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Modified ProjectionType Methods For Monotone Variational Inequalities
 SIAM Journal on Control and Optimization
, 1996
"... . We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods may be viewed as projectiontype methods in which the projection direction is modified by a strongly monotone mapping of the form I \Gamma ffF or, if F is affine with un ..."
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. We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods may be viewed as projectiontype methods in which the projection direction is modified by a strongly monotone mapping of the form I \Gamma ffF or, if F is affine with underlying matrix M , of the form I + ffM T , with ff 2 (0; 1). We show that these methods are globally convergent and, if in addition a certain error bound based on the natural residual holds locally, the convergence is linear. Computational experience with the new methods is also reported. Key words. Monotone variational inequalities, projectiontype methods, error bound, linear convergence. AMS subject classifications. 49M45, 90C25, 90C33 1. Introduction. We consider the monotone variational inequality problem of finding an x 2 X satisfying F (x ) T (x \Gamma x ) 0 8x 2 X; (1) where X is a closed convex set in ! n and F is a monotone and continuous function from ! n to ...
A Class Of Globally Convergent Algorithms For Pseudomonotone Variational Inequalities
, 2001
"... We describe a fairly broad class of algorithms for solving variational inequalities, global convergence of which is based on the strategy of generating a hyperplane separating the current iterate from the solution set. The methods are shown to converge under very mild assumptions. Specifically, the ..."
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Cited by 3 (3 self)
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We describe a fairly broad class of algorithms for solving variational inequalities, global convergence of which is based on the strategy of generating a hyperplane separating the current iterate from the solution set. The methods are shown to converge under very mild assumptions. Specifically, the problem mapping is only assumed to be continuous and pseudomonotone with respect to at least one solution. The strategy to obtain (super)linear rate of convergence is also discussed. The algorithms in this class di#er in the tools which are used to construct the separating hyperplane. Our general scheme subsumes an extragradienttype projection method, a globally and locally superlinearly convergent JosephyNewtontype method, a certain minimizationbased method, and a splitting technique. 1 INTRODUCTION Given a function F : R n # R n and a set C # R n , the classical variational inequality problem [1, 3, 8, 9, 5], abbreviated VIP(F, C), is to find a point x such that x # C, #F ...
Error Bounds for Inconsistent Linear Inequalities and Programs
 Operations Research Letters
, 1994
"... For any system of linear inequalities, consistent or not, the norm of the violations of the inequalities by a given point, multiplied by a condition constant that is independent of the point, bounds the distance between the point and the nonempty set of points that minimize these violations. Similar ..."
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Cited by 2 (1 self)
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For any system of linear inequalities, consistent or not, the norm of the violations of the inequalities by a given point, multiplied by a condition constant that is independent of the point, bounds the distance between the point and the nonempty set of points that minimize these violations. Similarly, for a dual pair of possibly infeasible linear programs, the norm of violations of primaldual feasibility and primaldual objective equality, when multiplied by a condition constant, bounds the distance between a given point and the nonempty set of minimizers of these violations. These results extend error bounds for consistent linear inequalities and linear programs to inconsistent systems. Keywords error bounds; linear inequalities; linear programs Error bounds are playing an increasingly important role in mathematical programming. Beginning with Hoffman's classical error bound for linear inequalities [3], many papers have examined error bounds for linear and convex inequalities, line...
Error estimation for nonlinear complementarity problems via linear systems with interval data
 NUMER. FUNCT. ANAL. OPTIM
, 2008
"... For the nonlinear complementarity problem we derive norm bounds for the error of an approximate solution, generalizing the known results for the linear case. Furthermore, we present a linear system with interval data, whose solution set contains the error of an approximate solution. We perform exten ..."
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For the nonlinear complementarity problem we derive norm bounds for the error of an approximate solution, generalizing the known results for the linear case. Furthermore, we present a linear system with interval data, whose solution set contains the error of an approximate solution. We perform extensive numerical tests and compare the different approaches.
Error Bounds of Pmatrix Linear
"... The linear complementarity problem is to find a vector $x\in R^{n} $ such that $Mx+q\geq 0 $ , $x\geq 0 $ , $x^{T}(Mx+q)=0 $, or to show that no such vector exists, where $M\in R^{n\mathrm{x}n} $ and $q\in R^{n} $. We denote this problem by LCP(f, $q $). A matrix $M $ is called a $\mathrm{P}$matrix ..."
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The linear complementarity problem is to find a vector $x\in R^{n} $ such that $Mx+q\geq 0 $ , $x\geq 0 $ , $x^{T}(Mx+q)=0 $, or to show that no such vector exists, where $M\in R^{n\mathrm{x}n} $ and $q\in R^{n} $. We denote this problem by LCP(f, $q $). A matrix $M $ is called a $\mathrm{P}$matrix if
The Auxiliary Problem Algorithm for Generalized Linear Complementarity Problem Over a Polyhedral Cone
, 2007
"... In this paper, we consider an auxiliary problem algorithm for solving the generalized linear complementarity problem over a polyhedral cone (GLCP). First, we equivalently reformulate the GLCP as an affine variational inequalities problem over a polyhedral cone via a linearly constrained quadratic p ..."
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In this paper, we consider an auxiliary problem algorithm for solving the generalized linear complementarity problem over a polyhedral cone (GLCP). First, we equivalently reformulate the GLCP as an affine variational inequalities problem over a polyhedral cone via a linearly constrained quadratic programming under suitable assumptions, based on which we propose an auxiliary problem method to solve the GLCP and establish its global convergence. A numerical experiments of the method are also reported in this paper.
The Convergence Analysis of the Auxiliary Problem Method for Monotone Affine Variational Inequalities
"... In this paper, we provide the convergence analysis of the auxiliary problem method for solving the monotone affine variational inequalities (M.C. Ferris and O.L. Mangasarian, Annals of Operations Research, 47, 1993, 293305). The convergence rate was also given under suitable conditions. ..."
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In this paper, we provide the convergence analysis of the auxiliary problem method for solving the monotone affine variational inequalities (M.C. Ferris and O.L. Mangasarian, Annals of Operations Research, 47, 1993, 293305). The convergence rate was also given under suitable conditions.