Perturbation of error bounds

Our aim in the current article is to extend the developments in Kruger, Ngai & Th\’era, SIAM J. Optim. 20(6), 3280–3296 (2010) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by proper lower semicontinuous under data perturbations. We propose new … Read more

Calmness modulus of linear semi-infinite programs

Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide … Read more

Slopes of multifunctions and extensions of metric regularity

This article aims to demonstrate how the definitions of slopes can be extended to multi-valued mappings between metric spaces and applied for characterizing metric regularity. Several kinds of local and nonlocal slopes are defined and several metric regularity properties for set-valued mappings between metric spaces are investigated. Citation Published in Vietnam Journal of Mathematics 40:2&3(2012) … Read more

Metric regularity of the sum of multifunctions and applications

In this work, we use the theory of error bounds to study of 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 Citation XLIM (UMR-CNRS … Read more

Implicit Multifunction Theorems in complete metric spaces

In this paper, we establish some new characterizations of the metric regularity of implicit multifunctions in complete metric spaces by using the lower semicontinuous envelopes of the distance functions for set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the … Read more

Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions

The main objective of this paper is to provide new explicit criteria to characterize weak lower semi-continuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by the means of the proximal … Read more

Minimizing irregular convex functions: Ulam stability for approximate minima

The main concern of this article is to study Ulam stability of the set of $\varepsilon$-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space $X$, when the objective function is subjected to small perturbations (in the sense of Attouch \& Wets). More precisely, we characterize the class all … Read more

Stability of error bounds for convex constraint systems in Banach spaces

This paper studies stability of error bounds for convex constraint systems in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems … Read more

Stability of error bounds for semi-infinite convex constraint systems

In this paper, we are concerned with the stability of the error bounds for semi-infinite convex constraint systems. Roughly speaking, the error bound of a system of inequalities is said to be stable if all its “small” perturbations admit a (local or global) error bound. We first establish subdifferential characterizations of the stability of error … Read more

An Approximate Lagrange Multiplier Rule

In this paper, we show that for a large class of optimization problems, the Lagrange multiplier rule can be derived from the so-called approximate multiplier rule. In establishing the link between the approximate and the exact multiplier rule we first derive an approximate multiplier rule for a very general class of optimization problems using the … Read more