Reformulations of mathematical programming problems as linear complementarity problems

A family of complementarity problems are defined as extensions of the well known Linear Complementarity Problem (LCP). These are (i.) Second Linear Complementarity Problem (SLCP) which is an LCP extended by introducing further equality restrictions and unrestricted variables, (ii.) Minimum Linear Complementarity Problem (MLCP) which is an LCP with additional variables not required to be complementary and with a linear objective function which is to be minimized, (iii.) Second Minimum Linear Complementarity Problem (SMLCP) which is an MLCP but the nonnegative restriction on one of each pair of complementary variables is relaxed so that it is allowed to be unrestricted in value. A number of well known mathematical programming problems, namely quadratic programming (convex, nonconvex, pseudoconvex nonconvex), bilinear programming, game theory, zero-one integer programming, the fixed charge problem, absolute value programming, variable separable programming are reformulated as members of this family of four complementarity problems

The extended linear complementarity problem

In this paper we define the Extended Linear Complementarity Problem (ELCP), an extension of the well-known Linear Complementarity Problem (LCP). We study the general solution set of an ELCP and we present an algorithm to find all its solutions. Finally we show that the ELCP can be used to solve some important problems in the max algebra

An enumerative method for the solution of linear complementarity problems

In this report an enumerative method for the solution of the Linear Complementarity Problem (LCP) is presented. This algorithm completely processes the LCP, and does not require any particular property of the LCP to apply. That is the algorithm terminates after either finding all the solutions of an LCP or establishing that no solution exists. The method is extended to also process the Second Linear Complementarity Problem (SLCP), a problem which has been introduced to represent the general quadratic program involving unrestricted variables

The Reduced Order Method for Solving the Linear Complementarity Problem with an M-Matrix

In this paper, by seeking the zero and the positive entry positions of the solution, we provide a direct method, called the reduced order method, for solving the linear complementarity problem with an M-matrix. By this method, the linear complementarity problem is transformed into a low order linear complementarity problem with some low order linear equations and the solution is constructed by the solution of the low order linear complementarity problem and the solutions of these low order linear equations in the transformations. In order to show the accuracy and the effectiveness of the method, the corresponding numerical experiments are performed

On a "stability" in the linear complementarity problem

In this work we rewrote the linear complementarity problem in a formulation based on unknown projector operators. In particular, this formulation allows the introduction of a concept of "stability" that, in a certain way, might explain the way block pivotal algorithm performs. (c) 2008 Elsevier Inc. All rights reserved.info:eu-repo/semantics/publishedVersio