Solving discretely constrained, mixed linear complementarity problems with applications in energy

This paper concerns the existence of solutions to the discrete nonlinear complementarity problem. The problem is that of finding an integer vector x in the n-dimensional Euclidean space such that both x and f(x) are nonnegative, and the inner product of x and f(x) is equal to zero, where f is a nonlinear function mapping from the n-dimensional Euclidean space into itself. Several sufficient conditions are introduced to show the existence of solutions or a unique solution to the problem. Meanwhile, we study the closely related discrete fixed point problem and provide sufficient conditions for the existence of discrete fixed points. In addition, two economic applications are discussed.

On the Solutions of Discrete Nonlinear Complementarity and Related Problems

We characterize the class of integral square matrices M having the property that for every integral vector q the linear complementarity problem with data M, q has only integral basic solutions. These matrices, called principally unimodular matrices?, are those for which every principal nonsingular submatrix is unimodular. As a consequence, we show that if M is rank-symmetric and principally unimodular, and q is integral, then the problem has an integral solution if it has a solution. Principal unimodularity can be regarded as an extension of total unimodularity, and our results can be regarded as extensions of well-known results on integral solutions to linear programs. We summarize what is known about principally unimodular symmetric and skew-symmetric matrices.

Integral Solutions of Linear Complementarity Problems

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.

A method for solving the general parametric linear complementarity problem

In this paper we present a novel formulation based on a median function to solve discretely constrained mixed complementarity problems (MCPs). Such problems seek to combine integer (discrete) solutions that are also equilibrium ones and have applications in engineering and economics. Several theoretical results show the correspondence between the easier-to-solve formulations presented and the original discretely constrained MCP. Lastly, the approach is successfully tested on a variety of illustrative examples.

Solving discretely constrained mixed complementarity problems using a median function

We consider the problem of finding an integer solution to a linear complementarity problem. We introduce the class I of matrices for which the corresponding linear complementarity problem has an integer complementary solution for every vector, q, for which it has a solution. Strong principal unimodularity forms a sufficient condition for inclusion in the class I. It is also shown to be necessary for some well-known classes of matrices, including the class of positive semidefinite matrices. This is used in deriving necessary and sufficient conditions for a convex quadratic program to have an integer optimal solution for all b and c for which it has an optimal solution. Characterizations are also derived for some other well-known classes of matrices in linear complementarity to belong to the class I. In the end, a peeling algorithm for finding integer solution for linear complementarity problems is given and related results are derived.

Integer Solution for Linear Complementarity Problem

The concept of Δ-matroid is a nontrivial, proper generalization of the concept of matroid and has been further generalized to the concept of jump system. In this paper, we show that jump systems are, in some sense, equivalent to Δ-matroids. Using this equivalence and the Δ-matroid theory, we give simple proofs and extensions of many of the results on jump systems.

R. Sridhar

Papers

Integer Solution for Linear Complementarity Problem

Delta-matroid and jump system.