We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a P 0 -function we prove that every stationary point of the unconstrained problem is a global solution; furthermore, if the complementarity problem is defined by a uniform P -function, the level sets of the merit function are bounded. The properties of the new merit function are compared with those of the Mangasarian-Solodov's implicit Lagrangian and Fukushima's regularized gap function. We also introduce a new, simple, active-set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function.

this paper we consider the minimization problem min ff(x); x 2 !

: We investigate the numerical behavior of a new, simple algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a recently proposed merit function which possesses some interesting theoretical properties. One of the aims of the paper is to show that algorithms based on this merit function can be viable also from the numerical point of view. Key Words: Nonlinear complementarity problem, merit function, SC 1 function, global convergence, superlinear convergence. 1. INTRODUCTION We consider the nonlinear complementarity problem: F (x) 0; x 0; F (x) T x = 0; (NC) where F : IR n ! IR n is continuously differentiable. Recent research on the numerical solution of Problem (NC) has focused on the development of globally convergent algorithms. To this end, two approaches have been investigated: the transformation of the nonlinear complementarity problem into a system of (nonsmooth) equations and the use of continuation methods. Strictly related t...

This paper investigates inexact Uzawa methods for nonlinear saddle point problems. We prove that the inexact Uzawa method converges globally and superlinearly even if the derivative of the nonlinear mapping does not exist. We show that the Newton-type decomposition method for saddle point problems is a special case of a Newton-Uzawa method. We discuss applications of inexact Uzawa methods to separable convex programming problems and coupling of finite elements/boundary elements for nonlinear interface problems. Key words. saddle point, nonsmooth, Uzawa, Newton, inexact, inner/outer, convergence. AMS subject classifications. 65H10 Abbreviated title. Inexact Uzawa Method This work is supported by the Australian Research Council. 1 Introduction We consider the nonlinear saddle point problem H(x; y) = " F (x) + B T y \Gamma p Bx \Gamma Cy \Gamma q # = 0; (1.1) where B is an m \Theta n matrix, C is an m \Theta m symmetric positive semidefinite matrix, p is a vector in ! n ...

Stochastic programming is concerned with practical procedures for decision-making under uncertainty, by modelling uncertainties and risks associated with decisions in a form suitable for optimization. The field is developing rapidly with contributions from many disciplines such as operations research, probability and statistics, and economics. A stochastic linear program with recourse can equivalently be formulated as a convex programming problem. The problem is often large-scale as the objective function involves an expectation, either over a discrete set of scenarios or as a multidimensional integral. Moreover, the objective function is possibly nondifferentiable. This paper provides a brief overview of recent developments on smooth approximation techniques and Newton-type methods for solving two-stage stochastic linear programs with recourse, and parallel implementation of these methods. A simple numerical example is used to signal the potential of smoothing approaches. 1 Introducti...

A new algorithm for large-scale nonlinear programs with box constraints is introduced. The algorithm is based on an efficient identification technique of the active set at the solution and on a nonmonotone stabilization technique. It possesses global and superlinear convergence properties under standard, mild assumptions. A new technique for generating test problems with known characteristics is also introduced. The implementation of the method is described along with computational results for large-scale problems. 1 Introduction In this paper we consider the solution of the box constrained nonlinear programming problem min x2K f(x) (1) where K = fx 2 IR n : l i x i u i ; i = 1; : : : ; ng (2) is a nonempty set. We assume that the lower and upper bounds may be finite or infinite and that f is a twice continuously differentiable function in an open set containing K. A vector ¯ x 2 K is said to be a stationary point for Problem (1) if it satisfies 8 ? ? ! ? ? : l i = ¯ x i =) ...

. This paper proposes a BFGS-SQP method for linearly constrained optimization where the objective function f is only required to have a Lipschitz gradient. The KKT system of the problem is equivalent to a system of nonsmooth equations F (v) = 0. At every step a quasi-Newton matrix is updated if kF (v k )k satisfies a rule. This method converges globally and the rate of convergence is superlinear when f is twice strongly differentiable at a solution of the optimization problem. No assumptions on the constraints are required. This generalizes classical convergence theory of the BFGS method which requires a twice continuous differentiability assumption on the objective function. Applications to stochastic programs with recourse are discussed on a CM5 parallel computer. Key words: quasi-Newton methods, convex programming, nonsmooth equations. AMS(MOS) subject classification : 90C30, 90C25 Abbreviated title : BFGS method for LC 1 optimization 1 This work is supported by the Australian...