A Truly Globally Convergent Newton-Type Method for the Monotone Nonlinear Complementarity Problem

Abstract

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.

A New Projection Method for Variational Inequality Problems

“…Some of the projection ideas presented here also proved to be useful in devising truly globally convergent (i.e., the whole sequence of iterates is globally convergent to a solution without any regularity assumptions) and locally superlinearly convergent inexact Newton methods for solving systems of monotone equations [30] and monotone NCPs [31].…”

“…In the case when F (·) is strongly monotone and/or the feasible set C has some special structure (e.g., C is the nonnegative orthant or, more generally, a box), there exist many efficient methods that can be used to solve those special cases of VI(F, C) (see [4,5,14,18,19,20,22,23,34,36,2,16,29,15,28,27,31]). In some of those methods, F (·) is further assumed to be differentiable, or Lipschitz continuous, or affine.…”

Forcing strong convergence of proximal point iterations in a Hilbert space

“…Another extension using certain enlargements (outer approximations) of the operator defining the problem can be found in [28]. Using also the linesearch technique of [26], the framework of [31,28] led to the development of truly globally convergent inexact Newton methods for monotone equations [30] and complementarity problems [25]. However, by itself, the method of [31] does not attain the goal of the present paper.…”

A General Iterative Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

“…It is very clear that our algorithm with a variational regularization parameter {r n } has certain advantages over the algorithm with a fixed regularization parameter r. In some setting, when the regularization parameter {r n } depends on the iterative step n, the algorithm may converge to some solution Q-superlinearly, that is, the algorithm has a faster convergence rate when the regularization parameter {r n } depends on n, see [13] and the references therein for more information.…”