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

Abstract

• We survey recent progress in conic optimization. • We discuss second order cone programs (SOCP), semidefinite problems (SDP), and copositive problems. • Special emphasis is given to modelling issues. • The survey contains more than 110 references. A conic optimization problem is a problem involving a constraint that the optimization variable be in some closed convex cone. Prominent examples are linear programs (LP), 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, and copositive optimization. We review how tight bounds can be obtained by relaxing the copositivity constraint to semidefiniteness, and we discuss the effect that different modelling techniques have on the quality of the bounds. We also provide some new techniques for lifting linear constraints and show how these can be used for stable set and coloring relaxations.