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On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility an ..."
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
Superlinear convergence of an algorithm for monotone linear complementarity problems, when no strictly complementary solution exists
 Math. Oper. Res
, 1999
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On Smoothing Methods for the P 0 Matrix Linear Complementarity Problem
, 1998
"... In this paper, we propose a BigM smoothing method for solving the P 0 matrix linear complementarity problem. We study the trajectory defined by the augmented smoothing equations and global convergence of the method under an assumption that the original P 0 matrix linear complementarity problem has ..."
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Cited by 4 (1 self)
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In this paper, we propose a BigM smoothing method for solving the P 0 matrix linear complementarity problem. We study the trajectory defined by the augmented smoothing equations and global convergence of the method under an assumption that the original P 0 matrix linear complementarity problem has a solution. Key words. linear complementarity problem, P 0 matrix, smoothing algorithm. AMS subject classifications. 65H10, 90C30, 90C33. This author's work was supported by the Australian Research Council. y This author is supported in part by NSF Grants DMI9522507 and DMS9703490. 2 1 Introduction In this paper we consider the linear complementarity problem (LCP) t T s = 0; s = Mt + q; and t; s 0; where M is an n \Theta n P 0 matrix and q is an n dimensional vector. A matrix M 2 R n\Thetan is called a P 0 matrix if max i:t i 6=0 t i (Mt) i 0; for all t 2 R n ; t 6= 0: A linear complementarity problem is called a P 0 matrix LCP if the matrix M is a P 0 matrix. The clas...
Unified analysis of kernelbased interiorpoint methods for P∗(κ)LCP
 SIAM Journal on Optimization
, 2010
"... Part of the Mathematics Commons This article is brought to you for free and open access by the Department of Mathematical Sciences at Digital Commons@Georgia Southern. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of Digital Commons@G ..."
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Part of the Mathematics Commons This article is brought to you for free and open access by the Department of Mathematical Sciences at Digital Commons@Georgia Southern. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of Digital Commons@Georgia Southern.
An Application Of Carver's Theorem To Monotone Linear Complementarity Problems
"... h is sufficient for the boundedness of the solution set F , see Mangasarian [3]. Corollary 3 in Ye [6] states that the existence of such x and s is also necessary in the case of monotone LCPs with a socalled negative qvalue. To see that it is necessary in general, suppose that there exist no ..."
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h is sufficient for the boundedness of the solution set F , see Mangasarian [3]. Corollary 3 in Ye [6] states that the existence of such x and s is also necessary in the case of monotone LCPs with a socalled negative qvalue. To see that it is necessary in general, suppose that there exist no such positive solution pair (x; s). Then, by Carver's theorem [1], there exists a direction u 6= 0 such that M T u 0; u 0; h T u 0: If h T u ! 0 then F = ;, by Farkas' lemma. Suppose now that there exist (x ; s ) 2 F , and consequently h T u = 0. Since u 0 and M T u 0, w
A HOMOGENEOUS MODEL FOR MONOTONE MIXED HORIZONTAL LINEAR COMPLEMENTARITY PROBLEMS
"... Abstract. We propose a homogeneous model for the class of mixed horizontal linear complementarity problems. The proposed homogeneous model is always solvable and provides the solution of the original problem if it exists, or a certificate of infeasibility otherwise. Our formulation preserves the s ..."
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Abstract. We propose a homogeneous model for the class of mixed horizontal linear complementarity problems. The proposed homogeneous model is always solvable and provides the solution of the original problem if it exists, or a certificate of infeasibility otherwise. Our formulation preserves the sparsity of the original formulation and does not reduce to the homogeneous model of the equivalent standard linear complementarity problem. We study the properties of the model and show that interiorpoint methods can be used efficiently for the numerical solutions of the homogeneous problem. Numerical experiments show convincingly that it is much more efficient to use the proposed homogeneous model for the mixed horizontal linear complementarity problem than to use known homogeneous models for the equivalent standard linear complementarity problem.
LongStep . . . P*NONLINEAR COMPLEMENTARITY PROBLEMS
, 2002
"... A P *Nonlinear Complementarity Problem as a generalization of the P *Linear Complementarity Problem is considered. We show that the longstep version of the homogeneous selfdual interiorpoint algorithm could be used to solve such a problem. The algorithm achieves linear global convergence and ..."
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A P *Nonlinear Complementarity Problem as a generalization of the P *Linear Complementarity Problem is considered. We show that the longstep version of the homogeneous selfdual interiorpoint algorithm could be used to solve such a problem. The algorithm achieves linear global convergence and quadratic local convergence under the following assumptions: the function satisfies a modified scaled Lipschitz condition, the problem has a strictly complementary solution, and certain submatrix of the Jacobian is nonsingular on some compact set.