Semidefinite Programming in the Space of Partial Positive Semidefinite Matrices

We build upon the work of Fukuda et al.\ \cite{FuKoMuNa01} and Nakata et al.\ \cite{NaFuFuKoMu01}, in which the theory of partial positive semidefinite matrices has been applied to the semidefinite programming (SDP) problem as a technique for exploiting sparsity in the data. In contrast to their work, which improves an existing algorithm that is based … Read more

Solving a Class of Semidefinite Programs via Nonlinear Programming

In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems includes instances of … Read more

A Computational Study of a Gradient-Based Log-Barrier Algorithm for a Class of Large-Scale SDPs

The authors of this paper recently introduced a transformation \cite{BuMoZh99-1} that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based … Read more

A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Low-rank Factorization

In this paper, we present a nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form. The algorithm’s distinguishing feature is a change of variables that replaces the symmetric, positive semidefinite variable X of the SDP with a rectangular variable R according to the factorization X = RR^T. The rank of the factorization, i.e., … Read more