A Class of Semidefinite Programs with rank-one solutions

We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite matrix of rank at most $r$, where $r$ is the rank of the matrix involved in the objective function of the SDP. The optimization problems of this class are semidefinite packing problems, which are the SDP analogs to vector packing problems. Of particular interest is the case in which our result guarantees the existence of a solution of rank one: we show that the computation of this solution actually reduces to a Second Order Cone Program (SOCP). We point out an application in statistics, in the optimal design of experiments.