Error bounds for rank constrained optimization problems and applications

TL;DR

This paper develops error bounds for rank constrained optimization problems and demonstrates their applications, including the exactness of penalty methods and convergence analysis of convex relaxation algorithms.

Contribution

It establishes Lipschitzian error bounds for rank constrained problems and proves the exactness of penalty formulations, advancing understanding of solution stability and algorithm convergence.

Findings

Error bounds are established under calmness conditions.

Penalty problems are shown to be exact when penalty parameters are sufficiently large.

Error bounds for iterative algorithms decrease with more stages.

Abstract

This paper is concerned with the rank constrained optimization problem whose feasible set is the intersection of the rank constraint set $\mathcal{R}=\!\big\{X\in\mathbb{X}\ |\ {\rm rank}(X)\le \kappa\big\}$ and a closed convex set $\Omega$. We establish the local (global) Lipschitzian type error bounds for estimating the distance from any $X\in \Omega$ ($X\in\mathbb{X}$) to the feasible set and the solution set, respectively, under the calmness of a multifunction associated to the feasible set at the origin, which is specially satisfied by three classes of common rank constrained optimization problems. As an application of the local Lipschitzian type error bounds, we show that the penalty problem yielded by moving the rank constraint into the objective is exact in the sense that its global optimal solution set coincides with that of the original problem when the penalty parameter is over a certain threshold. This particularly offers an affirmative answer to the open question whether the penalty problem (32) in (Gao and Sun, 2010) is exact or not. As another application, we derive the error bounds of the iterates generated by a multi-stage convex relaxation approach to those three classes of rank constrained problems and show that the bounds are nonincreasing as the number of stages increases.