On Some Regularity Properties in Variational Analysis

Some properties, connected with recent generalizations of the classic notion of Lipschitz continuity for multifunctions, are investigated with reference to variational systems, that is to solution maps associated to parametrized generalized equations. The latter ones are a convenient framework to address several questions, mainly related to the stability and sensitivity analysis, arising in mathematical programming, optimal control, equilibrium and variational inequality theory. Global and local criteria for metric regularity and Lipschitz-likeness of variational systems are obtained. Some applications to the exact penalization of mathematical programs with equilibrium constraints and to the Lipschitzian stability of fixed points for multivalued contractions are then considered.     

On Metric Pseudo-(sub)Regularity of Multifunctions and Optimality Conditions for Degenerated Mathematical Programs

This paper is devoted to the analysis of a special kind of regularity of a multifunction which we call metric pseudo-(sub)regularity, when the multifunction is not metrically (sub)regular at a given point but is metrically (sub)regular at certain points in a neighborhood with moduli possibly tending to infinity with a certain order. By using advanced techniques of generalized differentiation we derive conditions both necessary and sufficient for this property. As a byproduct we obtain a new sufficient condition for metric subregularity. Then we apply these results to multifunctions composed by a smooth mapping and a generalized polyhedral multifunction and obtain explicit formulas for this case. Finally we show how the theory can be used to obtain necessary optimality conditions when the constraint qualification condition of metric (sub)regularity is violated.     

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.     

Metric regularity of a positive order for generalized equations

This paper focuses on the metric regularity of a positive order for generalized equations. More concretely, we establish verifiable sufficient conditions for a generalized equation to achieve the metric regularity of a positive order at its a given solution. The provided conditions are expressed in terms of the Fréchet coderivative/or the Mordukhovich coderivative/or the Clarke one of the corresponding multifunction formulated the generalized equation. In addition, we show that such sufficient conditions turn out to be also necessary for the metric regularity of a positive order of the generalized equation in the case where the multifunction established the generalized equation is closed and convex.     

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.     

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.     

Implicit multifunction theorems for the sensitivity analysis of variational conditions

We study implicit multifunctions (set-valued mappings) obtained from inclusions of the form 0∈M(p,x), whereM is a multifunction. Our basic implicit multifunction theorem provides an approximation for a generalized derivative of the implicit multifunction in terms of the derivative of the multifunctionM. Our primary focus is on three special cases of inclusions 0∈M(p,x) which represent different kinds of generalized variational inequalities, called “variational conditions”. Appropriate versions of our basic implicit multifunction theorem yield approximations for generalized derivatives of the solutions to each kind of variational condition. We characterize a well-known generalized Lipschitz property in terms of generalized derivatives, and use our implicit multifunction theorems to state sufficient conditions (and necessary in one case) for solutions of variational conditions to possess this Lipschitz, property. We apply our results to a general parameterized nonlinear programming problem, and derive a new second-order condition which guarantees that the stationary points associated with the Karush-Kuhn-Tucker conditions exhibit generalized Lipschitz continuity with respect to the parameter.     

Implicit multifunction theorems in complete metric spaces

In this paper, we establish some new characterizations of metric regularity of implicit multifunctions in complete metric spaces by using lower semicontinuous envelopes of the distance functions to 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 perturbation stability of implicit multifunctions.     

Metric regularity and Lipschitzian stability of parametric variational systems

The paper concerns the study of variational systems described by parameterized generalized equations/variational conditions important for many aspects of nonlinear analysis, optimization, and their applications. Focusing on the fundamental properties of metric regularity and Lipschitzian stability, we establish various qualitative and quantitative relationships between these properties for multivalued parts/fields of parametric generalized equations and the corresponding solution maps for them in the framework of arbitrary Banach spaces of decision and parameter variables.     

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.     

Coderivative Analysis of Variational Systems

The paper mostly concerns applications of the generalized differentiation theory in variational analysis to Lipschitzian stability and metric regularity of variational systems in infinite-dimensional spaces. The main tools of our analysis involve coderivatives of set-valued mappings that turn out to be proper extensions of the adjoint derivative operator to nonsmooth and set-valued mappings. The involved coderivatives allow us to give complete dual characterizations of certain fundamental properties in variational analysis and optimization related to Lipschitzian stability and metric regularity. Based on these characterizations and extended coderivative calculus, we obtain efficient conditions for Lipschitzian stability of variational systems governed by parametric generalized equations and their specifications.     

