- New lower bounds and asymptotics for the cp-rank Immanuel M. Bomze (immanuel.bomzeunivie.ac.at) Werner Schachinger (werner.schachingerunivie.ac.at) Reinhard Ullriche (reinhard.ullrichunivie.ac.at) Abstract: Let $p_n$ denote the largest possible cp-rank 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 $s_n=\binom{n+1}2-4$, the current best upper bound, and the Drew-Johnson-Loewy (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 NI14048-POP, Isaac Newton Institute for Mathematical Sciences, University of Cambridge UK, submitted Download: [PDF]Entry Submitted: 06/13/2014Entry Accepted: 06/13/2014Entry Last Modified: 06/18/2014Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.