On alternating direction methods for monotropic semidefinite programming

Abstract

This thesis studies a new optimization model called monotropic semidefinite programming. This model extends the monotropic programming model from vectors to matrices on one hand, and the linear semidefinite programming model to the convex case on the other hand.We propose several modified alternating direction methods for solving monotropic semidefinite programming problems. In order to avoid solving difficult sub-variational inequality problems on matrix space at each iteration, we establish a set of projection-based algorithms. We discuss alternating direction methods in different ways to deal with quadratic objective and general nonlinear objective. These methods can be used to solve convex nonlinear semidefinite programming problems. A practical application comes from the covariance matrix estimation problem. Another practical application is the matrix completion problem. Both of them can be modeled as convex matrix optimization problems and certain modified alternating direction methods apply. We also conduct numerical tests for problems arising from these applications.