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.     

Strong Abadie CQ,ACQ, calmness and linear regularity

The Abadie CQ (ACQ) for convex inequality systems is a fundamental notion in optimization and approximation theory. In terms of the contingent cone and tangent derivative, we extend the Abadie CQ to more general convex multifunction cases and introduce the strong ACQ for both multifunctions and inequality systems. Some seemly unrelated notions are unified by the new ACQ and strong ACQ. Relationships among ACQ, strong ACQ, basic constraint qualification (BCQ) and strong BCQ are discussed. Using the strong ACQ, we study calmness of a closed and convex multifunction between two Banach spaces and, different from many existing dual conditions for calmness, establish several primal characterizations of calmness. As applications, some primal characterizations for error bounds and linear regularity are established; in particular, some existing results are improved.     

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.     

DC Optimization Approach to Metric Regularity of Convex Multifunctions with Applications to Infinite Systems

The paper develops a new approach to the study of metric regularity and related well-posedness properties of convex set-valued mappings between general Banach spaces by reducing them to unconstrained minimization problems with objectives given as the difference of convex (DC) functions. In this way, we establish new formulas for calculating the exact regularity bound of closed and convex multifunctions and apply them to deriving explicit conditions ensuring well-posedness of infinite convex systems described by inequality and equality constraints.     

Characterizing the Single-Valuedness of Multifunctions

We characterize the local single-valuedness and continuity of multifunctions (set-valued mappings) in terms of their premonotonicity and lower semicontinuity. This result completes the well-known fact that lower semicontinuous, monotone multifunctions are single-valued and continuous. We also show that a multifunction is actually a Lipschitz single-valued mapping if and only if it is premonotone and has a generalized Lipschitz property called Aubin continuity. The possible single-valuedness and continuity of multifunctions is at the heart of some of the most fundamental issues in variational analysis and its application to optimization. We investigate the impact of our characterizations on several of these issues; discovering exactly when certain generalized subderivatives can be identified with classical derivatives, and determining precisely when solutions to generalized variational inequalities are locally unique and Lipschitz continuous. As an application of our results involving generalized variational inequalities, we characterize when the Karush–Kuhn–Tucker pairs associated with a parameterized optimization problem are locally unique and Lipschitz continuous.     

Lipschitz-like property of an implicit multifunction and its applications

The aim of this work is twofold. First, we use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of an implicit multifunction. More explicitly, new sufficient conditions in terms of the Fréchet coderivative and the normal/Mordukhovich coderivative of parametric multifunctions for this implicit multifunction to have the Lipschitz-like property at a given point are established. Then we derive sufficient conditions ensuring the Lipschitz-like property of an efficient solution map in parametric vector optimization problems by employing the above implicit multifunction results.     

Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets

Motivated by the subsmoothness of a closed set introduced by Aussel et al. (2005) [8], we introduce and study the uniform subsmoothness of a collection of infinitely many closed subsets in a Banach space. Under the uniform subsmoothness assumption, we provide an interesting subdifferential formula on distance functions and consider uniform metric regularity for a kind of multifunctions frequently appearing in optimization and variational analysis. Different from the existing works, without the restriction of convexity, we consider several fundamental notions in optimization such as the linear regularity, CHIP, strong CHIP and property (G) for a collection of infinitely many closed sets. We establish relationships among these fundamental notions for an arbitrary collection of uniformly subsmooth closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of closed convex sets to the nonconvex setting.     

Metric regularity of epigraphical multivalued mappings and applications to vector optimization

In this work we combine in a meaningful way two techniques of variational analysis and nonsmooth optimization. On one hand, we use the error bound approach to study the metric regularity of some special types of multifunctions and, on the other hand, we exploit the incompatibility between the metric regularity and the Pareto minimality. This method allows us to present some $\varepsilon $ -Fermat rules for set-valued optimization problem in the setting of general Banach spaces. Our results are comparable to several recent results in literature.     

Existence and continuity of solution trajectories of generalized equations with application in electronics

We consider a special form of parametric generalized equations arising from electronic circuits with AC sources and study the effect of perturbing the input signal on solution trajectories. Using methods of variational analysis and strong metric regularity property of an auxiliary map, we are able to prove the regularity properties of the solution trajectories inherited by the input signal. Furthermore, we establish the existence of continuous solution trajectories for the perturbed problem. This can be achieved via a result of uniform strong metric regularity for the auxiliary map.