Global Newton-type methods and semismooth reformulations for NCP
type: Item
instance of: scholarly article
title: Global Newton-type methods and semismooth reformulations for NCP
zbMATH Open document ID: 1016.65042
DOI: 10.1016/S0168-9274(02)00169-1
published in: Applied Numerical Mathematics
publication date: 2003-03-10T00:00:00Z
review text: A popular way to reformulate a nonlinear complementarity problem (NCP) \[ x_if_i(x_1,\ldots,x_n)=0,\quad x_i\geq 0,\qquad f_i(x_1,\ldots,x_n)\geq 0,\quad i=1\ldots,n \] as a nonlinear system consists in choosing an NCP-function, i. e., a function \(\varphi\) of two variables such that \(\varphi(a,b)=0\) iff \(a\geq 0, b\geq 0\) and \(ab=0\), and then obtaining the equivalent nonlinear system \[ \varphi(x_i,f_i(x_1,\ldots,x_n))=0,\quad i=1\ldots,n. \] A hybrid Newton-type method is presented which combines a finite-difference Newton-type method for solving the last system with a cheap direct search method for the minimization of a continuously differentiable merit function; the Newton-type method is enhanced by a line-search strategy based on a continuous merit function. Two different merit functions are used, one for the line search strategy in a primary Newton-type method and the other for the auxiliary direct search method: both of them have to be equivalent to the natural merit function. Under standard assumptions on the level sets of the merit functions, the method is proved to be globally convergent with fast local rate of convergence. The method is implemented with several choices for the reformulation and the merit function used in the line-search procedure. The numerical results show a good performance of the method with all the choices.
reviewed by: Horst Hollatz
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 90C53
zbMATH DE Number: 1879015
zbMATH Keywords: nonlinear complementarity problems
zbMATH Keywords: NCP-functions
zbMATH Keywords: semismooth systems
zbMATH Keywords: globally convergence
zbMATH Keywords: numerical examples
zbMATH Keywords: Newton-type method
zbMATH Keywords: direct search method
author: Sandra Pieraccini
author: M. G. Gasparo
author: Aldo Pasquali
describes a project that uses: GQTPAR
MaRDI profile type: Publication
cites work: A penalized Fischer-Burmeister NCP-function
cites work: Q4000274
cites work: A semismooth equation approach to the solution of nonlinear complementarity problems
cites work: A theoretical and numerical comparison of some semismooth algorithms for complementarity problems
cites work: A New Merit Function For Nonlinear Complementarity Problems And A Related Algorithm
cites work: Engineering and Economic Applications of Complementarity Problems
cites work: A special newton-type optimization method
cites work: On the resolution of monotone complementarity problems
cites work: Test examples for nonlinear programming codes
cites work: A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems
cites work: Q4252822
cites work: A new class of semismooth Newton-type methods for nonlinear complementarity problems
cites work: Direct search methods: Then and now
cites work: On the local convergence of quasi-Newton methods for nonlinear complementarity problems
cites work: Inexact trust region method for large sparse systems of nonlinear equations
cites work: Mathematical Programs with Equilibrium Constraints
cites work: Equivalence of the Complementarity Problem to a System of Nonlinear Equations
cites work: Computing a Trust Region Step
cites work: A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems
cites work: Hybrid Newton-type method for a class of semismooth equations
cites work: Optimization. Algorithms and consistent approximations
cites work: Secant methods for semismooth equations
cites work: Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations
cites work: A nonsmooth version of Newton's method
cites work: Newton and Quasi-Newton Methods for a Class of Nonsmooth Equations and Related Problems
cites work: Growth behavior of a class of merit functions for the nonlinear complementarity problem
full work available at URL: s0168-9274(02)00169-1
OpenAlex ID: W1993739881
description: scientific article; zbMATH DE number 1879015