Variational pairs and applications to stability in nonsmooth analysis.
type: Item
instance of: scholarly article
title: Variational pairs and applications to stability in nonsmooth analysis.
zbMATH Open document ID: 1035.49014
DOI: 10.1016/S0362-546X(01)00129-8
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-01-01T00:00:00Z
review text: In the paper, it is pointed out that many of the basic results of nonsmooth analysis (e.g., subdifferential calculus, mean-value inequalities and optimality conditions) can be derived directly by general variational principles. In this manner, the authors provide a general definition of a variational pair \((X,\delta)\) where \(X\) is a complete metric space and \(\delta\) associates to any lower semicontinuous function \(f:X\to \mathbb{R}\cup \{+\infty\}\) and any point \(x\in X\), a nonnegative extended real number \(\delta f(x)\). The main property which is required on \(\delta\) is that for every \(\overline{x}\in X\) and for every numbers \(\sigma,r>0\) with \[ f(\overline{x})< \inf_{d(x,\overline{x})< r}f(x)+ \sigma r \] one can find some \(x\in X\) with \(d(x,\overline{x})< r\) and \(\delta f(x)< \sigma\). It is shown that with \[ \delta_1f(x)= \begin{cases} 0, &\text{ if \(x\) is a local minimizer of \(f\)},\\ \limsup_{y\to x} \frac{f(y)-f(x)} {y-x}, &\text{ otherwise}, \end{cases} \] and \[ \delta_2f(x)= d(0,\partial f(x)) \] (where \(\partial\) is a suitable subdifferential operator), \((X,\delta_1)\) and \((X,\delta_2)\) are examples of a variational pair. The both main theorems of the paper are consequences of the notion of variational pair. The first one can be seen as a general (global) metric regularity result for a lower semicontinuous function. The second one is basically a local and parametric refinement of the previous one, dealing with the local solvability of an inequality of the type \(f(x,p)\leq 0\) with parameter \(p\) of a topological space. The results are applied for the characterization of so-called asymptotically well-behaved functions, for the discussion of metric regularity properties for parametrized families of multifunctions and for the derivation of a stability assertion for parametrized optimization problems assuming an extended Mangasarian-Fromowitz type constraint qualification.
reviewed by: Jörg Thierfelder
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 1751625
zbMATH Keywords: variational pair
zbMATH Keywords: variational principle
zbMATH Keywords: subdifferential
zbMATH Keywords: godd asymptotical behavior
author: D. Azé
author: Jean-Noël Corvellec
author: Roberto Lucchetti
MaRDI profile type: Publication
cites work: Q5186236
cites work: The topology of the \(\rho\)-Hausdorff distance
cites work: Lipschitz Behavior of Solutions to Convex Minimization Problems
cites work: Stable approximations of set-valued maps
cites work: Stability in Mathematical Programming with Nondifferentiable Data
cites work: Well behaved asymptotical convex functions
cites work: Mean Value Property and Subdifferential Criteria for Lower Semicontinuous Functions
cites work: Nonsmooth Constrained Optimization and Multidirectional Mean Value Inequalities
cites work: On a Newton Type Iterative Method for Solving Inclusions
cites work: Subtraction theorems and approximate openness for multifunctions: Topological and infinitesimal viewpoints
cites work: Q4277838
cites work: Stability and regular points of inequality systems
cites work: A Smooth Variational Principle With Applications to Subdifferentiability and to Differentiability of Convex Functions
cites work: Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps
cites work: Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity
cites work: A survey of subdifferential calculus with applications
cites work: Optimization and nonsmooth analysis
cites work: Mean Value Inequalities in Hilbert Space
cites work: Qualitative properties of trajectories of control systems: a survey
cites work: Convex composite functions in Banach spaces and the primal lower-nice property
cites work: Q4895394
cites work: An introduction to \(\Gamma\)-convergence
cites work: Q3917070
cites work: Evolution equations with lack of convexity
cites work: A critical point theory for nonsmooth functionals
cites work: A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions
cites work: Q3134552
cites work: When a well behaving convex function is well-conditioned
cites work: Nonconvex minimization problems
cites work: Q3200121
cites work: Refinements of necessary optimality conditions in nondifferentiable programming. I
cites work: Regular Points of Lipschitz Functions
cites work: Nonsmooth Analysis: Differential Calculus of Nondifferentiable Mappings
cites work: On the local surjection property
cites work: Approximate subdifferentials and applications 3: the metric theory
cites work: Q4524521
cites work: Fuzzy principles and characterization of trustworthiness
cites work: Q4263363
cites work: Verifiable Conditions for Openness and Regularity of Multivalued Mappings in Banach Spaces
cites work: First-order rules for nonsmooth constrained optimization
cites work: Q3952709
cites work: Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces
cites work: Nonsmooth sequential analysis in Asplund spaces
cites work: Stablity of Set-Valued Mappings In Infinite Dimensions: Point Criteria and Applications
cites work: Metric regularity, openness and Lipschitzian behavior of multifunctions
cites work: Q4485618
cites work: Generalized Directional Derivatives and Subgradients of Nonconvex Functions
cites work: Proximal Subgradients, Marginal Values, and Augmented Lagrangians in Nonconvex Optimization
cites work: Clarke-Ledyaev mean value inequalities in smooth Banach spaces
cites work: The equivalence of several basic theorems for subdifferentials
description: scientific article; zbMATH DE number 1751625