Calmness and inverse image characterizations for Asplund spaces

We establish new characterizations of Asplund spaces in terms of conditions ensuring the calmness property for constraint set mappings and the validity of inverse image formula for a general constrained system.     

On Lipschitz Semicontinuity Properties of Variational Systems with Application to Parametric Optimization

In this paper, two properties of recognized interest in variational analysis, known as Lipschitz lower semicontinuity and calmness, are studied with reference to a general class of variational systems, i.e. to solution mappings to parameterized generalized equations. In the consideration of the metric nature of such properties, some related sufficient conditions are established, which are expressed via nondegeneracy conditions on derivative-like objects appropriate for a metric space analysis. For certain classes of generalized equations in Asplund spaces, it is shown how such conditions can be formulated by using the Fréchet coderivative of the field and the derivative of the base. Applications to the stability analysis of parametric constrained optimization problems are proposed.     

On Hölder calmness of solution mappings in parametric equilibrium problems

We consider parametric equilibrium problems in metric spaces. Sufficient conditions for the Hölder calmness of solutions are established. We also study the Hölder well-posedness for equilibrium problems in metric spaces.     

Well-Posedness Under Relaxed Semicontinuity for Bilevel Equilibrium and Optimization Problems with Equilibrium Constraints

Bilevel equilibrium and optimization problems with equilibrium constraints are considered. We propose a relaxed level closedness and use it together with pseudocontinuity assumptions to establish sufficient conditions for well-posedness and unique well-posedness. These conditions are new even for problems in one-dimensional spaces, but we try to prove them in general settings. For problems in topological spaces, we use convergence analysis while for problems in metric cases we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescu’s measures of noncompactness of approximate solution sets. Besides some new results, we also improve or generalize several recent ones in the literature. Numerous examples are provided to explain that all the assumptions we impose are very relaxed and cannot be dropped.     

Calculus of directional coderivatives and normal cones in Asplund spaces

We study the directional Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings in Asplund spaces and establish extensive calculus results on these constructions under various operations of sets and mappings. We also develop calculus of the directional sequential normal compactness both in general Banach spaces and in Asplund spaces.     

Necessary optimality conditions for minimax programming problems with mathematical constraints

In this paper, necessary optimality conditions in terms of upper and/or lower subdifferentials of both cost and constraint functions are derived for minimax optimization problems with inequality, equality and geometric constraints in the setting of non-differentiatiable and non-Lipschitz functions in Asplund spaces. Necessary optimality conditions in the fuzzy form are also presented. An application of the fuzzy necessary optimality condition is shown by considering minimax fractional programming problem.     

Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems

In this paper, we introduce a new concept of ϵ-efficiency for vector optimization problems. This extends and unifies various notions of approximate solutions in the literature. Some properties for this new class of approximate solutions are established, and several existence results, as well as nonlinear scalarizations, are obtained by means of the Ekeland’s variational principle. Moreover, under the assumption of generalized subconvex functions, we derive the linear scalarization and the Lagrange multiplier rule for approximate solutions based on the scalarization in Asplund spaces.     

Constraint qualifications for constrained Lipschitz optimization problems and applications to a MPCC

Three constraint qualifications (the weak generalized Robinson constraint qualification, the bounded constraint qualification, and the generalized Abadie constraint qualification), which are weaker than the generalized Robinson constraint qualification (GRCQ) given by Yen (1997) [1], are introduced for constrained Lipschitz optimization problems. Relationships between those constraint qualifications and the calmness of the solution mapping are investigated. It is demonstrated that the weak generalized Robinson constraint qualification and the bounded constraint qualification are easily verifiable sufficient conditions for the calmness of the solution mapping, whereas the proposed generalized Abadie constraint qualification, described in terms of graphical derivatives in variational analysis, is weaker than the calmness of the solution mapping. Finally, those constraint qualifications are written for a mathematical program with complementarity constraints (MPCC), and new constraint qualifications ensuring the C-stationary point condition of a MPCC are obtained.     

The Fermat rule for multifunctions for super efficiency

Using variational analysis, we study the vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In terms of the coderivatives and normal cones, we present Fermat’s rules as necessary or sufficient conditions for a super efficient solution of the above problems.     

