# 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}
}
• O. Mangasarian
• Published 1 December 1976
• Mathematics, Computer Science
• Mathematical Programming
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

## Topics from this paper

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
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
• J. Pang
• Mathematics, Computer Science
Math. 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
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
• J. Pang
• Mathematics, Computer Science
Math. 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
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
• Mathematics
• 1986
Direct complementary pivot algorithms for the linear complementarity problem with P-matrices are known to have exponential computational complexity. The analog of Gauss-Seidel and SOR iteration for
Degeneracy in linear complementarity problems: a survey
• S. Mohan
• Mathematics, Computer Science
Ann. 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
• 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
• Mathematics, Computer Science
J. 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.

## References

SHOWING 1-10 OF 16 REFERENCES
Minimality and complementarity properties associated with Z-functions and M-functions
• A. Tamir
• Mathematics, Computer Science
Math. Program.
• 1974
A nonlinear generalization of square matrices with non-positive off-diagonal elements is presented, and an algorithm to solve the corresponding complementarity problem is suggested and a potential application in extending the well-known linear Leontief input—output systems is discussed.
Polyhedral sets having a least element
• Mathematics, Computer Science
Math. Program.
• 1972
It is shown the linear complementarity problem always has a unique solution which is at the same time a least element of the corresponding polyhedron if and only if its matrix is square, Leontief, and has positive diagonals.
On the solution of large, structured linear complementarity problems: III.
• Mathematics
• 1974
This paper addresses the problem of solving a class of specially-structured linear complementarity problems of potentially very large size. An efficient method which couples a modification of the
The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation
Let ${\bf A}$ be a real symmetric positive definite $n \times n$ matrix and ${\bf b}$ a real column n-vector. We consider the following problem: Find real column n-vectors ${\bf x}$ and ${\bf y}$ s...
On the Alass of Complementary Cones and Lemke’s Algorithm
In this paper, a geometrical description of Lemke’s algorithm is presented for solving the linear complementarily problem: Find nonnegative vectors x and y satisfying $x = My + q$, $x^T y = 0$. This
Nonlinear Programming
It is shown that if A is closed for all k → x x, k → y y, where ( k A ∈ ) k y x , then ( ) A ∉ y x .