Openness stability and implicit multifunction theorems: Applications to variational systems

In this paper we aim to present two general results regarding, on one hand, the openness stability of set-valued maps and, on the other hand, the metric regularity behavior of the implicit multifunction related to a generalized variational system. Then, these results are applied in order to obtain, in a natural way, and in a widely studied case, several relations between the metric regularity moduli of the field maps defining the variational system and the solution map. Our approach allows us to complete and extend several very recent results from the literature.     

Openness properties for parametric set-valued mappings and implicit multifunctions

The aim of this paper is to obtain some openness results in terms of normal coderivative for parametric set-valued mappings acting between infinite dimensional spaces. Then, implicit multifunction results are obtained by simply specializing the openness results. Moreover, we study a kind of metric regularity of the implicit multifunction. The results of the paper generalize several recent results in literature.     

Lipschitzian stability of fully parameterized generalized equations

This paper concerns the study of Lipschitzian stability of fully parameterized generalized equations in which both single-valued and set-valued functions depend on parameters. Various relationships between the Lipschitz-like and metric regularity properties of the solution mapping, the base mapping, or field mapping in the fully perturbed generalized equations are established by using the Dontchev–Hager Fixed Point Theorem. The implicit mapping theorem for metric regularity is also extended to fully parameterized generalized equations.     

Stability for trust-region methods via generalized differentiation

We obtain necessary and sufficient conditions for local Lipschitz-like property and sufficient conditions for local metric regularity in Robinson’s sense of Karush–Kuhn–Tucker point set maps of trust-region subproblems in trust-region methods. The main tools being used in our investigation are dual criteria for fundamental properties of implicit multifunctions which are established on the basis of generalized differentiation of normal cone mappings.     

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.     

Local convergence of quasi-Newton methods under metric regularity

We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results.     

Point-based sufficient conditions for metric regularity of implicit multifunctions

We obtain some point-based sufficient conditions for the metric regularity in Robinson’s sense of implicit multifunctions in a finite-dimensional setting. The new implicit function theorem (which is very different from the preceding results of Ledyaev and Zhu [Yu.S. Ledyaev, Q.J. Zhu, Implicit multifunctions theorems, Set-Valued Anal. 7 (1999) 209–238], Ngai and Théra [H.V. Ngai, M. Théra, Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Anal. 12 (2004) 195–223], Lee, Tam and Yen [G.M. Lee, N.N. Tam, N.D. Yen, Normal coderivative for multifunctions and implicit function theorems, J. Math. Anal. Appl. 338 (2008) 11–22]) can be used for analyzing parametric constraint systems as well as parametric variational systems. Our main tools are the concept of normal coderivative due to Mordukhovich and the corresponding theory of generalized differentiation.     

Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds

This paper is focused on the problem of finding a singularity of the sum of two vector fields defined on a Hadamard manifold, or more precisely, the study of a generalized equation in a Riemannian setting. We extend the concept of metric regularity to the Riemannian setting and investigate its relationship with the generalized equation in this new context. In particular, a version of Graves’s theorem is presented and we also define some concepts related to metric regularity, including the Aubin property and the strong metric regularity of set-valued vector fields. A conceptual method for finding a singularity of the sum of two vector fields is also considered. This method has as particular instances: the proximal point method, Newton’s method, and Zincenko’s method on Hadamard manifolds. Under the assumption of metric regularity at the singularity, we establish that the methods are well defined in a suitable neighborhood of the singularity. Moreover, we also show that each sequence generated by these methods converges to this singularity at a superlinear rate.     

Stability of Implicit Multifunctions in Banach Spaces

This paper is devoted to present new sufficient conditions for both the metric regularity in the Robinson??s sense and the Lipschitz-like property in the Aubin??s sense of implicit multifunctions in general Banach spaces. The basic tools of our analysis involve the Clarke subdifferential, the Clarke coderivative of set-valued mappings, and the Ekeland variational principle. The metric regularity of implicit multifunction is compared with the Lipschitz-like property.     

