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This paper addresses the study of subdifferentials for set-valued mappings/multifunctions, which take values in ordered spaces. First we obtain the main calculus (sum and chain) rules for such subdifferentials. Then the developed subdifferential calculus is applied to establishing existence theorems for the so-called relative Pareto minimizers in general problems of set-valued optimization with constraints of various types. [ABSTRACT FROM AUTHOR]

The numerical algorithm by Cristofari et al. modifies the augmented Lagrangian method ALGENCAN proposed by Andreani and his collaborators by incorporating certain second-order information into the augmented Lagrangian framework. Professor Franco Giannessi, University of Pisa, is an outstanding mathematician whose contributions to optimization theory and its applications and to the world optimization community are difficult to overstate. The paper by Izmailov and Solodov introduces and develops a novel perturbed augmented Lagrangian method framework for constrained optimization problems. [Extracted from the article]

The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way, we develop extended calculus rules for first-order and second-order generalized differential constructions while paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second-order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers, strong metric subregularity of Karush-Kuhn-Tucker systems in parametric optimization, and so on. [ABSTRACT FROM AUTHOR]

This paper addresses problems of second-order cone programming important in optimization theory and applications. The main attention is paid to the augmented Lagrangian method (ALM) for such problems considered in both exact and inexact forms. Using generalized differential tools of second-order variational analysis, we formulate the corresponding version of second-order sufficiency and use it to establish, among other results, the uniform second-order growth condition for the augmented Lagrangian. The latter allows us to justify the solvability of subproblems in the ALM and to prove the linear primal–dual convergence of this method. [ABSTRACT FROM AUTHOR]

This is the Preface to the Special Issue of JOTA dedicated to the 60th birthday of Professor Aram V. Arutyunov [ABSTRACT FROM AUTHOR]

This paper brieffy overviews some recent and very fresh results on a rather new class of dynamic optimization problems governed by the so-called sweeping (Moreau) processes with controlled moving sets. Uncontrolled sweeping processes have been known in dynamical systems and applications starting from 1970s while control problems for them have drawn attention of mathematicians, applied scientists, and practitioners quite recently. We discuss here such problems and major results achieved in their theory and applications. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notions of critical and noncritical multipliers for variational systems and extend to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush-Kuhn-Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal-dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain calmness property of a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitz-like/Aubin property of solution maps yields their robust isolated calmness. [ABSTRACT FROM AUTHOR]

The paper addresses an optimal control problem for a perturbed sweeping process of the rate-independent hysteresis type described by a controlled "play-and stop" operator with separately controlled perturbations. This problem can be reduced to dynamic optimization of a state-constrained unbounded differential inclusion with highly irregular data that cannot be treated by means of known results in optimal control theory for differential inclusions. We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of well-posed finite-dimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. To solve the discretized control systems, we derive effective necessary optimality conditions by using second-order generalized differential tools of variational analysis that explicitly calculated in terms of the given problem data. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of tilt stability of local minimizers for classical nonlinear programs with equality and inequality constraints in finite dimensions described by twice continuously differentiable functions. The importance of tilt stability has been well recognized from both theoretical and numerical perspectives of optimization, and this area of research has drawn much attention in the literature, especially in recent years. Based on advanced techniques of variational analysis and generalized differentiation, we derive here complete pointbased second-order characterizations of tilt-stable minimizers entirely in terms of the initial program data under the new qualification conditions, which are the weakest ones for the study of tilt stability. [ABSTRACT FROM AUTHOR]

This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming, important for optimization theory and its applications. First, we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/co-derivatives of the projection operator associated with the circular cone. Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs. [ABSTRACT FROM AUTHOR]

In this paper we study the concept of nondomination in problems of set and vector optimization with variable ordering structures, which reduces to Pareto efficiency when the ordering structure is constant/nonvariable. Based on advanced tools of variational analysis and generalized differentiation, we develop verifiable necessary conditions for nondominated points of sets and for nondominated solutions to vector optimization problems with general geometric constraints that are new in both finite and infinite dimensions. Many examples are provided to illustrate and highlight the major features of the obtained results. [ABSTRACT FROM AUTHOR]

The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of general nonsmooth problems of cone-constrained optimization (or conic programming) important for various aspects of optimization theory and applications. Based on advanced constructions and techniques of variational analysis and generalized differentiation, we derive new necessary optimality conditions (in both «exact» and «fuzzy» forms) for nonsmooth conic programs, establish characterizations of well-posedness for cone-constrained systems, and develop new applications to semi-infinite programming. [ABSTRACT FROM AUTHOR]

