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．     

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.     

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.     

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.     

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.     

On the Convergence of Coderivative of SAA Solution Mapping for a Parametric Stochastic Generalized Equation

The aim of this paper is to investigate the convergence properties for Mordukhovich’s coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic generalized equation. It is demonstrated that, under suitable conditions, both the cosmic deviation and the ρ-deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic generalized equation converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric generalized equations is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic mathematical program with complementarity constraints.     

On regular coderivatives in parametric equilibria with non-unique multipliers

This paper deals with the computation of regular coderivatives of solution maps associated with a frequently arising class of generalized equations (GEs). The constraint sets are given by (not necessarily convex) inequalities, and we do not assume linear independence of gradients to active constraints. The achieved results enable us to state several versions of sharp necessary optimality conditions in optimization problems with equilibria governed by such GEs. The advantages are illustrated by means of examples.     

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.     

Metric subregularity for composite-convex generalized equations in Banach spaces

In this paper we consider a convex-composite generalized constraint equation in Banach spaces. Using variational analysis technique, in terms of normal cones and coderivatives, we first establish sufficient conditions for such an equation to be metrically subregular. Under the Robinson qualification, we prove that these conditions are also necessary for the metric subregularity. In particular, some existing results on error bound and metric subregularity are extended to the composite-convexity case from the convexity case.     

On Several Types of Basic Constraint Qualifications via Coderivatives for Generalized Equations

In this paper, we study several types of basic constraint qualifications in terms of Clarke/Fréchet coderivatives for generalized equations. Several necessary and/or sufficient conditions are given to ensure these constraint qualifications. It is proved that basic constraint qualification and strong basic constraint qualification for convex generalized equations can be obtained by these constraint qualifications, and the existing results on constraint qualifications for the inequality system can be deduced from the given conditions in this paper. The main work of this paper is an extension of the study on constraint qualifications from inequality systems to generalized equations.     

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.     

On proto-differentiability of generalized perturbation maps

This paper is devoted to the sensitivity analysis in optimization problems and variational inequalities. The concept of proto-differentiability of set-valued maps (see [R.T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 449-482]) plays the key role in our investigation. It is proved that, under some suitable qualification conditions, the generalized perturbation maps (that is, the solution set map to a parameterized constraint system, to a parameterized variational inequality, or to a parameterized optimization problem) are proto-differentiable.     

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.     

Sensitivity Analysis of Multiobjective Optimization Problems with Parameterized Quasi-Variational Inequalities

This paper mainly establishes the sensitivity analysis of a multiobjective optimization problem with parameterized quasi-variational inequalities (QVIs). Using the (regular) coderivative of the associated epigraphical multifunction, the (regular) subdifferentials of the efficient frontier maps are estimated, which involve the (regular) coderivatives of the solution mapping to the parameterized QVIs. Under the linear independent constraint qualification, the defined auxiliary set-valued mappings in the parameterized QVIs are clam. The detailed formulae of subdifferentials of the efficient frontier maps are obtained and examples are simultaneously provided for analyzing and illustrating the obtained results.     

一类矩阵对的广义特征值的扰动界限

关于矩阵特征值的扰动,下面的结果是熟知的:若A与C皆为n阶正规矩阵,它们的特征值分别为α_1,…,α_n与γ_1,…,γ_n,则据Wielandt-Hoffman定理,存在1,…,n的一个排列k_1,…,k_n,使得     

Nondifferentiable minimax fractional programming in complex spaces with parametric duality

In this paper, by using the second-order proto-differentiability and second-order lower semidifferentiability, second-order differential properties of a class of set-valued maps are investigated and an explicit expression of the second-order derivatives is obtained. Then, second-order sensitivity properties are discussed for generalized perturbation maps.     

Calmness of efficient solution maps in parametric vector optimization

The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained.     

集值映射及其应用

考虑集值映射的动力学,证明了对于上半连续的集值映射在一定条件下吸引子的存在性及吸引子在扰动下的上半连续性,进一步考虑集值映射在微分方程数值模拟中的应用．利用集值映射的吸引子在扰动下的上半连续性,阐明微分方程数值模拟中的次分算法及区间算法的合理性．     

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.     

On the Aubin property of a class of parameterized variational systems

The paper deals with a new sharp condition ensuring the Aubin property of solution maps to a class of parameterized variational systems. This class encompasses various types of parameterized variational inequalities/generalized equations with fairly general constraint sets. The new condition requires computation of directional limiting coderivatives of the normal-cone mapping for the so-called critical directions. The respective formulas have the form of a second-order chain rule and extend the available calculus of directional limiting objects. The suggested procedure is illustrated by means of examples.