New Lower Bounds and Asymptotics for the cp-Rank

Let pₙ denote the largest possible cp-rank of an n× 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 pₙ are sₙ=n+12-4, the current best upper bound, and the Drew--Johnson--Loewy (DJL) lower bound dₙ=n²4. The famous DJL conjecture (1994) states that pₙ=dₙ. Here we show pₙ= n²2 + O(n3/2) = 2dₙ+ O(n3/2)pₙ= n²2 + O(n3/2) = 2dₙ+ O(n3/2), and construct counterexamples to the DJL conjecture for all n≥ 12 (for orders seven through eleven counterexamples were already given in [I. M. Bomze, W. Schachinger, and R. Ullrich, Linear Algebra Appl., 459 (2014), pp. 208--221].