# Linear complementarity problems solvable by A single linear program

@article{Mangasarian1976LinearCP, title={Linear complementarity problems solvable by A single linear program}, author={Olvi L. Mangasarian}, journal={Mathematical Programming}, year={1976}, volume={10}, pages={263-270} }

It is shown that the linear complementarity problem of finding az inRn such thatMz + q ⩾ 0, z ⩾ 0 andzT(Mz + q) = 0 can be solved by a single linear program in some important special cases including those whenM or its inverse is a Z-matrix, that is a real square matrix with nonpositive off-diagonal elements. As a consequence certain problems in mechanics, certain problems of finding the least element of a polyhedral set and certain quadratic programming problems, can each be solved by a single…

## 110 Citations

L-Matrices and solvability of linear complementarity problems by a linear program-Matrices and solvability of linear complementarity problems by a linear program

- Mathematics
- 2003

The classes ofL1-matrices,L2-matrices,L3-matrices andW-matrices are introduced to study solvability of a linear complementarity problem via solving a linear program. Three sufficient conditions are…

Characterization of linear complementarity problems as linear programs

- Mathematics
- 1978

It is shown that the linear complementarity problem of finding an n-by-1 vector x such that Mx+q≧0, x≧0, and x T(Mx+q)=0, where M is a given n-by-n real matrix and q is a given n-by-1 vector, is…

A note on an open problem in linear complementarity

- MathematicsMath. Program.
- 1977

This note proves a result which improves on a characterization obtained by Mangasarian of the class ofn × n matricesM such that for everyn-vectorq for which the linear complementarity problem (q, M) is feasible, then the problem has a solution.

Simplified Characterizations of Linear Complementarity Problems Solvable as Linear Programs

- MathematicsMath. Oper. Res.
- 1979

New and simplified characterizations are given for solving, as a linear program, the linear complementarity problem of finding an x in Rn such that Mx + q ≥ 0, x ≥ 0 and x1Mx + q = 0. The simplest…

On a class of least-element complementarity problems

- MathematicsMath. Program.
- 1979

Two least-element characterizations of solutions to the above linear complementarity problem are established and a new and direct method to solve this class of problems, which depends on the idea of “least-element solution” is presented.

On characterizing linear complementarity problems as linear programs

- Economics
- 1989

Using a new bilinear programming formulation of the linear complementarity problem, we simplify Mangasabian's necessary and sufficient conditions under which these problems can be solved as linear…

Iterative algorithms for the linear complementarity problem

- Computer Science, Mathematics
- 1986

Some work of van Bokhoven is extended to a class of nonsymmetric P-matrices, and several new iterative algorithms for the linear complementarity problem are developed and compared.

Degeneracy in linear complementarity problems: a survey

- MathematicsAnn. Oper. Res.
- 1993

The literature on the implications of degeneracy to the linear complementarity theory is reviewed, finding that ifLCP(0,M) has a nontrivial solution, a condition related to degeneracy, then unless certain other conditions are satisfied the algorithm may not be able to decide about the solvability of the given LCP(q, M).

Linear complementarity problems solvable by a polynomially bounded pivoting algorithm

- Mathematics, Computer Science
- 1985

A sufficient condition is given under which the parametric principal pivoting algorithm will compute the unique solution to a linear complementarity problem defined by an n by n P-matrix in no more…

The Generalized Linear Complementarity Problem: Least Element Theory and Z-Matrices

- MathematicsJ. Glob. Optim.
- 1997

The concept of sufficient matrices of class Z is investigated to obtain additional properties of the solution set and it is shown that if solutions exist, then one must be the least element of the feasible region.

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