Immanuel M. Bomze We show that the maximal cp-rank of $n\times n$ completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of $n\times n$ completely positive matrices, thus answering a long standing question. We also show that the maximal cp-rank of $5\times 5$ matrices equals six, which proves the famous Drew-Johnson-Loewy conjecture … Read more
Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of … Read more
Copositive optimization is a quickly expanding scientific research domain with wide-spread applications ranging from global nonconvex problems in engineering to NP-hard combinatorial optimization. It falls into the category of conic programming (optimizing a linear functional over a convex cone subject to linear constraints), namely the cone of all completely positive symmetric nxn matrices, and its … Read more
We provide Completely Positive and Copositive Programming formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which are used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, … Read more
We propose a fast population game dynamics, motivated by the analogy with infection and immunization processes within a population of “players,” for finding dominant sets, a powerful graph-theoretical notion of a cluster. Each step of the proposed dynamics is shown to have a linear time/space complexity and we show that, under the assumption of symmetric … Read more
We present three new copositivity tests based upon difference-of-convex (d.c.) decompositions, and combine them to a branch-and-bound algorithm of $\omega$-subdivision type. The tests employ LP or convex QP techniques, but also can be used heuristically using appropriate test points. We also discuss the selection of efficient d.c.~decompositions and propose some preprocessing ideas based on the … Read more
Copositive optimization problems are particular conic programs: extremize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone … Read more
Using a bordering approach, and building upon an already known factorization of a principal block, we establish sufficient conditions under which we can extend this factorization to the full matrix. Simulations show that the approach is promising also in higher dimensions. CitationPreprint, Univ.of Vienna (2017), submittedArticleDownload View PDF
In an important paper, Burer recently showed how to reformulate general mixed-binary quadratic optimization problems (QPs) into copositive programs where a linear functional is minimized over a linearly constrained subset of the cone of completely positive matrices. In this note we interpret the implication from a topological point of view, showing that the Minkowski sum … Read more
A so-called Standard Bi-Quadratic Optimization Problem (StBQP) consists in minimizing a bi-quadratic form over the Cartesian product of two simplices (so this is different from a Bi-Standard QP where a quadratic function is minimized over the same set). An application example arises in portfolio selection. In this paper we present a bi-quartic formulation of StBQP, … Read more