EP theorems and linear complementarity problems

Komei Fukuda, Makoto Namiki, Akihisa Tamura

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

Let A be a rational n × n square matrix and b be a rational n-vector for some positive integer n. The linear complementarity problem (abbreviated by LCP) is to find a vector (x, y) ∈R2n satisfying y = Ax + b (x, y) ≥ 0 and the complementarity condition: xi · yi = 0 for all i = 1,..., n. The LCP is known to be NP-complete, but there are some known classes of matrices A for which the LCP is polynomially solvable, for example the class of positive semi-definite (PSD-) matrices. In this paper, we study the LCP from the view point of EP (existentially polynomial time) theorems due to Cameron and Edmonds. In particular, we investigate the LCP duality theorem of Fukuda and Terlaky in EP form, and show that this immediately yields a simple modification of the criss-cross method with a nice practical feature. Namely, this algorithm can be applied to any given A and b, and terminates in one of the three states: (1) a solution x is found; (2) a solution to the dual LCP is found (implying the nonexistence of a solution to the LCP); or (3) a succinct certificate is given to show that the input matrix A is not "sufficient". Note that all PSD-matrices and P-matrices are sufficient matrices.

Original languageEnglish
Pages (from-to)107-119
Number of pages13
JournalDiscrete Applied Mathematics
Volume84
Issue number1-3
DOIs
Publication statusPublished - 1998 May 15
Externally publishedYes

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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