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.     

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.     

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.     

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 and Sensitivity Analysis of Solutions to Weak Vector Variational Inequalities

The paper mainly concerns the study of a parametric weak vector variational inequality. Robinson’s metric regularity and Lipschitzian stability of the solution mapping for the parametric weak vector variational inequality are firstly established. Then sensitivity analysis of the solution mapping for the parametric weak vector variational inequality is discussed. As applications, Lipschitzian continuity and differentiability of the solution mapping are also investigated for a parametric variational inequality.     

Coderivative calculus and metric regularity for constraint and variational systems

This paper provides new developments in generalized differentiation theory of variational analysis with their applications to metric regularity of parameterized constraint and variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity for these two major classes of parametric systems is based on appropriate coderivative constructions for set-valued mappings and on extended calculus rules supporting their computation and estimation. The main attention is paid in this paper to the so-called reversed mixed coderivative, which is of crucial importance for efficient pointwise characterizations of metric regularity in the general framework of set-valued mappings between infinite-dimensional spaces. We develop new calculus results for the latter coderivative that allow us to compute it for large classes of parametric constraint and variational systems. On this basis we derive verifiable sufficient conditions, necessary conditions as well as complete characterizations for metric regularity of such systems with computing the corresponding exact bounds of metric regularity constants/moduli. This approach allows us to reveal general settings in which metric regularity fails for major classes of parametric variational systems. Furthermore, the developed coderivative calculus leads us also to establishing new formulas for computing the radius of metric regularity for constraint and variational systems, which characterize the maximal region of preserving metric regularity under linear (and other types of) perturbations and are closely related to conditioning aspects of optimization.     

Convex Difference Criteria for the Quantitative Stability of Parametric Quasidifferentiable Systems

In this paper solvability and Lipschitzian stability properties for a special class of nonsmooth parametric generalized systems defined in Banach are studied via a variational analysis approach. Verifiable sufficient conditions for such properties to hold under scalar quasidifferentiability assumptions are formulated by combining *-difference and Demyanov difference of convex compact subsets of the dual space with classic quasidifferential calculus constructions. Applications to the formulation of sufficient conditions for metric regularity/open covering of nonsmooth maps, along with their employment in deriving optimality conditions for quasidifferentiable extremum problems, as well as an application to the study of semicontinuity of the optimal value function in parametric optimization are discussed. In memory of Aleksandr Moiseevich Rubinov (1940–2006).     

Stability Analysis for Composite Optimization Problems and Parametric Variational Systems

This paper aims to provide various applications for second-order variational analysis of extended-real-valued piecewise linear functions recently obtained by the authors. We mainly focus here on establishing relationships between full stability of local minimizers in composite optimization and Robinson’s strong regularity of associated (linearized and nonlinearized) KKT systems. Finally, we address Lipschitzian stability of parametric variational systems with convex piecewise linear potentials.     

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.     

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.     

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.     

Differentiability and regularity of Lipschitzian mappings

We introduce new differentiability properties of functions between Banach spaces and establish their relationships with graphical regularity of Lipschitzian single-valued and set-valued mappings. The proofs are based on advanced tools of nonsmooth variational analysis including new results on coderivative scalarization and normal cone calculus.

     

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.     

Generalized Newton’s method based on graphical derivatives

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness and Lipschitzian assumptions, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and B-differentiable versions of Newton’s method for nonsmooth Lipschitzian equations.     

Localization of Generalized Normal Maps and Stability of Variational Inequalities in Reflexive Banach Spaces

In this paper, we introduce a localized version of generalized normal maps as well as generalized natural mappings. By using these concepts, we study continuity properties of the solution map of parametric variational inequalities in reflexive Banach spaces. This localization permits us to deal with variational conditions posed on sets that may not be convex and to establish existence and continuity of solutions. We also establish homeomorhisms between the solution set of variational inequalities and the solution set of generalized normal maps. Using these homeomorphisms and the degree theory, we show that the solution map of parametric variational inequalities is lower semicontinuous. Our results extend some results of Robinson (Set-Valued Anal 12:259–274, 2004). The authors wish to express their sincere appreciation to Professor Stephen M. Robinson, Department of Industrial and Systems Engineering, University of Wisconsin-Madison, for his valuable comments and suggestions. This research was partially supported by a grant from National Science Council of Taiwan, ROC.     

Robinson metric regularity of parametric variational systems

In this paper, the Robinson metric regularity of a parametric variational system is investigated. Some applications to the contingent derivative of parametric variational system and to the Robinson metric regularity of a parametric vector optimization problem are then studied.     

Lower Semicontinuity of the Solution Map to a Parametric Generalized Variational Inequality in Reflexive Banach Spaces

This paper is concerned with the study of solution stability of a parametric generalized variational inequality in reflexive Banach spaces. Under the requirements that the operator of a unperturbed problem is of class (S)_?+? and operators under consideration are pseudo-monotone and demicontinuous, we show that the solution map of a parametric generalized variational inequality is lower semicontinuous. The obtained results are proved without conditions related to the degree theory and the metric projection.     

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.     

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.     

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.