An Adaptive Linear Approximation Algorithm for Copositive Programs

We study linear optimization problems over the cone of copositive matrices. These problems appear in nonconvex quadratic and binary optimization; for instance, the maximum clique problem and other combinatorial problems can be reformulated as such problems. We present new polyhedral inner and outer approximations of the copositive cone which we show to be exact in the limit. In contrast to previous approximation schemes, our approximation is not necessarily uniform for the whole cone but can be guided adaptively through the objective function, yielding a good approximation in those parts of the cone that are relevant for the optimization and only a coarse approximation in those parts that are not. Using these approximations, we derive an adaptive linear approximation algorithm for copositive programs. Numerical experiments show that our algorithm gives very good results for certain nonconvex quadratic problems

Algorithmic copositivity detection by simplicial partition

AbstractWe present new criteria for copositivity of a matrix, i.e., conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant. These criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof. We show that, as the partition gets finer and finer, the conditions eventually capture all strictly copositive matrices. We propose an algorithmic implementation which considers several numerical aspects. As an application, we present results on the maximum clique problem. We also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones

Topological and symbolic dynamics for hyperbolic systems with holes

Bundfuss S, Krüger T, Troubetzkoy S. Topological and symbolic dynamics for hyperbolic systems with holes. Ergodic Theory and Dynamical Systems. 2011;31(05):1305-1323.We consider an axiom A diffeomorphism or a Markov map of an interval and the invariant set Omega* of orbits which never falls into a fixed hole. We study various aspects of the symbolic representation of Omega* and of its non-wandering set Omega(nw). Our results are on the cardinality of the set of topologically transitive components of Omega(nw) and their structure. We also prove that Omega* is generically a subshift of finite type in several senses

An improved algorithm to test copositivity

Copositivity plays a role in combinatorial and nonconvex quadratic optimization. However, testing copositivity of a given matrix is a co-NP-complete problem. We improve a previously given branch-and-bound type algorithm for testing copositivity and discuss its behavior in particular for the maximum clique problem. Numerical experiments indicate that the speedup is considerable.