A modified simplex partition algorithm to test copositivity

Abstract

A real symmetric matrix A is copositive if x^top Axge 0 for all xge 0. As A is copositive if and only if it is copositive on the standard simplex, algorithms to determine copositivity, such as those in Sponsel et al. (J Glob Optim 52:537–551, 2012) and Tanaka and Yoshise (Pac J Optim 11:101–120, 2015), are based upon the creation of increasingly fine simplicial partitions of simplices, testing for copositivity on each. We present a variant that decomposes a simplex bigtriangleup , say with n vertices, into a simplex bigtriangleup _1 and a polyhedron varOmega _1; and then partitions varOmega _1 into a set of at most (n-1) simplices. We show that if A is copositive on varOmega _1 then A is copositive on bigtriangleup _1, allowing us to remove bigtriangleup _1 from further consideration. Numerical results from examples that arise from the maximum clique problem show a significant reduction in the time needed to establish copositivity of matrices.