 
Summary: SIAM J. OPTIM. c 2010 Society for Industrial and Applied Mathematics
Vol. 20, No. 5, pp. 23272351
LOWRANK OPTIMIZATION ON THE CONE OF POSITIVE
SEMIDEFINITE MATRICES
M. JOURN´EE, F. BACH, P.A. ABSIL§, AND R. SEPULCHRE
Abstract. We propose an algorithm for solving optimization problems defined on a subset
of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization
X = Y Y T , where the number of columns of Y fixes an upper bound on the rank of the positive
semidefinite matrix X. It is thus very effective for solving problems that have a lowrank solution.
The factorization X = Y Y T leads to 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 secondorder optimization method with guaranteed quadratic convergence. It furthermore
provides some conditions on the rank of the factorization to ensure equivalence with the original
problem. In contrast to existing methods, the proposed algorithm converges monotonically to the
sought solution. Its numerical efficiency is evaluated on two applications: the maximal cut of a graph
and the problem of sparse principal component analysis.
Key words. lowrank constraints, cone of symmetric positive definite matrices, Riemannian
quotient manifold, sparse principal component analysis, maximumcut algorithms, largescale algo
rithms
AMS subject classifications. 65K05, 90C30, 90C25, 90C22, 90C27, 62H25, 58C05, 49M15
