Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints∗ Houyuan Jiang CSIRO Mathematical and Information Science Private Bag 2 Glen Osmond, SA 5064, Australia Email:
[email protected] Daniel Ralph The University of Melbourne Department of Mathematics and Statistics Parkville, Vic. 3052, Australia Email:
[email protected] December 29, 1997; Revised August, 1999 and January 2000. The final version of this paper appears in Computational Optimization and Applications, 25 (2002), pp. 123–150
Abstract Mathematical programs with nonlinear complementarity constraints are reformulated using better-posed but nonsmooth constraints. We introduce a class of functions, parameterized by a real scalar, to approximate these nonsmooth problems by smooth nonlinear programs. This smoothing procedure has the extra benefits that it often improves the prospect of feasibility and stability of the constraints of the associated nonlinear programs and their quadratic approximations. We present two globally convergent algorithms based on sequential quadratic programming, SQP, as applied in exact penalty methods for nonlinear programs. Global convergence of the implicit smooth SQP method depends on existence of a lower-level nondegenerate (strictly complementary) limit point of the iteration sequence. Global convergence of the explicit smooth SQP method depends on a weaker property, i.e. existence of a limit point at which a generalized constraint qualification holds. We also discuss some practical matters relating to computer implementations.
Key Words. Mathematical programs with equilibrium constraints, bilevel optimization, complementarity problems, sequential quadratic programming, exact penalty, generalized constraint qualification, global convergence, smoothing method. AMS (MOS) Subject Classifications. 90C, 90D65. ∗
This work is supported by the Australian Research Council.
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Introduction
Mathematical programs with equilibrium constraints (MPEC for short) form a relatively new and interesting class of optimization problems. The roots of MPEC lie in game theory and especially bilevel optimization. MPEC include a number of significant applications in economics and engineering. See the monograph [28] for comprehensive theoretical treatment, applications and references. The MPEC considered in this paper is a mathematical program with nonlinear complementarity problem (NCP) constraints: min f (x, y) x,y
subject to g(x, y) ≥ 0 0 ≤ F (x, y) ⊥ y ≥ 0
(1)
where f :
