Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form

Srinadh Bhojanapalli; Nicolas Boumal; Prateek Jain; Praneeth Netrapalli

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arXiv · 1803.00186

Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form

Srinadh Bhojanapalli·Nicolas Boumal·Prateek Jain·Praneeth Netrapalli

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Abstract

Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the penalty formulation of appropriately rank-constrained SDPs as long as the number of constraints scales sub-quadratically with the desired rank of the optimal solution. Our result is based on a simple penalty function formulation of the rank-constrained SDP along with a smoothed analysis to avoid worst-case cost matrices. We particularize our results to two applications, namely, Max-Cut and matrix completion.

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Srinadh Bhojanapalli, Nicolas Boumal, Prateek Jain, Praneeth Netrapalli. 2018-03-01. Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form. https://arxiv.org/abs/1803.00186

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