The Ekeland variational principle for Henig proper minimizers and super minimizers

In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of the Ekeland variational principle for these points involving coderivatives in the sense of Ioffe, Clarke and Mordukhovich. As applications we obtain sufficient conditions for F to have a Henig proper minimizer or a super minimizer under the Palais-Smale type conditions. The techniques are essentially based on the characterizations of Henig proper efficient points and super efficient points by mean of the Henig dilating cones and the Hiriart-Urruty signed distance function.     

Lagrange multipliers for set-valued optimization problems associated with coderivatives

In this paper we investigate a vector optimization problem (P) where objective and constraints are given by set-valued maps. We show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of (P) as solutions of a scalar optimization problem (SP) with single-valued objective and constraint functions. Then applying some classical or recent results in optimization theory to (SP) and using estimates of subdifferentials of marginal functions, we obtain optimality conditions for (P) expressed in terms of Lagrange or sequential Lagrange multipliers associated with various coderivatives of the set-valued data.     

Sensitivity analysis of gap functions for vector variational inequality via coderivatives

The aim of this article is to investigate codifferential properties of a class of set-valued maps and gap function involving vector variational inequality. Relationships between their coderivatives are discussed. Formulae for computing coderivatives of the gap function are established. Optimality conditions of solutions for vector variational inequalities are obtained. The finite-dimensional cases are also discussed.     

A Sard theorem for tame set-valued mappings

If F is a set-valued mapping from Rn into Rm with closed graph, then y∈Rm is a critical value of F if for some x with y∈F(x), F is not metrically regular at (x,y). We prove that the set of critical values of a set-valued mapping whose graph is a definable (tame) set in an o-minimal structure containing additions and multiplications is a set of dimension not greater than m−1 (respectively a σ-porous set). As a corollary of this result we get that the collection of asymptotically critical values of a set-valued mapping with a semialgebraic graph has dimension not greater than m−1. We also give an independent proof of the fact that a definable continuous real-valued function is constant on components of the set of its subdifferentiably critical points.     

广义扰动映射的上导数

主要讨论了两集值映射和的上导数．在比标准约束品性弱的条件下得到了两个集值映射和的上导数与两集值映射上导数的和之间的包含关系,并将此结论用于讨论广义扰动映射的上导数,得到广义扰动映射的上导数的上界估计．     

Riemann-measurable selections

LetX be a Polish space equipped with a σ-finite regular Borel measure μ. IfE is a metric space andF a set-valued function:X → 2 ^E with complete values, and ifF is lower semicontinuous at almost all points ofX, we prove that there exists a Riemann-measurable selections ofF.     

Hadamard Inequality and a Refinement of Jensen Inequality for Set—Valued Functions

We present the following set-valued analogue of the Hadamard inequality: Let Y be a Banach space, I be an open interval and let F: I ? cl(Y) be a continuous and convex set-valued function. Then $${F(a)+F(b)\over 2}\subset{1\over b-a}\int_a^b\ F(x)dx\subset F \bigg({a+b\over 2}\bigg)$$ , for every a, b ∈ I, a < b. Some refinement of Jensen inequality for set-valued functions is also given.     

Geometric approach to convex subdifferential calculus

In this paper, we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic results of convex subdifferential calculus in full generality and also derive new results of convex analysis concerning optimal value/marginal functions, normals to inverse images of sets under set-valued mappings, calculus rules for coderivatives of single-valued and set-valued mappings, and calculating coderivatives of solution maps to parameterized generalized equations governed by set-valued mappings with convex graphs.     

Codifferential Calculus

In this paper, some exact calculus rules are obtained for calculating the coderivatives of the composition of two multivalued maps. Similar rules are displayed for sums. A crucial role is played by an intermediate set-valued map called the resolvent. We first establish inclusions for contingent, Fréchet and limiting coderivatives. Combining them, we get equality rules. The qualification conditions we present are natural and less exacting than classical conditions.     

A viability approach to the inverse set-valued map theorem

The purpose of this paper is to characterize by means of viability tools the pseudo-lipschitzianity property of a set-valued map F in a neighborhood of a point of its graph in terms of derivatives of this set-valued map F in a neighborhood of a point of its graph, instead of using the transposes of the derivatives. On the way, we relate these properties to the calmness index of a set-valued map, an extensions of Clarke’s calmness of a function, as well as Doyen’s Lipschitz kernel of a set-valued map, which is the largest Lipschitz submap.     

