Total dual integrality and integral solutions of the linear complementarity problem

Article Type

Research Article

Publication Title

Linear Algebra and Its Applications

Abstract

This paper deals with the problem of finding an integer solution to a linear complementarity problem (LCP). Chandrasekaran et al. [1] introduced the class I of integral matrices for which the corresponding LCP has an integer solution for every integral vector q, for which it has a solution and proved that for some well-known matrix classes principal unimodularity forms a necessary and sufficient condition for inclusion in the class I. In this paper, we identify some more well-known matrix classes for which principal unimodularity forms a necessary and sufficient condition for inclusion in the class I. The concept of total dual integrality is utilized to obtain a necessary and sufficient condition for existence of an integer solution to LCP with a hidden K-matrix. We interconnect the concept of Hilbert basis with principal unimodularity of a matrix and the corresponding complementary cones. A necessary and sufficient condition is given for the existence of an integer solution of a linear fractional programming problem by using its LCP formulation.

First Page

359

Last Page

374

DOI

10.1016/j.laa.2018.08.004

Publication Date

11-15-2018

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