Convergence of the proximal point method for metrically regular mappings

In this paper we consider the following general version of the proximal point algorithm for solving the inclusion T(x) 3 0, where T is a set-valued mapping acting from a Banach space X to a Banach space Y . First, choose any sequence of functions gn : X → Y with gn(0) = 0 that are Lipschitz continuous in a neighborhood of the origin. Then pick an initial guess x0 and find a sequence xn by applying the iteration gn(xn+1-xn)+T(xn+1) 3 0 for n = 0, 1,... We prove that if the Lipschitz constants of gn are bounded by half the reciprocal of the modulus of regularity of T, then there exists a neighborhood O of x (x being a solution to T(x) 3 0) such that for each initial point x0 2 O one can find a sequence xn generated by the algorithm which is linearly convergent to x. Moreover, if the functions gn have their Lipschitz constants convergent to zero, then there exists a sequence starting from x0 2 O which is superlinearly convergent to x. Similar convergence results are obtained for the cases when the mapping T is strongly subregular and strongly regular

Douglas–Rachford Feasibility Methods for Matrix Completion Problems

In this paper we give general recommendations for successful application of the Douglas-Rachford reflection method to convex and non-convex real matrix-completion problems. These guidelines are demonstrated by various illustrative examples

Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems ∗

We discuss recent positive experiences applying convex feasibility algorithms of Douglas–Rachford type to highly combinatorial and far from convex problems.

Metric regularity of Newton’s iteration

For a version of Newton's method applied to a generalized equation with a parameter, we extend the paradigm of the Lyusternik–Graves theorem to the framework of a mapping acting from the pair “parameter-starting point” to the set of corresponding convergent Newton sequences. Under ample parameterization, metric regularity of the mapping associated with convergent Newton sequences becomes equivalent to the metric regularity of the mapping associated with the generalized equation. We also discuss an inexact Newton method and present an application to discretized optimal control