This paper contains selected applications of the new tangential extremal principles and related results developed in Mordukhovich and Phan (Math Program ) to calculus rules for infinite intersections of sets and optimality conditions for problems of semi-infinite programming and multiobjective optimization with countable constraints. [ABSTRACT FROM AUTHOR]

In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. The first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi-infinite programming and multiobjective optimization. [ABSTRACT FROM AUTHOR]

This paper is mainly devoted to the study and applications of H61der metric sub-regularity (or metric q-subregularity of order q ∈E (0, 1]) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and point-based sufficient conditions as well as necessary conditions for q-metric subregularity with evaluating the exact subregularity bound, which are new even for the conventional (first-order) metric subregularity in both finite and infinite dimensions. In this way we also obtain new fractional error bound results for composite polynomial systems with explicit calculating fractional exponents. Finally, metric q-subregularity is applied to conduct a quantitative convergence analysis of the classical proximal point method (PPM) for finding zeros of maximal monotone operators on Hilbert spaces. [ABSTRACT FROM AUTHOR]

This article is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to n given nonempty subsets of this space is minimal. This is a generalized version of the classical Heron problem: on a given straight line, find a point C such that the sum of the distances from C to the given points A and B is minimal. We show that the advanced variational techniques allow us to solve optimal location problems of this type completely in some important settings. [ABSTRACT FROM AUTHOR]

In this paper we derive new sufficient conditions for global weak Pareto solutions to set-valued optimization problems with general geometric constraints of the type where $$F: X\rightrightarrows Z$$ is a set-valued mapping between Banach spaces with a partial order on $$Z$$. Our main results are established by using advanced tools of variational analysis and generalized differentiation; in particular, the extremal principle and full generalized differential calculus for the subdifferential/coderivative constructions involved. Various consequences and refined versions are also considered for special classes of problems in vector optimization including those with Lipschitzian data, with convex data, with finitely many objectives, and with no constraints. [ABSTRACT FROM AUTHOR]

This paper pursues a twofold goal. First goal is to derive new results on generalized differentiation in variational analysis focusing mainly on a broad class of intrinsically nondifferentiable marginal/value functions. Then the results established in this direction are applied to deriving necessary optimality conditions for the optimistic version of bilevel programs, which occupy a remarkable place in optimization theory and its various applications. We obtain new sets of optimality conditions in both smooth and nonsmooth settings of finite-dimensional and infinite-dimensional spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials, or tangent cones and subderivatives, or their mixture in finite-dimensional spaces. The results obtained not only extend previous ones in the literature, but also allow us to provide new types of criteria for local weak sharpness. Applications of the developed theory are given to semi-infinite programming and to a new class of semi-infinite complementarity problems. [Copyright &y& Elsevier]

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and Hadamard manifolds. A number of the results obtained in this paper are also new for the case of conventional problems in finite-dimensional Euclidean spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper. [ABSTRACT FROM AUTHOR]

In this article we introduce new notions of local extremality for finite and infinite systems of closed sets and establish the corresponding extremal principles for them, here called rated extremal principles. These developments are in the core geometric theory of variational analysis. We present their applications to calculus and optimality conditions for problems with infinitely many constraints. [ABSTRACT FROM AUTHOR]

In this paper we develop new applications of variational analysis and generalized differentiation to the following optimization problem and its specifications: given n closed subsets of a Banach space, find such a point for which the sum of its distances to these sets is minimal. This problem can be viewed as an extension of the celebrated Fermat-Torricelli problem: given three points on the plane, find another point that minimizes the sum of its distances to the designated points. The generalized Fermat-Torricelli problem formulated and studied in this paper is of undoubted mathematical interest and is promising for various applications including those frequently arising in location science, optimal networks, etc. Based on advanced tools and recent results of variational analysis and generalized differentiation, we derive necessary as well as necessary and sufficient optimality conditions for the extended version of the Fermat-Torricelli problem under consideration, which allow us to completely solve it in some important settings. Furthermore, we develop and justify a numerical algorithm of the subgradient type to find optimal solutions in convex settings and provide its numerical implementations. [ABSTRACT FROM AUTHOR]