Chain rules for linear openness in metric spaces and applications

In this work we present a general theorem concerning chain rules for linear openness of set-valued mappings acting between metric spaces. As particular cases, we obtain classical and also some new results in this field of research, including the celebrated Lyusternik–Graves Theorem. The applications deal with the study of the well-posedness of the solution mappings associated to parametric systems. Sharp estimates for the involved regularity moduli are given.     

Lusin's theorem on fuzzy measure spaces

In this paper, we show that weakly null-additive fuzzy measures on metric spaces possess regularity. Lusin's theorem, which is well-known in classical measure theory, is generalized to fuzzy measure space by using the regularity and weakly null-additivity. A version of Egoroff's theorem for the fuzzy measure defined on metric spaces is given. An application of Lusin's theorem to approximation in the mean of measurable function on fuzzy measure spaces is presented.     

On mappings covering at a nonlinear rate and their perturbation stability

The present paper contains a study of covering (alias, openness) properties at a nonlinear rate for set-valued mappings between metric spaces. Such study is focussed on the stability of these properties in the presence of perturbations. A crucial result valid for linear openness, known as Milyutin’s theorem, is extended to set-valued mappings covering at a nonlinear rate under possibly non-Lipschitz perturbations. Consequently, a Lyusternik type theorem is derived from such extension and a general penalization principle for constrained optimization problems, which exploits nonlinear covering properties, is presented.     

Metric Regularity of the Sum of Multifunctions and Applications

The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported.     

Enhanced metric regularity and Lipschitzian properties of variational systems

This paper mainly concerns the study of a large class of variational systems governed by parametric generalized equations, which encompass variational and hemivariational inequalities, complementarity problems, first-order optimality conditions, and other optimization-related models important for optimization theory and applications. An efficient approach to these issues has been developed in our preceding work (Aragón Artacho and Mordukhovich in Nonlinear Anal 72:1149–1170, 2010) establishing qualitative and quantitative relationships between conventional metric regularity/subregularity and Lipschitzian/calmness properties in the framework of parametric generalized equations in arbitrary Banach spaces. This paper provides, on one hand, significant extensions of the major results in op.cit. to partial metric regularity and to the new hemiregularity property. On the other hand, we establish enhanced relationships between certain strong counterparts of metric regularity/hemiregularity and single-valued Lipschitzian localizations. The results obtained are new in both finite-dimensional and infinite-dimensional settings.     

On Generalized Douglas–Weyl(α, β)-Metrics

In this paper, we study generalized Douglas–Weyl(α, β)-metrics. Suppose that a regular(α, β)-metric F is not of Randers type. We prove that F is a generalized Douglas–Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas–Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.     

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.     

Metric regularity of composition set-valued mappings: Metric setting and coderivative conditions

The paper concerns a new method to obtain a proof of the openness at linear rate/metric regularity of composite set-valued maps on metric spaces by the unification and refinement of several methods developed somehow separately in several works of the authors. In fact, this work is a synthesis and a precise specialization to a general situation of some techniques explored in the last years in the literature. In turn, these techniques are based on several important concepts (like error bounds, lower semicontinuous envelope of a set-valued map, local composition stability of multifunctions) and allow us to obtain two new proofs of a recent result having deep roots in the topic of regularity of mappings. Moreover, we make clear the idea that it is possible to use (co)derivative conditions as tools of proof for openness results in very general situations.     

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.     

On the Stability of the Directional Regularity

In this paper we select two tools of investigation of the classical metric regularity of set-valued mappings, namely the Ioffe criterion and the Ekeland Variational Principle, which we adapt to the study of the directional setting. In this way, we obtain in a unitary manner new necessary and/or sufficient conditions for directional metric regularity. As an application, we establish stability of this property at composition and sum of set-valued mappings. In this process, we introduce directional tangent cones and the associated generalized primal differentiation objects and concepts. Moreover, we underline several links between our main assertions by providing alternative proofs for several results.

     

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.