Matrix analysis
Method of successive projections for finding a common point of sets in metric spaces
Journal of Optimization Theory and Applications
Eclatement de Contraintes en Parallèle pour la Minimisation d'une Forme Quadratique
Proceedings of the 7th IFIP Conference on Optimization Techniques: Modeling and Optimization in the Service of Man, Part 2
Alternating Projections on Manifolds
Mathematics of Operations Research
Local Linear Convergence for Alternating and Averaged Nonconvex Projections
Foundations of Computational Mathematics
Mathematics of Operations Research
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We present an analysis of sets of matrices with rank less than or equal to a specified number s. We provide a simple formula for the normal cone to such sets, and use this to show that these sets are prox-regular at all points with rank exactly equal to s. The normal cone formula appears to be new. This allows for easy application of prior results guaranteeing local linear convergence of the fundamental alternating projection algorithm between sets, one of which is a rank constraint set. We apply this to show local linear convergence of another fundamental algorithm, approximate steepest descent. Our results apply not only to linear systems with rank constraints, as has been treated extensively in the literature, but also nonconvex systems with rank constraints.