Asplund空间中方向上导数和序列法紧性质分析法则

研究了广义微分结构中的集合方向Mordukhovich法锥、集值映射的方向上导数,以及集合和集值映射的方向序列法紧性的分析法则. 基于集合方向Mordukhovich法锥的交集法则,在方向内半紧性假设下,建立了集合的方向Mordukhovich法锥、集值映射的方向上导数的分析法则.此外,借助Asplund乘积空间中集合的方向序列法紧性的交集法则, 在方向内半紧性和相应的规范条件下,建立了集合和集值映射的(部分)方向序列法紧性的加法、逆像、复合等法则.     

Bernstein–Doetsch-type criteria for the continuity and Lipschitz continuity of convex set-valued mappings

The Bernstein–Doetsch criterion (for convex and midconvex functionals) has been repeatedly generalized to convex and midconvex set-valued mappings F: X → 2 ^Y ; continuity and local Lipschitz continuity were understood in the sense of the Hausdorff distance. However, all such results imposed restrictive additional boundedness-type conditions on the images F(x). In this paper, the Bernstein–Doetsch criterion is generalized to arbitrary convex and midconvex set-valued mappings acting on normed linear spaces X,Y.     

On the local existence of solutions to a class of nonconvex evolution inclusions

We prove the local existence of solutions to the Cauchy problemx'∈-?_F V(x)+F(x+f(t,x),x(0)=x ₀, where? _FV is the Fréchet subdifferential of a functionV with aψ-monotone subdifferential of order 2,F is an upper semicontinuous set-valued map contained in the Fréchet subdifferential of aφ- convex function of order two andf is a Carathéodory mapping.     

Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces

Let (X,?) be a partially ordered set and d be a complete metric on X. Let F,G be two set-valued mappings on X. We obtained sufficient conditions for the existence of common fixed point of F and G satisfying an implicit relation in partially ordered set X.     

Generalized penalization and maximization of vectorial nonsmooth functions

We use directional Lipschitz concepts and a minimal time function with respect to a set of directions in order to derive generalized penalization results for Pareto minimality in set-valued constrained optimization. Then, we obtain necessary optimality conditions for maximization in constrained vector optimization in terms of generalized differentiation objects. To the latter aim, we deduce first some enhanced calculus rules for coderivatives of the difference of two mappings. All the main results of this paper are tailored to model directional features of the optimization problem under study.     

Calmness of Set-Valued Mappings Between Asplund Spaces and Application to Equilibrium Problems

The paper deals with the calmness of two classes of nonconvex set-valued mappings in Asplund spaces and its application to equilibrium problems. Its main part is devoted to establish new sufficient conditions for calmness, which are derived in terms of coderivatives and w* boundaries of normal cones to constraint sets. In order to achieve this goal, a new concept so-called “sequential normal smoothness” for the sets in Asplund spaces is introduced and compared with two well-known notions of convexity and semismoothness. Finally, the results are applied to prove necessary optimality conditions for nonparametric equilibrium problems under new weak constraint qualifications.     

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.     

Viscosity Coderivatives and Their Limiting Behavior in Smooth Banach Spaces

This paper concerns with generalized differentiation of set-valued and nonsmooth mappings between Banach spaces. We study the so-called viscosity coderivatives of multifunctions and their limiting behavior under certain geometric assumptions on spaces in question related to the existence of smooth bump functions of any kind. The main results include various calculus rules for viscosity coderivatives and their topological limits. They are important in applications to variational analysis and optimization.     

Subgradient of distance functions with applications to Lipschitzian stability

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. Dedicated to Terry Rockafellar in honor of his 70th birthday. This research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451158.     

Existence of equilibria of set-valued maps on bounded epi-Lipschitz domains in Hilbert spaces without invariance conditions

In this paper, we provide a new result of the existence of equilibria for set-valued maps on bounded closed subsets K of Hilbert spaces. We do not impose either convexity or compactness assumptions on K but we assume that K has epi-Lipschitz sections, i.e. its intersection with suitable finite dimensional spaces is locally the epigraph of Lipschitz functions. In finite dimensional spaces, the famous Brouwer theorem asserts the existence of a fixed point for a continuous function from a compact convex set K to itself. Our result could be viewed as a kind of generalization of this classical result in the context of Hilbert spaces and when the function (or the set-valued map) does not necessarily map K into itself (K is not invariant under the map). Our approach is based firstly on degree theory for compact and for condensing set-valued maps and secondly on flows generated by trajectories of differential inclusions.