Strong metric subregularity of mappings in variational analysis and optimization

Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form $f+F$, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.     

Banach空间集值映射的度量正则性与变分方程的Lipschitz稳定性

综述了集值映射的某些概念,例如度量正则性、伪Lipschitz性质(Aubin性质)、度量次正则性和Calm性质和这些概念的相互关系以及某些判据.也给出了他们在变分方程解的鲁棒Lipschitz稳定性、约束优化问题的最优性条件、集合族的线性正则性质和广义方程迭代过程的收敛性.     

Metric subregularity for composite-convex generalized equations in Banach spaces

In this paper we consider a convex-composite generalized constraint equation in Banach spaces. Using variational analysis technique, in terms of normal cones and coderivatives, we first establish sufficient conditions for such an equation to be metrically subregular. Under the Robinson qualification, we prove that these conditions are also necessary for the metric subregularity. In particular, some existing results on error bound and metric subregularity are extended to the composite-convexity case from the convexity case.     

On Metric and Calmness Qualification Conditions in Subdifferential Calculus

The paper contains two groups of results. The first are criteria for calmness/subregularity for set-valued mappings between finite-dimensional spaces. We give a new sufficient condition whose subregularity part has the same form as the coderivative criterion for “full” metric regularity but involves a different type of coderivative which is introduced in the paper. We also show that the condition is necessary for mappings with convex graphs. The second group of results deals with the basic calculus rules of nonsmooth subdifferential calculus. For each of the rules we state two qualification conditions: one in terms of calmness/subregularity of certain set-valued mappings and the other as a metric estimate (not necessarily directly associated with aforementioned calmness/subregularity property). The conditions are shown to be weaker than the standard Mordukhovich–Rockafellar subdifferential qualification condition; in particular they cover the cases of convex polyhedral set-valued mappings and, more generally, mappings with semi-linear graphs. Relative strength of the conditions is thoroughly analyzed. We also show, for each of the calculus rules, that the standard qualification conditions are equivalent to “full” metric regularity of precisely the same mappings that are involved in the subregularity version of our calmness/subregularity condition. The research of Jiří V. Outrata was supported by the grant A 107 5402 of the Grant Agency of the Academy of Sciences of the Czech Republic.     

Regularity and Conditioning of Solution Mappings in Variational Analysis

Concepts of conditioning have long been important in numerical work on solving systems of equations, but in recent years attempts have been made to extend them to feasibility conditions, optimality conditions, complementarity conditions and variational inequalities, all of which can be posed as solving ‘generalized equations’ for set-valued mappings. Here, the conditioning of such generalized equations is systematically organized around four key notions: metric regularity, subregularity, strong regularity and strong subregularity. Various properties and characterizations already known for metric regularity itself are extended to strong regularity and strong subregularity, but metric subregularity, although widely considered, is shown to be too fragile to support stability results such as a radius of good behavior modeled on the Eckart–Young theorem.     

Hölder metric subregularity for multifunctions in type Banach spaces

Using the Borwein–Preiss variational principle and in terms of the proximal coderivative, we provide a new type of sufficient conditions for the Hölder metric subregularity and Hölder error bounds in a class of smooth Banach spaces. As an application, new characterizations for the tilt stability of Hölder minimizers are established.     

Metric subregularity for proximal generalized equations in Hilbert spaces

In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert spaces. We also establish the results of Robinson-Ursescu theorem type for prox-regular multifunctions.     

BCQ and Strong BCQ for Nonconvex Generalized Equations with Applications to Metric Subregularity

In this paper, based on basic constraint qualification (BCQ) and strong BCQ for convex generalized equation, we are inspired to further discuss constraint qualifications of BCQ and strong BCQ for nonconvex generalized equation and then establish their various characterizations. As applications, we use these constraint qualifications to study metric subregularity of nonconvex generalized equation and provide necessary and/or sufficient conditions in terms of constraint qualifications considered herein to ensure nonconvex generalized equation having metric subregularity.     

Metric Subregularity for a Multifunction

Metric subregularity is an important and active area in modern variational analysis and nonsmooth optimization. Many existing results on the metric subregularity were established in terms of coderivatives of the multifunctions concerned. This note tries to give a survey of the metric subregularity theory related to the coderivatives and normal cones.     

