Chapter 8 Copositive Programming

A symmetric matrix S is copositive if yT S y≥0 for all y≥0, and the set of all copositive matrices, denoted C∗, is a closed, pointed, convex cone; see [25] for a recent survey. Researchers have realized how to model many NP-hard optimization problems as copositive programs, that is, programs over C∗ for which the objective and all other constraints are linear [7, 9, 13, 16, 32–34]. This makes copositive programming NP-hard itself, but the models are nevertheless interesting because copositive programs are convex, unlike the problems which they model. In addition, C∗ can be approximated up to any accuracy using a sequence of polyhedralsemidefinite cones of ever larger sizes [13, 30], so that an underlying NP-hard problem can be approximated up to any accuracy if one is willing to spend the computational effort. In actuality, most of these NP-hard problems are modeled as linear programs over the dual cone C of completely positive matrices, that is, matrices Y that can be written as the sum of rank-1 matrices yyT for y ≥ 0 [4]. These programs are called completely positive programs, and the aforementioned copositive programs are constructed using standard duality theory. Currently the broadest class of problems known to be representable as completely positive programs are those with nonconvex quadratic objective and linear constraints over binary and continuous variables [9]. In addition, complementarity constraints on bounded, nonnegative variables can be incorporated. In this chapter, we recount and extend this result using the more general notion of matrices that are completely positive over a closed, convex cone.