Rank optimality for the Burer-Monteiro factorization

Abstract : When solving large scale semidefinite programs that admit a low-rank solution, a very efficient heuristic is the Burer-Monteiro factorization: Instead of optimizing over the full matrix, one optimizes over its low-rank factors. This strongly reduces the number of variables to optimize, but destroys the convexity of the problem, thus possibly introducing spurious second-order critical points which can prevent local optimization algorithms from finding the solution. Boumal, Voroninski, and Bandeira [2018] have recently shown that, when the size of the factors is of the order of the square root of the number of linear constraints, this does not happen: For almost any cost matrix, second-order critical points are global solutions. In this article, we show that this result is essentially tight: For smaller values of the size, second-order critical points are not generically optimal, even when considering only semidefinite programs with a rank 1 solution.
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Submitted on : Tuesday, December 18, 2018 - 11:37:31 AM
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Irène Waldspurger, Alden Waters. Rank optimality for the Burer-Monteiro factorization. 2018. ⟨hal-01958814⟩

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