact smoothing of the well-known min function is also introduced and shown to possess desirable theoretical properties. It is shown how the new formulations can be integrated into a sequential quadratic programming solver, and their practical performance is compared on a range of test problems.

[1]  R. Fletcher.  Practical Methods of Optimization , 1988 .

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

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

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

[5]  A. Fischer.  A Newton-type method for positive-semidefinite linear complementarity problems , 1995 .

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

[7]  Patrick T. Harker,et al.  Smooth Approximations to Nonlinear Complementarity Problems , 1997, SIAM J. Optim..

[8]  Michael C. Ferris,et al.  Engineering and Economic Applications of Complementarity Problems , 1997, SIAM Rev..

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

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

[12]  Xiaojun Chen,et al.  A penalized Fischer-Burmeister NCP-function , 2000, Math. Program..

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

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

[15]  Francisco Facchinei,et al.  The Semismooth Algorithm for Large Scale Complementarity Problems , 2001, INFORMS J. Comput..

[16]  Todd Munson,et al.  Benchmarking optimization software with COPS. , 2001 .

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

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

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

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

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

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

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

[24]  Nicholas I. M. Gould,et al.  Global Convergence of a Trust-Region SQP-Filter Algorithm for General Nonlinear Programming , 2002, SIAM J. Optim..

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

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

[27]  Jie Sun,et al.  Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints , 2004, Math. Program..

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

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

[30]  Lorenz T. Biegler,et al.  An Interior Point Method for Mathematical Programs with Complementarity Constraints (MPCCs) , 2005, SIAM J. Optim..

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

[32]  Robert J. Vanderbei,et al.  Interior-Point Algorithms, Penalty Methods and Equilibrium Problems , 2006, Comput. Optim. Appl..