A relaxed interior point method for low-rank semidefinite programming problems with applications to matrix completion

TL;DR

This paper introduces a novel relaxed interior point method tailored for low-rank semidefinite programming, enabling efficient solutions for matrix completion problems by relaxing optimality conditions and allowing flexible Newton direction computations.

Contribution

It proposes a new relaxed interior point framework that relaxes optimality conditions and supports both first- and second-order methods for low-rank SDP problems.

Findings

Convergence of the proposed method is established.

Preliminary computational results are encouraging.

The method effectively solves matrix completion problems.

Abstract

A new relaxed variant of interior point method for low-rank semidefinite programming problems is proposed in this paper. The method is a step outside of the usual interior point framework. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) structure, the first order optimality conditions have to be relaxed and are therefore approximated by solving an auxiliary least-squares problem. The relaxed interior point framework opens numerous possibilities how primal and dual approximated Newton directions can be computed. In particular, it admits the application of both the first- and the second-order methods in this context. The convergence of the method is established. A prototype implementation is discussed and encouraging preliminary computational results are reported for solving the SDP-reformulation of matrix-completion problems.