Asplund operators and holomorphic maps

LetE andF be locally convex topological vector spaces. A holomorphic mapf: E→F is defined to be an Asplund map if it takes the separable subsets of a neighbourhood of eacha∈E into absolutely convex weakly metrisable subsets ofF; a Banach space is an Asplund space if and only if its identity map has this property. We show that a continuous linear map from a quasinormable locally convex spaceE into a Banach spaceF is an Asplund map if and only if it factors through an Asplund space. IfE andF are both Banach spaces, then a holomorphic mapf: E→F is an Asplund map if and only if its derivative maps factor through Asplund spaces for eacha∈E. This is true if and only if such a factorisation holds ata=0. Part of this research was done during a visit to the University of Namibia, whose financial support is gratefully acknowledged This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

Calmness and Exact Penalization in Vector Optimization with Cone Constraints

In this paper, a (local) calmness condition of order α is introduced for a general vector optimization problem with cone constraints in infinite dimensional spaces. It is shown that the (local) calmness is equivalent to the (local) exact penalization of a vector-valued penalty function for the constrained vector optimization problem. Several necessary and sufficient conditions for the local calmness of order α are established. Finally, it is shown that the local calmness of order 1 implies the existence of normal Lagrange multipliers. Presented at the 6th International Conference on Optimization: Techniques and Applications, Ballarat, Australia, December 9–11, 2004 This work is supported by the Postdoctoral Fellowship of Hong Kong Polytechnic University.     

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.     

System of Generalized Vector Quasi-Equilibrium Problems in Locally FC-Spaces

A new class of locally finite continuous topological spaces （for short, locally FC-spaces） and a class of system of generalized vector quasi-equilibrium problems are introduced. By applying a generalized Himmelberg type fixed point theorem for a set-valued mapping with KKM-property due to the author, a collectively fixed point and an equilibrium existence theorem of generalized game are first proved in locally FC-spaces. By applying our equilibrium existence theorem of generalized game, some new existence theorems of equilibrium points for the system of generalized vector quasi-equilibrium problems are proved in locally FC-spaces. These theorems improve, unify and generalize many known results in the literatures.     

Calmness of constraint systems with applications

The paper is devoted to the analysis of the calmness property for constraint set mappings. After some general characterizations, specific results are obtained for various types of constraints, e.g., one single nonsmooth inequality, differentiable constraints modeled by polyhedral sets, finitely and infinitely many differentiable inequalities. The obtained conditions enable the detection of calmness in a number of situations, where the standard criteria (via polyhedrality or the Aubin property) do not work. Their application in the framework of generalized differential calculus is explained and illustrated by examples associated with optimization and stability issues in connection with nonlinear complementarity problems or continuity of the value-at-risk. This research was supported by the DFG Research center Matheon Mathematics for key technologies in Berlin Support by grant IAA1030405 of the Grant Agency of the Academy of Sciences of the Czech Republic is acknowledged     

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.     

A nondegenerate fuzzy optimality condition for constrained optimization problems without qualification conditions

Fuzzy optimization conditions in terms of the Fréchet subdifferential for reflexive spaces were investigated by Borwein, Treiman and Zhu (1998) in [1]. To achieve the nondegenerate form, it is well known that some qualification conditions should be assumed. In this paper, we are going to prove that the nondegenerate fuzzy optimality condition even holds with no qualification conditions in Asplund spaces (in particular, reflexive spaces) for optimization problems with semi-continuous and continuous data. The results are even new in finite-dimensional frameworks.     

Pareto Optimality in Nonconvex Economies with Infinite-dimensional Commodity Spaces

The paper contains applications of variational analysis to the study of Pareto optimality in nonconvex economies with infinite-dimensional commodity spaces satisfying the Asplund property. Our basic tool is a certain extremal principle that provides necessary conditions for set extremality and can be treated as a variational extension of the classical convex separation principle to systems of nonconvex sets. In this way we obtain new versions of the generalized second welfare theorem for nonconvex economies in terms of appropriate normal cones of variational analysis.     

Exact penalty functions and calmness for mathematical programming under nonlinear perturbations

In the present paper, the effects of nonlinear perturbations of constraint systems are considered over the relationship between calmness and exact penalization, within the context of mathematical programming with equilibrium constraints. Two counterexamples are provided showing that the crucial link between the existence of penalty functions and the property of calmness for perturbed problems is broken in the presence of general perturbations. Then, some properties from variational analysis are singled out, which are able to restore to a certain extent the broken link. Consequently, conditions on the value function associated to perturbed optimization problems are investigated in order to guarantee the occurrence of the above properties.     

Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting.     

Asplund空间中非凸向量均衡问题近似解的最优性条件

在Asplund空间中,研究了非凸向量均衡问题近似解的最优性条件.借助Mordukhovich次可微概念,在没有任何凸性条件下获得了向量均衡问题εe-拟弱有效解,εe-拟Henig有效解,εe-拟全局有效解以及εe-拟有效解的必要最优性条件.作为它的应用,还给出了非凸向量优化问题近似解的最优性条件.