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Polynomial time guarantees for the Burer-Monteiro method
Diego Cifuentes · Ankur Moitra

Thu Dec 01 09:30 AM -- 11:00 AM (PST) @ Hall J #841
The Burer-Monteiro method is one of the most widely used techniques for solving large-scale semidefinite programs (SDP). The basic idea is to solve a nonconvex program in $Y$, where $Y$ is an $n \times p$ matrix such that $X = Y Y^T$. We show that this method can solve SDPs in polynomial time in a smoothed analysis setting. More precisely, we consider an SDP whose domain satisfies some compactness and smoothness assumptions, and slightly perturb the cost matrix and the constraints. We show that if $p \gtrsim \sqrt{2(1{+}\eta)m}$, where $m$ is the number of constraints and $\eta>0$ is any fixed constant, then the Burer-Monteiro method can solve SDPs to any desired accuracy in polynomial time, in the setting of smooth analysis. The bound on $p$ approaches the celebrated Barvinok-Pataki bound in the limit as $\eta$ goes to zero, beneath which it the nonconvex program can be suboptimal. Our main technical contribution, which is key for our tight bound on $p$, is to connect spurious approximately critical points of the nonconvex program to tubular neighborhoods of certain algebraic varieties, and then estimate the volume of such tubes.

Author Information

Diego Cifuentes (Georgia Institute of Technology)
Ankur Moitra (MIT)

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