Let p n 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 p n are s n = n+1 2 − 4, the current best upper bound, and the Drew-Johnson-Loewy (DJL) lower bound d n = n 2 4. The famous DJL conjecture (1994) states that p n = d n. Here we show
In a Standard Quadratic Optimization Problem (StQP), a possibly indefinite quadratic form (the simplest nonlinear function) is extremized over the standard simplex, the simplest polytope. Despite this simplicity, the nonconvex instances of this problem class allow for remarkably rich patterns of coexisting local solutions, which are closely related to practical difficulties in solving StQPs globally. In this study, we improve on existing lower bounds for the number of strict local solutions by a new technique to construct instances with a rich solution structure. Furthermore we provide extensive case studies where the system of supports (the so-called pattern) of solutions are analyzed in detail. Note that by naive simulation, in accordance to theory, most of the interesting patterns would not be encountered, since random instances have, with a high probability, quite sparse solutions (with singleton or doubleton supports), and likewise their expected numbers are considerably lower than in the worst case. Hence instances with a rich solution pattern are rather rare. On the other hand, by concentrating on (thin) subsets of promising instances, we are able to give an empirical answer on the size distribution of supports of strict local solutions to the StQP and their patterns, complementing average-case analysis of this NP-hard problem class.
Recently a new evolutionary game dynamics, the Infection-Immunization Dynamics,has been introduced for discrete time. In this paper a continuous time version of this model is derived and the existence and structure of solutions is analysed. This is a very challenging task, since standard technique existence theorems for Differential Inclusions do not hold in general. An extended solution concept, the notion of Krasovsky solutions, can be applied though. Some stability results are stated and discussed.
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