Metric Subregularity in Generalized Equations

In this article, we study the metric subregularity of generalized equations using a new tool of nonsmooth analysis. We obtain a sufficient condition for a generalized equation to be metrically subregular, which is not a necessary condition for metric regularity, using a subtle adjustment of the Mordukhovich coderivative. We apply these results to the study of the metric subregularity in a Cournot duopoly game.     

On Directional Metric Regularity, Subregularity and Optimality Conditions for Nonsmooth Mathematical Programs

This paper mainly deals with the study of directional versions of metric regularity and metric subregularity for general set-valued mappings between infinite-dimensional spaces. Using advanced techniques of variational analysis and generalized differentiation, we derive necessary and sufficient conditions, which extend even the known results for the conventional metric regularity. Finally, these results are applied to non-smooth optimization problems. We show that that at a locally optimal solution M-stationarity conditions are fulfilled if the constraint mapping is subregular with respect to one critical direction and that for every critical direction a M-stationarity condition, possibly with different multipliers, is fulfilled.     

Metric Subregularity of Composition Set-Valued Mappings with Applications to Fixed Point Theory

In this paper we underline the importance of the parametric subregularity property of set-valued mappings, defined with respect to fixed sets. We show that this property appears naturally for some very simple mappings which play an important role in the theory of metric regularity. We prove a result concerning the preservation of metric subregularity at generalized compositions. Then we obtain, in purely metric setting, several fixed point assertions for set-valued mappings in local and global frameworks.     

Perturbations of Moore–Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper,the perturbations of the Moore–Penrose metric generalized inverses of linear operators in Banach spaces are described.The Moore–Penrose metric generalized inverse is homogeneous and nonlinear in general,and the proofs of our results are different from linear generalized inverses.By using the quasi-additivity of Moore–Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition,we show some error estimates of perturbations for the singlevalued Moore–Penrose metric generalized inverses of bounded linear operators.Furthermore,by means of the continuity of the metric projection operator and the quasi-additivity of Moore–Penrose metric generalized inverse,an expression for Moore–Penrose metric generalized inverse is given.     

On metric subregularity for convex constraint systems by primal equivalent conditions

In this paper, we mainly study metric subregularity for a convex constraint system defined by a convex set-valued mapping and a convex constraint subset. The main work is to provide several primal equivalent conditions for metric subregularity by contingent cone and graphical derivative. Further it is proved that these primal equivalent conditions can characterize strong basic constraint qualification of convex constraint system given by Zheng and Ng (SIAM J Optim 18:437–460, 2007).     

A tale of two conformally invariant metrics

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.     

Characterization of Stability for Cone Increasing Constraint Mappings

We investigate stability (in terms of metric regularity) for the specific class of cone increasing constraint mappings. This class is of interest in problems with additional knowledge on some nondecreasing behavior of the constraints (e.g. in chance constraints, where the occurring distribution function of some probability measure is automatically nondecreasing). It is demonstrated, how this extra information may lead to sharper characterizations. In the first part, general cone increasing constraint mappings are studied by exploiting criteria for metric regularity, as recently developed by Mordukhovich. The second part focusses on genericity investigations for global metric regularity (i.e. metric regularity at all feasible points) of nondecreasing constraints in finite dimensions. Applications to chance constraints are given.     

On Necessary Optimality Conditions with Higher-Order Complementarity Slackness for Set-Valued Optimization Problems

Set-Valued and Variational Analysis - We aim to establish Karush-Kuhn-Tucker multiplier rules involving higher-order complementarity slackness under Hölder metric subregularity. These rules...     

Linear Convergence of Prox-SVRG Method for Separable Non-smooth Convex Optimization Problems under Bounded Metric Subregularity

Journal of Optimization Theory and Applications - With the help of bounded metric subregularity which is weaker than strong convexity, we show the linear convergence of proximal stochastic...     

Quadratic Growth and Strong Metric Subregularity of the Subdifferential for a Class of Non-prox-regular Functions

Journal of Optimization Theory and Applications - This paper mainly studies the quadratic growth and the strong metric subregularity of the subdifferential of a function that can be represented as...     

Hölder metric subregularity for constraint systems in Asplund spaces

Positivity - This paper deals with the Hölder metric subregularity property of a certain constraint system in Asplund space. Using the techniques of variational analysis, its main part is...