On solving linear complementarity problems as linear programs

Journal of Global Optimization, 1995

We de ne a minimization problem with simple bounds associated to the horizontal linear complementarity problem (HLCP). When the HLCP is solvable, its solutions are the global minimizers of the associated problem. When the HLCP is feasible, we are able to prove a number of properties of the stationary points of the associated problem. In many cases, the stationary points are solutions of the HLCP. The theoretical results allow us to conjecture that local methods for box constrained optimization applied to the associated problem are e cient tools for solving linear complementarity problems. Numerical experiments seem to con rm this conjecture.

Siam Journal on Optimization, 2008

This paper presents a parameter-free integer-programming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPCC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes-infeasibility, unboundedness, or solvability-of an LPCC. An extreme point/ray generation scheme in the spirit of Benders decomposition is developed, from which valid inequalities in the form of satisfiability constraints are obtained. The feasibility problem of these inequalities and the carefully guided linear programming relaxations of the LPCC are the workhorse of the algorithm, which also employs a specialized procedure for the sparsification of the satifiability cuts. We establish the finite termination of the algorithm and report computational results using the algorithm for solving randomly generated LPCCs of reasonable sizes. The results establish that the algorithm can handle infeasible, unbounded, and solvable LPCCs effectively.

1989

In this paper, we first propose three practlca} algorithms for linear complementarity probleins, whic]i are based on the polynomial time method of Kojima, Mizuno and Yoshise [5l , and compare thern by showing the computational complexities. Then we modify two of the algorithms in order to accelerate them. Through the computational experiments for three types of linear complementarity problerns, we cempare the proposed algorithms in praetice and see the efficiency of the modified algorithms. We also esrimate the practical cornputational complexity ofeach algorithm fbr each type ofproblems.

European Journal of Operational Research, 1989

The problem addressed in this paper is the linear complementarity problem with upper bounds. First we show, using only combinatorial arguments, that the number of solutions is odd. Secondly a variable dimension algorithm based on complementary pivoting is given to find such a solution.

Lecture Notes in Computer Science, 2013

In this paper, we study the sparse linear complementarity problem, denoted by k-LCP: the coefficient matrix has at most k nonzero entries per row. It is known that 1-LCP is solvable in linear time, while 3-LCP is strongly NP-hard. We show that 2-LCP is strongly NP-hard, while it can be solved in O(n 3 log n) time if it is sign-balanced, i.e., each row has at most one positive and one negative entries, where n is the number of constraints. Our second result matches with the currently best known complexity bound for the corresponding sparse linear feasibility problem. In addition, we show that an integer variant of sign-balanced 2-LCP is weakly NP-hard and pseudo-polynomially solvable, and the generalized 1-LCP is strongly NP-hard.

Mathematical Programming, 1990

It has been shown by Lemke that if a matrix is copositive plus on IR n , then feasibility of the corresponding Linear Complementarity Problem implies solvability. In this article we show, under suitable conditions, that feasibility of a Generalized Linear Complementarity Problem (i.e., de ned over a more general closed convex cone in a real Hilbert Space) implies solvability whenever the operator is copositive plus on that cone. We show that among all closed convex cones in a nite dimensional real Hilbert Space, polyhedral cones are the only ones with the property that every copositive plus, feasible GLCP is solvable. We also prove a perturbation result for Generalized Linear Complementarity Problems.

Journal of Computational and Applied Mathematics, 2000

This paper provides an introduction to complementarity problems, with an emphasis on applications and solution algorithms. Various forms of complementarity problems are described along with a few sample applications, which provide a sense of what types of problems can be addressed e ectively with complementarity problems. The most important algorithms are presented along with a discussion of when they can be used e ectively. We also provide a brief introduction to the study of matrix classes and their relation to linear complementarity problems. Finally, we provide a brief summary of current research trends.

1996

Linear Algebra Appl, 1993

Specially structured Linear Complementarity Problems (LCP's) and their solution by the criss{ cross method are examined in this paper. The criss{cross method is known to be nite for LCP's with positive semide nite bisymmetric matrices and with P{matrices. It is also a simple nite algorithm for oriented matroid programming problems. Recently Cottle, Pang and Venkateswaran identi ed the class of (column, row) su cient matrices. They showed that su cient matrices are a common generalization of P{ and PSD{matrices. Cottle also showed that the principal pivoting method (with a clever modi cation) can be applied to row su cient LCP's. In this paper the niteness of the criss{cross method for su cient LCP's is proved. Further it is shown that a matrix is su cient if and only if the criss{cross method processes all the LCP's de ned by this matrix and all the LCP's de ned by the transpose of this matrix and any parameter vector.