# Several classes of stationary points for rank regularized minimization problems

## TL;DR

This paper introduces various stationary points for rank regularized minimization problems through different reformulations, providing a relation chart to guide low-rank solution search and characterizing conditions for local minimizers.

## Contribution

It defines multiple stationary points for the problem and its reformulations, establishing their relations and offering conditions for local optimality in the PSD cone context.

## Key findings

- Established a relation chart for stationary points across reformulations
- Provided weaker conditions for local minimizers to be M-stationary
- Characterized the directional limiting normal cone for the PSD cone

## Abstract

For the rank regularized minimization problem, we introduce several kinds of stationary points by the problem itself and its equivalent reformulations including the mathematical program with an equilibrium constraint (MPEC), the global exact penalty of the MPEC,the surrogate yielded by eliminating the dual part in the exact penalty. A clear relation chart is established for these stationary points, which guides the user to choose an appropriate reformulation for seeking a low-rank solution. As a byproduct, we also provide a weaker condition for a local minimizer of the MPEC to be the M-stationary point by characterizing the directional limiting normal cone to the graph of the normal cone mapping of the positive semidefinite (PSD) cone.