Maximizing Products of Linear Forms, and The Permanent of Positive Semidefinite Matrices
release_ra37myimgrfn3e5ro5iqqqonrq
by
Chenyang Yuan, Pablo A. Parrilo
2021
Abstract
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.
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