Calculus of Tangent Sets and Derivatives of Set Valued Maps under Metric Subregularity Conditions

In this paper we intend to give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second order objects are envisaged and the assumptions we impose in order to get the calculus are in terms of metric subregularity of the assembly of the initial data. This approach is different from those used in alternative recent papers in literature and allows us to avoid compactness conditions. A special attention is paid for the case of perturbation set-valued maps which appear naturally in optimization problems.Comment: 17 page

A Lyusternik–Graves theorem for the proximal point method

We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion y∈T(x), where y is a perturbation element near 0 and T is a set-valued mapping acting from a Banach space X to a Banach space Y which is metrically regular around some point (xˉ,0) in its graph. We study the behavior of the convergent iterates generated by the algorithm and we prove that they inherit the regularity properties of T, and vice versa. We analyze the cases when the mapping T is metrically regular and strongly regular.Research of the first author was partially supported by Ministerio de Ciencia e Innovación (Spain), grant MTM2008-06695-C03-01 and program “Juan de la Cierva”. Research of the second author was partially supported by Contract EA4540 (France)

Enhanced metric regularity and Lipschitzian properties of variational systems

Variational analysis and optimization, Parametric variational systems, Generalized equations, Set-valued mappings, Metric regularity, Lipschitzian properties,

ON THE LOCAL CONVERGENCE OF THE DOUGLAS–RACHFORD ALGORITHM

International audienceWe discuss the Douglas–Rachford algorithm to solve the feasibility problem for two closed sets A, B in R d. We prove its local convergence to a fixed point when A, B are finite unions of convex sets. We also show that for more general nonconvex sets the scheme may fail to converge and start to cycle, and may then even fail to solve the feasibility problem

Nonlinear local error bounds via a change of metric

International audienceIn this work, we improve the approach of the second author and V. Motreanu [Math. Program. 114 (2008), 291–319] to nonlinear error bounds for lower semicontinuous functions on complete metric spaces, an approach consisting in reducing the nonlinear case to the linear one through a change of metric. This improvement is basically a technical one, and it allows dealing with local error bounds in an appropriate way. We present some consequences of the general results in the framework of classical nonsmooth analysis, involving Banach spaces and subdifferential operators. In particular, we describe connections between local quadratic growth of a function and metric regularity of its subdifferential