Gap Continuity of Multimaps

We introduce a notion of continuity for multimaps (or set-valued maps) which is mild. It encompasses both lower semicontinuity and upper semicontinuity. We give characterizations and we consider some permanence properties. This notion can be used for various purposes. In particular, it is used for continuity properties of subdifferentials and of value functions in parametrized optimization problems. We also prove an approximate selection theorem.     

Variational Subdifferential for Quasiconvex Functions

It is shown that a convenient subdifferential for the class of quasiconvex functions is variational. This property combines a variational principle with a kind of weak fuzzy sum rule. It entails a number of useful properties. The subdifferential considered here is the lower subdifferential at the origin (in the sense of Plastria) of the incident derivative or inner epiderivative of the function.     

Compactness Properties, Openness Criteria and Coderivatives

We study the local surjectivity and openness properties of mappings and correspondences by using coderivatives. We recall local criteria but concentrate on point criteria. Our study relies on a compactness condition which is of independent interest, for instance, for the study of the behavior of the injectivity constant of a linear map or of a convex process under stabilization procedures. Several known criteria for openness are shown to be a consequences of this new compactness condition.     

Compact nets,filters, and relations

The notion of compact filter is introduced and compared to related notions of compact nets and compact relations. A characterization of compact nets is given in regular spaces. Several applications in optimization theory and nonlinear analysis are presented.     

Metric Estimates for the Calculus of Multimappings

The present paper completes previous results of Penot and Jourani dealing with continuity properties of multimappings obtained via intersections or inverse images. A key role is played by estimates of metric regularity type. These estimates have consequences on the calculus of generalized derivatives and on the computation of normal cones. The considered multimappings are not supposed to have convex values, but when this property holds, simple qualification conditions ensure these estimates.     

Infinitesimal calculus in metric spaces

We study the possibility of defining tangent vectors to a metric space at a given point and tangent maps to applications from a metric space into another metric space. Such infinitesimal concepts may help in analysing situations in which no obvious differentiable structure is at hand. Some examples are presented; our interest arises from hyperspaces in particular. Our approach is simple and relies on the selection of appropriate curves. Comparisons with other notions are briefly pointed out.     

Cutting Plane Algorithms and Approximate Lower Subdifferentiability

A notion of boundedly ε-lower subdifferentiable functions is introduced and investigated. It is shown that a bounded from below, continuous, quasiconvex function is locally boundedly ε-lower subdifferentiable for every ε>0. Some algorithms of cutting plane type are constructed to solve minimization problems with approximately lower subdifferentiable objective and constraints. In those algorithms an approximate minimizer on a compact set is obtained in a finite number of iterations provided some boundedness assumption be satisfied.     

Generalized Affine Functions and Generalized Differentials

We study some classes of generalized affine functions, using a generalized differential. We study some properties and characterizations of these classes and we devise some characterizations of solution sets of optimization problems involving such functions or functions of related classes.