Maximizing Products of Linear Forms, and The Permanent of Positive Semidefinite Matrices

We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices. By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices, as well as computationally efficient certificates for this approximation. We also propose an analog of van der Waerden's conjecture for HPSD matrices, where the polynomial optimization problem is interpreted as a relaxation of the permanent.Comment: 12 pages, 2 figure

On the Low Rank Solutions for Linear Matrix Inequalities

In this paper we present a polynomial-time procedure to find a low rank solution for a system of Linear Matrix Inequalities (LMI). The existence of such a low rank solution was shown in Au-Yeung and Poon [1] and Barvinok [3]. In Au-Yeung and Poon’s approach, an earlier unpublished manuscript of Bohnenblust [6] played an essential role. Both proofs in [1] and [3] are nonconstruc-tive in nature. The aim of this paper is to offer a constructive and polynomial-time procedure to find such a low rank solution approximatively. Extensions of our new results and their relations to some of the known results in the literature are discussed