Calmness of partial perturbation to composite rank constraint systems and its applications

TL;DR

This paper investigates the calmness property of partial perturbations in composite rank constraint systems, establishing error bounds and criteria for calmness, and applying these to derive exact penalties and surrogates for rank-constrained optimization.

Contribution

It introduces criteria for calmness of partial perturbations in rank systems and applies these to develop exact penalties and surrogates for optimization problems.

Findings

Calmness of partial perturbations is equivalent to local and global Lipschitz error bounds.

Criteria are provided for identifying sets with calm perturbations, verified for common rank constraints.

Derived global exact penalties and DC surrogates improve rank-constrained optimization methods.

Abstract

This paper is concerned with the calmness of a partial perturbation to the composite rank constraint system, an intersection of the rank constraint set and a general closed set, which is shown to be equivalent to a local Lipschitz-type error bound and also a global Lipschitz-type error bound under a certain compactness. Based on its lifted formulation, we derive two criteria for identifying those closed sets such that the associated partial perturbation possesses the calmness, and provide a collection of examples to demonstrate that the criteria are satisfied by common nonnegative and positive semidefinite rank constraint sets. Then, we use the calmness of this perturbation to obtain several global exact penalties for rank constrained optimization problems, and a family of equivalent DC surrogates for rank regularized problems.