This article mainly concerns deriving first-order and second-order necessary (and partly sufficient) optimality conditions for a general class of constrained optimization problems via smoothing regularization procedures based on infimal-like convolutions/envelopes. In this way, we obtain first-order optimality conditions of both lower subdifferential and upper subdifferential types and then second-order conditions of three kinds involving, respectively, generalized second-order directional derivatives, graphical derivatives of first-order subdifferentials and second-order subdifferentials defined via coderivatives of first-order constructions. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study enhanced notions of relative Pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers for general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces. [ABSTRACT FROM AUTHOR]

The paper mostly concerns the study of generalized differential properties of the so-called minimal time functions associated, in particular, with constant dynamics and arbitrary closed target sets in control theory. Functions of this type play a significant role in many aspects of optimization, control theory, and Hamilton–Jacobi partial differential equations. We pay the main attention to computing and estimating limiting subgradients of the minimal value functions and to deriving the corresponding relations for Fréchet type ε-subgradients in arbitrary Banach spaces. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

This article studies new applications of advanced methods of variational analysis and generalized differentiation to constrained problems of multiobjective/vector optimization. We pay most attention to general notions of optimal solutions for multiobjective problems that are induced by geometric concepts of extremality in variational analysis, while covering various notions of Pareto and other types of optimality/efficiency conventional in multiobjective optimization. Based on the extremal principles in variational analysis and on appropriate tools of generalized differentiation with well-developed calculus rules, we derive necessary optimality conditions for broad classes of constrained multiobjective problems in the framework of infinite-dimensional spaces. Applications of variational techniques in infinite dimensions require certain 'normal compactness' properties of sets and set-valued mappings, which play a crucial role in deriving the main results of this article. [ABSTRACT FROM AUTHOR]

The paper is devoted to new applications of advanced tools of modern variational analysis and generalized differentiation to the study of broad classes of multiobjective optimization problems subject to equilibrium constraints in both finite-dimensional and infinite-dimensional settings. Performance criteria in multiobjective/vector optimization are defined by general preference relationships satisfying natural requirements, while equilibrium constraints are described by parameterized generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and are handled in this paper via appropriate normal/coderivative/subdifferential constructions that exhibit full calculi. Most of the results obtained are new even in finite dimensions, while the case of infinite-dimensional spaces is significantly more involved requiring in addition certain “sequential normal compactness” properties of sets and mappings that are preserved under a broad spectrum of operations. [ABSTRACT FROM AUTHOR]

The article is devoted to applications of modern variational analysis to the study of constrained optimization and equilibrium problems in infinite-dimensional spaces. We pay a particular attention to the remarkable classes of optimization and equilibrium problems identified as mathematical programs with equilibrium constraints (MPECs) and equilibrium problems with equilibrium constraints (EPECs) treated from the viewpoint of multiobjective optimization. Their underlying feature is that the major constraints are governed by parametric generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically non-smooth and can be handled by using an appropriate machinery of generalized differentiation exhibiting a rich/full calculus. The case of infinite-dimensional spaces is significantly more involved in comparison with finite dimensions, requiring in addition a certain sufficient amount of compactness and an efficient calculus of the corresponding 'sequential normal compactness' (SNC) properties. [ABSTRACT FROM AUTHOR]

Abstract: The paper is devoted to the study of metric regularity, which is a remarkable property of set-valued mappings playing an important role in many aspects of nonlinear analysis and its applications. We pay the main attention to metric regularity of the so-called parametric variational systems that contain, in particular, various classes of parameterized/perturbed variational and hemivariational inequalities, complementarity systems, sets of optimal solutions and corresponding Lagrange multipliers in problems of parametric optimization and equilibria, etc. On the basis of the advanced machinery of generalized differentiation, we surprisingly reveal that metric regularity fails for certain major classes of parametric variational systems, which admit conventional descriptions via subdifferentials of convex as well as prox-regular extended-real-valued functions. [Copyright &y& Elsevier]

This article concerns new aspects of generalized differentiation theory that plays a crucial role in many areas of modern variational analysis, optimization and their applications. In contrast to the majority of previous developments, we focus here on generalized differentiation of parameter-dependent objects (sets, set-valued mappings and nonsmooth functions), which naturally appear, e.g. in parametric optimization and related topics. The basic generalized differential constructions needed in this case are different for those known in parameter-independent settings, while they still enjoy comprehensive calculus rules developed in this article. [ABSTRACT FROM AUTHOR]

