ON THE NON-HOMOGENEITY OF COMPLETELY POSITIVE CONES * M. SEETHARAMA GOWDA † AND R. SZNAJDER ‡ Abstract. For a closed cone C in R n , the completely positive cone of C is the convex cone K in S n generated by {uu T : u ∈ C}. Completely positive cones arise, for example, in the conic LP reformulation of a nonconvex quadratic minimization problem over an arbitrary set with linear and binary constraints. Motivated by the useful and desirable properties of the nonnegative orthant and the positive semidefinite cone (and more generally of symmetric cones in Euclidean Jordan algebras), in this paper, we investigate when (or whether) K can be self-dual, irreducible, or homogeneous. Key words. copositive and completely positive cones, self-dual, irreducible cone, homogeneous cone 1. Introduction. Consider R n with the usual inner product. Given a closed cone C in R n that is not necessarily convex, we consider two related cones in the space S n of all n × n real symmetric matrices: The completely positive cone of C defined by K :=   uu T : u ∈C  , (1.1) where the sum denotes a finite sum, and the copositive cone of C given by E := {A ∈S n : A is copositive on C}. (1.2) When C = R n , these two cones reduce to S n + , which is the underlying cone in semidefinite programming [16] and semidefinite linear complementarity problems [13], [12]. In the case of C = R n + , these cones reduce, respectively, to the cones of com- pletely positive matrices and copositive matrices which have appeared prominently in statistical and graph theoretic literature [3] and in copositive programming [7]. In a path-breaking work, Burer [4] showed that a nonconvex quadratic minimization prob- lem over the nonnegative orthant with some additional linear and binary constraints can be reformulated as a linear program over the cone of completely positive matri- ces. Since then, a number of authors have investigated the properties of the cone of * Research Report, November 3, 2011. † Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA; E-mail: gowda@math.umbc.edu, URL: http://www.math.umbc.edu/∼gowda ‡ Department of Mathematics, Bowie State University, Bowie, MD 20715, USA; rszna- jder@bowiestate.edu 1