A modified simplex partition algorithm to test copositivity
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Abstract
Abstract A real symmetric matrix A is copositive if $$x^\top Ax\ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>⊤</mml:mi> </mml:msup> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> for all $$x\ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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 $$\bigtriangleup $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>△</mml:mo> </mml:math> , say with n vertices, into a simplex $$\bigtriangleup _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>△</mml:mo> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and a polyhedron $$\varOmega _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> ; and then partitions $$\varOmega _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> into a set of at most $$(n-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> simplices. We show that if A is copositive on $$\varOmega _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> then A is copositive on $$\bigtriangleup _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>△</mml:mo> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , allowing us to remove $$\bigtriangleup _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>△</mml:mo> <mml:mn>1</mml:mn> </mml:msub> </mml:math> 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.
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