Conic optimization: a survey with special focus on copositive optimization and binary quadratic problems

A conic optimization problem is a problem involving a constraint that the optimization variable be in some closed convex cone. Prominent examples are second order cone programs (SOCP), semidefinite problems (SDP), and copositive problems. We survey recent progress made in this area. In particular, we highlight the connections between nonconvex quadratic problems, binary quadratic problems, … Read more

The Slater condition is generic in linear conic programming

We call a property generic if it holds for almost all problem instances. For linear conic problems, it has been shown in the literature that properties like uniqueness, strict complementarity or nondegeneracy of the optimal solution are generic under the assumption that Slater’s condition is fulfilled. The possibility that Slater’s condition generically fails has not … Read more

Irreducible elements of the copositive cone

An element $A$ of the $n \times n$ copositive cone $\copos{n}$ is called irreducible with respect to the nonnegative cone~$\NNM{n}$ if it cannot be written as a nontrivial sum $A = C+N$ of a copositive matrix $C$ and an elementwise nonnegative matrix $N$. This property was studied by Baumert~\cite{Baumert65} who gave a characterisation of irreducible … Read more