# A Penalty Method for Rank Minimization Problems in Symmetric Matrices

## TL;DR

This paper introduces a penalty method for solving rank minimization problems in symmetric matrices by reformulating them as a nonconvex semidefinite program, analyzing solution properties, and proposing an algorithm with computational validation.

## Contribution

It develops a penalty approach for symmetric rank minimization, analyzes solution calmness, and proposes a PALM-based algorithm with momentum for improved performance.

## Key findings

- Locally optimal solutions are KKT points.
- Calmness results support the penalty approach.
- Computational experiments demonstrate effectiveness.

## Abstract

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point. We develop a penalty formulation of the problem. We present calmness results for locally optimal solutions to the penalty formulation. We also develop a proximal alternating linearized minimization (PALM) scheme for the penalty formulation, and investigate the incorporation of a momentum term into the algorithm. Computational results are presented.