Metric Regularity of the Sum of Multifunctions and Applications

In this work, we use the theory of error bounds to study metric regularity of the sum of two multifunctions, as well as some important properties of variational systems. We use an approach based on the metric regularity of epigraphical multifunctions. Our results subsume some recent results by Durea and Strugariu.Comment: Submitted to JOTA 37 page

Newton's Method for Solving Inclusions Using Set-Valued Approximations

International audienceResults on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super)linear convergence of a Newton- type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton’s method, the nonsmooth Newton’s method for semismooth functions, the inexact proximal point algorithm, etc. Moreover, we also cover a forward-backward splitting algorithm for finding a zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton’s method is discussed

Directional Hölder Metric Regularity

This paper sheds new light on regularity of multifunctions through various characterizations of directional Hölder /Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations , we show that directional Hölder /Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Hölder /Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed

Variational analysis of paraconvex multifunctions

Our aim in this article is to study the class of so-called ρ- paraconvex multifunctions from a Banach space X into the subsets of another Banach space Y. These multifunctions are defined in relation with a modulus function ρ: X→ [0 , + ∞) satisfying some suitable conditions. This class of multifunctions generalizes the class of γ- paraconvex multifunctions with γ> 1 introduced and studied by Rolewicz, in the eighties and subsequently studied by A. Jourani and some others authors. We establish some regular properties of graphical tangent and normal cones to paraconvex multifunctions between Banach spaces as well as a sum rule for coderivatives for such class of multifunctions. The use of subdifferential properties of the lower semicontinuous envelope function of the distance function associated to a multifunction established in the present paper plays a key role in this study. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature

Directional metric pseudo subregularity of set-valued mappings: a general model

This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established. © 2019, Springer Nature B.V

Directional metric regularity of multifunctions

In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity

Directional Holder metric regularity

This paper sheds new light on regularity of multifunctions through various characterizations of directional Holder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Holder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directional Holder/Lipschitz metric regularity to investigate the stability and the sensitivity analysis of parameterized optimization problems are also discussed

Extensions of Fréchet ϵ-Subdifferential Calculus and Applications

AbstractIn this paper, we establish some calculus rules for the limiting Fréchet ϵ-subdifferentials of marginal functions and composite functions. Necessary conditions for approximate solutions of a constrained optimization problem are derived