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Immanuel M. Bomze (immanuel.bomzeunivie.ac.at) Abstract: Let $p_n$ denote the largest possible cprank of an $n\times n$ completely positive matrix. This matrix parameter has its significance both in theory and applications, as it sheds light on the geometry and structure of the solution set of hard optimization problems in their completely positive formulation. Known bounds for $p_n$ are $s_n=\binom{n+1}24$, the current best upper bound, and the DrewJohnsonLoewy (DJL) lower bound $d_n=\big\lfloor\frac{n^2}4\big\rfloor$. The famous DJL conjecture (1994) states that $p_n=d_n$. Here we show $\corr{p_n=\frac {n^2}2 +{\mathcal O}\big(n^{3/2}\big) = 2d_n+{\mathcal O}\big(n^{3/2}\big)}$, and construct counterexamples to the DJL conjecture for all $n\ge {12}$ Keywords: copositive optimization, completely positive matrices, nonnegative factorization Category 1: Linear, Cone and Semidefinite Programming (Other ) Citation: Preprint NI14048POP, Isaac Newton Institute for Mathematical Sciences, University of Cambridge UK, submitted Download: [PDF] Entry Submitted: 06/13/2014 Modify/Update this entry  
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