Prox-regularity of rank constraint sets and implications for algorithms

Details Prox-regularity of rank constraint sets and implications for algorithms Prox-regularity of rank constraint sets and implications for algorithms

2011-12-02

arxiv.org/abs/1112.0526v1

Mathematics

Optimization and Control

11 pages, 12 references

Scientific paper

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 on the boundary defined by points with rank exactly equal to s. 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.

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