From seven to eleven: Completely positive matrices with high cp-rank

Abstract We study n × n completely positive matrices M on the boundary of the completely positive cone, namely those orthogonal to a copositive matrix S which generates a quadratic form with finitely many zeroes in the standard simplex. Constructing particular instances of S , we are able to construct counterexamples to the famous Drew–Johnson–Loewy conjecture (1994) for matrices of order seven through eleven.

[1]  Charles R. Johnson,et al.  Completely positive matrices associated with M-matrices , 1994 .

[2]  Florian Jarre,et al.  On the cp-Rank and Minimal cp Factorizations of a Completely Positive Matrix , 2013, SIAM J. Matrix Anal. Appl..

[3]  Peter J. C. Dickinson,et al.  Irreducible elements of the copositive cone , 2013 .

[4]  Immanuel M. Bomze,et al.  Regularity versus Degeneracy in Dynamics, Games, and Optimization: A Unified Approach to Different Aspects , 2002, SIAM Rev..

[5]  Raphael Loewy,et al.  CP rank of completely positive matrices of order 5 , 2003 .

[6]  Charles R. Johnson,et al.  The No Long Odd Cycle Theorem for Completely Positive Matrices , 1996 .

[7]  Mirjam Dür,et al.  Copositive Programming – a Survey , 2010 .

[8]  Stanislav Busygin,et al.  Copositive Programming , 2009, Encyclopedia of Optimization.

[9]  Xu Chang.  REMARKS ON COMPLETELY POSITIVE MATRICES , 1999 .

[10]  Abraham Berman,et al.  Remarks on completey positive matrices , 1998 .

[11]  Roland Hildebrand.  Minimal zeros of copositive matrices , 2013 .

[12]  Roland Hildebrand,et al.  The extreme rays of the 5 × 5 copositive cone , 2012 .

[13]  Immanuel M. Bomze,et al.  Detecting all evolutionarily stable strategies , 1992 .

[14]  Immanuel M. Bomze,et al.  Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization , 2012, J. Glob. Optim..

[15]  Immanuel M. Bomze,et al.  Copositive optimization - Recent developments and applications , 2012, Eur. J. Oper. Res..

[16]  安藤 毅.  Completely positive matrices , 1991 .

[17]  Florian Jarre,et al.  New results on the cp-rank and related properties of co(mpletely )positive matrices , 2015 .

[18]  Mark Broom,et al.  On the number of local maxima of a constrained quadratic form , 1993, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.