Low-rank solutions of matrix inequalities with applications to polynomial optimization and matrix completion problems

This paper is concerned with the problem of finding a low-rank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a theory relating the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph. Furthermore, we propose two graph-theoretic convex programs to obtain a low-rank solution. The first convex optimization needs a tree decomposition of the sparsity graph. The second one does not rely on any computationally-expensive graph analysis and is always polynomial-time solvable. The results of this work can be readily applied to three separate problems of minimum-rank matrix completion, conic relaxation for polynomial optimization, and affine rank minimization. The results are finally illustrated on two applications of optimal distributed control and nonlinear optimization for electrical networks.