A modified simplex partition algorithm to test copositivity

Abstract

AbstractA real symmetric matrix A is copositive if $$x^\top Ax\ge 0$$ x ⊤ A x ≥ 0 for all $$x\ge 0$$ x ≥ 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 $$\bigtriangleup _1$$ △ 1 and a polyhedron $$\varOmega _1$$ Ω 1 ; and then partitions $$\varOmega _1$$ Ω 1 into a set of at most $$(n-1)$$ ( n - 1 ) simplices. We show that if A is copositive on $$\varOmega _1$$ Ω 1 then A is copositive on $$\bigtriangleup _1$$ △ 1 , allowing us to remove $$\bigtriangleup _1$$ △ 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.