This paper is devoted to applications of modem methods of variational analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main focus is on the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the "lack of compactness" in infinite dimensions, which leads to imposing certain "normal compactness" properties and developing their comprehensive calculus, together with appropriate calculus rules of generalized differentiation. On the other hand, one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems. In particular, we fully avoid the requirements of this type imposed on target endpoint sets in infinite-dimensional optimal control for discrete-time inclusions. [ABSTRACT FROM AUTHOR]

The paper is devoted to optimization problems of the Bolza and Mayer types for evolution systems governed by nonconvex Lipschitzian differential inclusions in Banach spaces under endpoint constraints described by finitely many equalities and inequalities with generally nonsmooth functions. We develop a variational analysis of such problems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of Asplund (and hence any reflexive Banach) spaces. In this way we establish extended results on stability of discrete approximations (with the strong W1,2-convergence of optimal solutions under consistent perturbations of endpoint constraints) and derive necessary optimality conditions for nonconvex discrete-time and continuous-time systems in the refined Euler--Lagrange and Weierstrass--Pontryagin forms accompanied by the appropriate transversality inclusions. In contrast to the case of geometric endpoint constraints in infinite dimensions, the necessary optimality conditions obtained in this paper do not impose any nonempty interiority/finite-codimension/ normal compactness assumptions. The approach and results developed in the paper make a bridge between optimal control/dynamic optimization and constrained mathematical programming problems in infinite-dimensional spaces. [ABSTRACT FROM AUTHOR]

We study equilibrium models governed by parameter-dependent quasi-variational inequalities important from the viewpoint of optimization/equilibrium theory as well as numerous applications. The main focus is on quasi-variational inequalities with parameters entering both single-valued mad multivalued parts of the corresponding generalized equations in the sense of Robinson. The main tools of our variational analysis involve coderivatives of solution maps to quasi-variational inequalities, which allow us to obtain efficient conditions for robust Lipschitzian stability of quasi- variational inequalities and also to derive new necessary optimality conditions for mathematical programs with quasi-variational constraints. To conduct this analysis, we develop new results on coderivative calculus for structural settings involved in our models. The results obtained are illustrated by applications to some optimization and equilibrium models related to parameterized Nash games of two players and to oligopolistic market equilibria. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider dynamic optimization problems for hyperbolic systems with boundary controls and pointwise state constraints. In contrast to parabolic dynamics, such systems have not been sufficiently studied in the literature. The reason is the lack of regularity in the case of hyperbolic dynamics. We present necessary optimality conditions for both Neumann and Dirichlet boundary control problems and discuss differences and relationships between them. [Copyright &y& Elsevier]

Robust Lipschitzian properties of set-valued mappings and marginal functions play a crucial role in many aspects of variational analysis and its applications, especially for issues related to variational stability and optimization. We develop an approach to variational stability based on generalized differentiation. The principal achievements of this paper include new results on coderivative calculus for set-valued mappings and singular subdifferentials of marginal functions in infinite dimensions with their extended applications to Lipschitzian stability. In this way we derive efficient conditions ensuring the preservation of Lipschitzian and related properties for set-valued mappings under various operations, with the exact bound/modulus estimates, as well as new sufficient conditions for the Lipschitz continuity of marginal functions. [ABSTRACT FROM AUTHOR]

The paper is devoted to studying generalized differential properties of distance functions that play a remarkable role in variational analysis, optimization, and their applications. The main object under consideration is the distance function of two variables in Banach spaces that signifies the distance from a point to a moving set. We derive various relationships between Fréchet-type subgradients and limiting (basic and singular) subgradients of this distance function and corresponding generalized normals to sets and coderivatives of set-valued mappings. These relationships are essentially different depending on whether or not the reference point belongs to the graph of the involved set-valued mapping. Our major results are new even for subdifferentiation of the standard distance function signifying the distance between a point and a fixed set in finite-dimensional spaces. The subdifferential results obtained are applied to deriving efficient dual-space conditions for the local Lipschitz continuity of distance functions generated by set-valued mappings, in particular, by those arising in parametric constrained optimization. [ABSTRACT FROM AUTHOR]

