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Rank Optimality for the Burer--Monteiro Factorization
SIAM Journal on Optimization ( IF 2.6 ) Pub Date : 2020-09-23 , DOI: 10.1137/19m1255318 Irène Waldspurger , Alden Waters
SIAM Journal on Optimization ( IF 2.6 ) Pub Date : 2020-09-23 , DOI: 10.1137/19m1255318 Irène Waldspurger , Alden Waters
SIAM Journal on Optimization, Volume 30, Issue 3, Page 2577-2602, January 2020.
When solving large-scale semidefinite programs that admit a low-rank solution, an efficient heuristic is the Burer--Monteiro factorization: instead of optimizing over the full matrix, one optimizes over its low-rank factors. This reduces the number of variables to optimize but destroys the convexity of the problem, thus possibly introducing spurious second-order critical points. The article [N. Boumal, V. Voroninski, and A. S. Bandeira, Deterministic Guarantees for Burer-Monteiro Factorizations of Smooth Semidefinite Programs, https://arxiv.org/abs/1804.02008, 2018] shows 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 the global solution is rank 1.
中文翻译：
Burer--Monteiro因子分解的秩最优
SIAM优化杂志，第30卷，第3期，第2577-2602页，2020年1月。
当求解允许低阶解决方案的大规模半定性程序时，一种有效的启发式方法是Burer-Monteiro因子分解：不是对整个矩阵进行优化，而是对其低阶因子进行了优化。这减少了要优化的变量数量，但破坏了问题的凸性，因此可能引入了虚假的二阶临界点。文章[N. Boumal，V.Voroninski和AS Bandeira，光滑半定程序的Burer-Monteiro因式分解的确定性保证，https：//arxiv.org/abs/1804.02008，2018]表明，当因子的大小约为线性约束数量的平方根不会发生：对于几乎所有成本矩阵，二阶临界点都是全局解决方案。在本文中，我们证明了这个结果实质上是严格的：
更新日期：2020-11-13
When solving large-scale semidefinite programs that admit a low-rank solution, an efficient heuristic is the Burer--Monteiro factorization: instead of optimizing over the full matrix, one optimizes over its low-rank factors. This reduces the number of variables to optimize but destroys the convexity of the problem, thus possibly introducing spurious second-order critical points. The article [N. Boumal, V. Voroninski, and A. S. Bandeira, Deterministic Guarantees for Burer-Monteiro Factorizations of Smooth Semidefinite Programs, https://arxiv.org/abs/1804.02008, 2018] shows 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 the global solution is rank 1.
中文翻译：
Burer--Monteiro因子分解的秩最优
SIAM优化杂志，第30卷，第3期，第2577-2602页，2020年1月。
当求解允许低阶解决方案的大规模半定性程序时，一种有效的启发式方法是Burer-Monteiro因子分解：不是对整个矩阵进行优化，而是对其低阶因子进行了优化。这减少了要优化的变量数量，但破坏了问题的凸性，因此可能引入了虚假的二阶临界点。文章[N. Boumal，V.Voroninski和AS Bandeira，光滑半定程序的Burer-Monteiro因式分解的确定性保证，https：//arxiv.org/abs/1804.02008，2018]表明，当因子的大小约为线性约束数量的平方根不会发生：对于几乎所有成本矩阵，二阶临界点都是全局解决方案。在本文中，我们证明了这个结果实质上是严格的：
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