Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties

• Published in: Mathematical programming Vol. 110; no. 2; pp. 337 - 371
• Main Authors: Anitescu, Mihai, Tseng, Paul, Wright, Stephen J.
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.07.2007; Springer Nature B.V
• Subjects: ALGORITHMS; Applied sciences; Calculus of variations and optimal control; CONVERGENCE; ELASTICITY; EQUILIBRIUM; Exact sciences and technology; GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; Laboratories; Mathematical analysis; MATHEMATICS; Nonlinear programming; Operational research and scientific management; Operational research. Management science; Regularization methods; Sciences and techniques of general use; Studies; 65K05; Strong stationarity; Minimization; Iterative method; Numerical method; Complementarity problem; Non linear programming; Equilibrium condition; Asymptotic convergence; Constrained optimization; Active constraint; Numerical analysis; 90C30; Nonlinear programming; Equilibrium constraints; Complementarity constraints; Elastic-mode formulation; 90C33; C-stationarity· M-stationarity; 49M30; 49M37; Regularity; Numerical convergence; Mathematical programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-006-0005-4

The elastic-mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first- and second-order necessary optimality conditions for the original problem are also first- and second-order points of the elastic-mode formulation. Here we study global convergence properties of methods based on this formulation, which involve generating an (exact or inexact) first- or second-order point of the formulation, for nondecreasing values of the penalty parameter. Under certain regularity conditions on the active constraints, we establish finite or asymptotic convergence to points having a certain stationarity property (such as strong stationarity, M-stationarity, or C-stationarity). Numerical experience with these approaches is discussed. In particular, our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elastic-mode formulation is solved inexactly. [PUBLICATION ABSTRACT]