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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.
In this paper, we show that the bundle method can be applied to solve semidefinite programming problems with a low rank solution without ever constructing a full matrix. To accomplish this, we use recent results from randomly sketching matrix optimiz
We describe a modular rewriting system for translating optimization problems written in a domain-specific language to forms compatible with low-level solver interfaces. Translation is facilitated by reductions, which accept a category of problems and
We present new results on optimization problems where the involved functions are evenly convex. By means of a generalized conjugation scheme and the perturbation theory introduced by Rockafellar, we propose an alternative dual problem for a general o
In this study, we present a general framework of outer approximation algorithms to solve convex vector optimization problems, in which the Pascoletti-Serafini (PS) scalarization is solved iteratively. This scalarization finds the minimum distance fro
This paper considers a general convex constrained problem setting where functions are not assumed to be differentiable nor Lipschitz continuous. Our motivation is in finding a simple first-order method for solving a wide range of convex optimization