Low-rank optimization for semidefinite convex problems

Details Low-rank optimization for semidefinite convex problems Low-rank optimization for semidefinite convex problems

2008-07-28

arxiv.org/abs/0807.4423v1

SIAM J. Optim. Volume 20, Issue 5, pp. 2327-2351 (2010)

Mathematics

Optimization and Control

submitted

Scientific paper

10.1137/080731359

We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It is thus very effective for solving programs that have a low rank solution. The factorization $X=Y Y^T$ evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem.

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