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.

Variational inequalities over sets defined by systems of equalities and inequalities are considered. A continuously differentiable merit function is proposed whose unconstrained minima coincide with the KKT-points of the variational inequality. A detailed study of its properties is carried out showing that under mild assumptions this reformulation possesses many desirable features. A simple algorithm is proposed for which it is possible to prove global convergence and a fast local convergence rate. Preliminary numerical results showing viability of the approach are reported.

. Recently, Peng considered a merit function for the variational inequality problem (VIP), which constitutes an unconstrained dierentiable optimization reformulation of VIP. In this paper, we generalize the merit function proposed by Peng and study various properties of the generalized function. We call this function the D-gap function. We give conditions under which any stationary point of the D-gap function is a solution of VIP and conditions under which it provides a global error bound for VIP. We also present a descent method for solving VIP based on the D-gap function. Key words: Variational inequality problems, unconstrained optimization reformulation, global error bound, descent method. 1 The authors are grateful to C. Kanzow for his comments on an earlier version of the paper. They also thank the referees for their constructive comments. y The work of this author was supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientist...

Abstract. The Josephy–Newton method for solving a nonlinear complementarity problem consists of solving, possibly inexactly, a sequence of linear complementarity problems. Under appropriate regularity assumptions, this method is known to be locally (superlinearly) convergent. To enlarge the domain of convergence of the Newton method, some globalization strategy based on a chosen merit function is typically used. However, to ensure global convergence to a solution, some additional restrictive assumptions are needed. These assumptions imply boundedness of level sets of the merit function and often even (global) uniqueness of the solution. We present a new globalization strategy for monotone problems which is not based on any merit function. Our linesearch procedure utilizes the regularized Newton direction and the monotonicity structure of the problem to force global convergence by means of a (computationally explicit) projection step which reduces the distance to the solution set of the problem. The resulting algorithm is truly globally convergent in the sense that the subproblems are always solvable, and the whole sequence of iterates converges to a solution of the problem without any regularity assumptions. In fact, the solution set can even be unbounded. Each iteration of the new method has the same order of computational cost as an iteration of the damped Newton method. Under natural assumptions, the local superlinear rate of convergence is also achieved. Key words. nonlinear complementarity problem, Newton method, proximal point method, projection method, global convergence, superlinear convergence

We consider a simply constrained optimization reformulation of the Karush-Kuhn-Tucker conditions arising from variational inequalities. Based on this reformulation, we present a new Newton-type method for the solution of variational inequalities. The main properties of this method are: (a) it is well-defined for an arbitrary variational inequality problem, (b) it is globally convergent at least to a stationary point of the constrained reformulation, (c) it is locally superlinearly/quadratically convergent under a certain regularity condition, (d) all iterates remain feasible with respect to the constrained optimization reformulation, and (e) it has to solve just one linear system of equations at each iteration. Some preliminary numerical results indicate that this method is quite promising.

The variational inequality problem (VIP) can be reformulated as an unconstrained minimization problem through the D-gap function. It is proved that the D-gap function has bounded level sets for the strongly monotone VIP. A hybrid Newton-type method is proposed for minimizing the D-gap function. Under some conditions, it is shown that the algorithm is globally convergent and locally quadratically convergent. Keywords: Variational inequality problem, D-gap function, Newton's method, unconstrained optimization, global convergence, quadratic convergence. 1 Introduction The variational inequality problem (VIP) is to nd a vector x 3 2 X such that hF (x 3 ); y x 3 i 0; 8 y 2 X; (1) where X is a nonempty closed convex subset of < n , F is a mapping from < n into itself, and h; i denotes the inner product in < n . If the constraint set X is the nonnegative orthant in < n , then the VIP reduces to the complementarity problem (CP). VIPs and CPs have been widely studied in var...

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This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to mixed complementarity problems, variational inequalities and mathematical programs with equilibrium constraints are also discussed.

The purpose of this paper is to provide a unified description of iterative algorithms for the solution of traffic equilibrium problems. We demonstrate that a large number of well known solution techniques can be described in a unified manner through the concept of partial linearization, and establish close relationships with other algorithmic classes for nonlinear programming and variational inequalities. In the case of nonseparable travel costs, the class of partial linearization algorithms are shown to yield new results in the theory of finite-dimensional variational inequalities. The possibility of applying truncated algorithms within the framework is also discussed.

. A box constrained variational inequality problem can be reformulated as an unconstrained minimization problem through the D-gap function. Some basic properties of the affine variational inequality subproblems in the classical Josephy-Newton method are studied. A hybrid Josephy-Newton method is then proposed for minimizing the D-gap function. Under suitable conditions, the algorithm is shown to be globally convergent and locally quadratically convergent. Some numerical results are also presented. Key words: Variational inequality problem, box constraints, D-gap function, Newton's method, unconstrained optimization, global convergence, quadratic convergence. 2 The research of this author was supported by Project 19601035 of NSFC in China. 4 Current address (October 1, 1997 --- September 30, 1998): Computer Sciences Department, University of Wisconsin --- Madison, 1210 West Dayton Street, 53706 Madison, WI; e-mail: kanzow@cs.wisc.edu. The research of this author was supported by th...