A Class of Semidefinite Programs with rank-one solutions

TL;DR

This paper identifies a class of semidefinite programs that always have low-rank solutions, often rank one, simplifying their computation to second order cone programs and impacting statistical experiment design.

Contribution

It establishes conditions under which certain SDPs have low-rank solutions, including rank-one, and links these solutions to more efficient SOCP computations.

Findings

SDPs in this class have solutions with rank at most r.

Rank-one solutions can be computed via SOCP.

Application demonstrated in optimal experimental design.

Abstract

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.