A modified simplex partition algorithm to test copositivity

Mohammadreza Safi (), Seyed Saeed Nabavi () and Richard J. Caron ()
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Mohammadreza Safi: University of Windsor
Seyed Saeed Nabavi: Semnan University
Richard J. Caron: University of Windsor

Journal of Global Optimization, 2021, vol. 81, issue 3, No 4, 645-658

Abstract: Abstract A 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.

Keywords: Copositive matrix; Maximum clique problem; Simplex; Semidefinite programming; Copositive programming; MSC 65K05; MSC 90C22; MSC 90C99 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10898-021-01092-1

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