集值映射的伪(*)连续与弱(*)连续性

本文引入了集值映射的伪（＊）连续性与弱（＊）连续性概念的定义,研究了伪（＊）连续、（＊）连续及弱（＊）连续的等价关系,最后研究了乘积空间中的集值映射成为伪（＊）连续和弱（＊）连续的充要条件。     

函数的次微分连续性与邻近正则性

研究了函数和集在某点的邻近正则性与次微分连续性,给出邻近正则函数的次微分在某种意义下的单调性及函数的邻近正则性与其上图的邻近正则性的关系．     

Supercalm Multifunctions For Convergence Analysis

Calmness of multifunctions is a well-studied concept of generalized continuity in which single-valued selections from the image sets of the multifunction exhibit a restricted type of local Lipschitz continuity where the base point is fixed as one point of comparison. Generalized continuity properties of multifunctions like calmness can be applied to convergence analysis when the multifunction appropriately represents the iterates generated by some algorithm. Since it involves an essentially linear relationship between input and output, calmness gives essentially linear convergence results when it is applied directly to convergence analysis. We introduce a new continuity concept called ‘supercalmness’ where arbitrarily small calmness constants can be obtained near the base point, which leads to essentially superlinear convergence results. We also explore partial supercalmness and use a well-known generalized derivative to characterize both when a multifunction is supercalm and when it is partially supercalm. To illustrate the value of such characterizations, we explore in detail a new example of a general primal sequential quadratic programming method for nonlinear programming and obtain verifiable conditions to ensure convergence at a superlinear rate.     

Nonsmooth Analysis of Singular Values. Part II: Applications

In this work we continue the nonsmooth analysis of absolutely symmetric functions of the singular values of a real rectangular matrix. Absolutely symmetric functions are invariant under permutations and sign changes of its arguments. We extend previous work on subgradients to analogous formulae for the proximal subdifferential and Clarke subdifferential when the function is either locally Lipschitz or just lower semicontinuous. We illustrate the results by calculating the various subdifferentials of individual singular values. Another application gives a nonsmooth proof of Lidskii’s theorem for weak majorization. Mathematics Subject Classifications (2000) Primary 90C31, 15A18; secondary 49K40, 26B05.Research supported by NSERC.     

Convergence of the method of chords for solving generalized equations

In this article, we study an iterative procedure of the following form

Nonsingularity Conditions for Multifunctions

We discuss three different characterizations of continuity properties for general multifunctions S : R^d Rⁿ. Each of these characterizations is given by the same simple nonsingularity condition, but stated in terms of three different generalized derivatives. Two of these characterizations are known, but the third is new to this paper. We discuss how all three have immediate analogues as generalized inverse mapping theorems, and we apply our new characterization to develop a fundamental and very broad sensitivity theorem for solutions to parameterized optimization problems.     

Generalized continuity for Multifunctions

The aim of this paper is to introduce two kinds of generalized continuity for multifunctions. Basic properties and characterizations of such multifunctions are established. These two generalized continuities include many of the variations of multifunction continuity already in the literature as special cases.      

Inverse of the Sum of Set-Valued Mappings and Applications

In this article, we develop results on the behavior of fixed points sets of set-valued pseudo-contraction mappings. Then, we investigate the notions related to the Aubin property and make use of connections between the two involved set-valued nonnecessarily Lipschitzian mappings to obtain results on the inverse of their sum similar to those in the literature generalizing Lyusternik and Graves theorems. By proximal convergence, we apply our results to the sensitivity analysis of variational inclusions.     

Subdifferential representation of homogeneous functions and extension of smoothness in Banach spaces

In this paper, we mainly consider proximal subdifferentials of lower semicontinuous functions defined on real Hilbert space and Clarke＇s subdifferentials of locally Lipschitzian functions defined on Banach space respectively, and obtain the generalized Euler identity of homogenous functions. Then, by introducing a multifunction F, we extend the smoothness of sphere and differentiability of norm function in Banach space.     

Iterative solving of variational inclusions under Wijsman perturbations

We prove that the metric regularity of set-valued mappings is stable under some Wijsman-type perturbations. Then, we solve a variational inclusion viewed as a limit-problem using assumptions on a sequence of associated problems. Finally, we apply our results to classical methods for solving variational inclusions.      

The Clarke and Michel-Penot Subdifferentials of the Eigenvalues of a Symmetric Matrix

We calculate the Clarke and Michel-Penot subdifferentials of the function which maps a symmetric matrix to its mth largest eigenvalue. We show these two subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized directional derivatives of the eigenvalue function, thereby completing earlier studies on the classical directional derivative.     

A proof of the Lyusternik–Graves theorem

In this paper, we give a new proof of the Lyusternik–Graves theorem, based on an intermediate result regarding linear openness inspired by works of Frankowska and Ursescu.     

An elementary new proof of the determination of a convex function by its subdifferential

Rockafellar proved that any closed, convex function is uniquely determined by its subdifferential mapping up to an additive constant. The aim of this article is to provide an elementary proof of the same result.     

The approximate and the Clarke subdifferentials can be different everywhere

We prove that, for a Lipschitz function on , n2, the approximate and the Clarke subdifferentials can differ everywhere. This completely answers a question by A.D. Ioffe, which was partially answered by G. Katriel.     

A characterization of smooth normed linear spaces

In this paper a new characterization of smooth normed linear spaces is discussed using the notion of proximal points of a pair of convex sets. It is proved that a normed linear space is smooth if and only if for each pair of convex sets, points which are mutually nearest to each other from the respective sets are proximal.     

Nonsmooth Analysis of Singular Values. Part I: Theory

The singular values of a rectangular matrix are nonsmooth functions of its entries. In this work we study the nonsmooth analysis of functions of singular values. In particular we give simple formulae for the regular subdifferential, the limiting subdifferential, and the horizon subdifferential, of such functions. Along the way to the main result we give several applications and in particular derive von Neumann’s trace inequality for singular values. Mathematics Subject Classifications (2000) Primary 90C31, 15A18; secondary 49K40, 26B05.Research supported by NSERC.     

Constraint Qualifications in Nonsmooth Multiobjective Optimization

For an inequality constrained nonsmooth multiobjective optimization problem involving locally Lipschitz functions, stronger KT-type necessary conditions and KT necessary conditions (which in the continuously differentiable case reduce respectively to the stronger KT conditions studied recently by Maeda and the usual KT conditions) are derived for efficiency and weak efficiency under several constraint qualifications. Stimulated by the stronger KT-type conditions, the notion of core of the convex hull of the union of finitely many convex sets is introduced. As main tool in the derivation of the necessary conditions, a theorem of the alternatives and a core separation theorem are also developed which are respectively extensions of the Motzkin transposition theorem and the Tucker theorem.     

An Extension of the Birkhoff Integrability for Multifunctions

A comparison between a set-valued Gould type and simple Birkhoff integrals of bf(X)-valued multifunctions with respect to a non-negative set function is given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.