This article concerns a new class of optimization-related problems called equilibrium problems with equilibrium constraints (EPEC). One may treat them as two-level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of mathematical programs with equilibrium constraints (MPEC). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This article presents necessary optimality conditions for EPECs in finite-dimensional spaces based on advanced generalized differential tools of variational analysis. The optimality conditions are derived in normal form under certain qualification requirements, which can be regarded as proper analogs of the classical Mangasarian-Fromovitz constraint qualification in the general settings under consideration. [ABSTRACT FROM AUTHOR]

This paper concerns optimal control problems for dynamical systems described by a parametric family of discrete/finite-difference approximations of continuous-time control systems. Control theory for parametric systems governed by discrete approximations plays an important role in both qualitative and numerical aspects of optimal control and occupies an intermediate position in dynamic optimization: between optimal control of discrete-time (with fixed steps) and continuous-time control systems. The central result in optimal control of discrete approximation systems is the approximate maximum principle (AMP), which gives the necessary optimality condition in a perturbed maximum principle form with no a priori convexity assumptions and thus ensures the stability of the Pontryagin maximum principle (PMP) under discrete approximation procedures. The AMP has been justified for optimal control problems of smooth dynamical systems with endpoint constraints under some properness assumption imposed on the sequence of optimal controls. in this paper we show, by a series of counterexamples, that the properness assumption is essential for the validity of the AMP, and that the AMP does not hold, in its expected (lower) subdifferential form, for nonsmooth problems. Moreover, a new upper subdifferential form of the AMP is established for ordinary and time-delay control systems. The results obtained surprisingly solve (in both negative and positive directions) a long-standing and well-recognized question about the possibility of extending the AMP to nonsmooth control problems, for which the affirmative answer has been expected in the conventional lower subdifferential form. [ABSTRACT FROM AUTHOR]

We consider parametric families of constrained problems in mathematical programming and conduct a local sensitivity analysis for multivalued solution maps. Coderivatives of set-valued mappings are our basic tool to analyze the parametric sensitivity of either stationary points or stationary point-multiplier pairs associated with parameterized optimization problems. An implicit mapping theorem for coderivatives is one key to this analysis for either of these objects, and in addition, a partial coderivative rule is essential for the analysis of stationary points. We develop general results along both of these lines and apply them to study the parametric sensitivity of stationary points alone, as well as stationary point-multiplier pairs. Estimates are computed for the coderivative of the stationary point multifunction associated with a general parametric optimization model, and these estimates are refined and augmented by estimates for the coderivative of the stationary point-multiplier multifunction in the case when the constraints are representable in a special composite form. When combined with existing coderivative formulas, our estimates are entirely computable in terms of the original data of the problem. [ABSTRACT FROM AUTHOR]

We develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization-related problems. The main difference between the conventional extremal principle and the extended version developed below is that, instead of the translation of sets involved in the extremal systems, we allow deformations. The new version seems to be more flexible in various applications and covers, in particular, multiobjective optimization problems with general preference relations. In this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games. [ABSTRACT FROM AUTHOR]

The paper aims to develop some basic principles and tools of nonconvex variational analysis with applications to necessary suboptimality and optimality conditions for constrained optimization problems in infinite dimensions. We establish a certain subdifferential variational principle as a new characterization of Asplund spaces. This result is different from conventional support forms of variational principles and appears to be convenient for applications to nonsmooth optimization. Based on the subdifferential variational principle, we obtain new necessary conditions for suboptimal solutions in general nonsmooth optimization problems with equality, inequality, and set constraints in Asplund spaces. In this way we establish the so-called sequential normal compactness properties of constraint sets that play an essential role in infinite-dimensional variational analysis and its applications. As a by-product of our approach, we derive various forms of necessary optimality conditions for nonsmooth constrained problems in infinite dimensions, which extend known results in that direction. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of optimization problems for dynamical systems governed by constrained delay-differential inclusions with generally nonsmooth and nonconvex data. We provide a variational analysis of the dynamic optimization problems based on their data perturbations that involve finite-difference approximations of time-derivatives matched with the corresponding perturbations of endpoint constraints. The key issue of such an analysis is the justification of an appropriate strong stability of optimal solutions under finite-dimensional discrete approximations. We establish the required pointwise convergence of optimal solutions and obtain necessary optimality conditions for delay-differential inclusions in intrinsic Euler-Lagrange and Hamiltonian forms involving nonconvex-valued subdifferentials and coderivatives of the initial data. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses various articles within the issue on topics including complementarity, multiobjective optimization and duality theory.