A metric approach to asymptotic analysis

We introduce a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. We relate this notion to a concept of “firm” asymptotic function. We use these notions to study boundedness properties which can be applied to continuity questions for some operations on sets and functions. Such questions arise in stability analysis of Hamilton-Jacobi equations. We present some other applications such as an extension of a theorem of Dieudonné and existence results in optimization and fixed point theory.     

Critical duality

We look for a general framework in which the Ekeland duality can be formulated. We propose a scheme in which the parameter sets are provided with a coupling function which induces a conjugacy. The decision spaces are not supposed to have any special structure. We examine several examples. In particular, we consider some special classes of generalized convex functions.     

Are dualities appropriate for duality theories in optimization?

We raise some questions about duality theories in global optimization. The main one concerns the possibility to extend the use of conjugacies to general dualities for studying dual optimization problems. In fact, we examine whether dualities are the most general concepts to get duality results. We also consider the passage from a Lagrangian approach to a perturbational approach and the reverse passage in the framework of general dualities. Since a notion of subdifferential can be defined for any duality, it is natural to examine whether the familiar interpretation of multipliers as generalized derivatives of the performance function associated with a dualizing parameterization of the given problem is still valid in the general framework of dualities.     

Infinite products of relations, set-valued series and uniform openness of multifunctions

Using the notations of convergent series of sets and convergent products of relations, general open mapping theorems are presented which encompass classical results of Banach, Ptak, Khanh, and others.     

On the Convergence of Descent Algorithms

It is proved that any cluster point of a sequence defined by a steepest descent algorithm in a general normed vector space is a critical point. The function is just assumed to be continuously differentiable. The class of algorithms we consider encompasses several choices such as the Cauchy steplength and the Curry steplength.     

Genericity of Well-Posedness, Perturbations and Smooth Variational Principles

We show that three important topics in nonlinear analysis and optimization are intimately related: the theory of perturbations, the notion of well-posedness and variational principles in the sense of Ekeland, Borwein–Preiss and Deville–Godefroy–Zizler. The concept of genericity and the new notion of flexible perturbation play a key role in these connections. This notion enables one to consider topologies on spaces of functions which have been introduced recently. A link between the Asplund and Ekeland–Lebourg methods and the Palais–Smale condition, another important topic in nonlinear analysis, is pointed out.     

Conjugate convex operators

Convex mappings from a locally convex space X into F^. = F ∪ {+∞} are considered, where F is an ordered topological vector space and + ∞ an arbitrary greatest element adjoined to F. In view of applications to the polarity theory of convex operators, the possibility is investigated of representing a convex mapping taking values in F^. as a supremum of continuous affine mappings.     

Mean-Value Theorem with Small Subdifferentials

We prove a mean-value theorem for lower semicontinuous functions on a large class of Banach spaces which contains the class of Asplund spaces, in particular reflexive Banach spaces and Banach spaces with a separable dual. It involves the lower subdifferential (or contingent subdifferential) and the Fréchet subdifferentials, which are among the smallest subdifferentials known to date. It follows that the estimates which it provides require weak assumptions and are accurate. When the function is locally Lipschitzian, we get a simple statement which refines the Lebourg mean-value theorem.     

Conditioning convex and nonconvex problems

Two ways of defining a well-conditioned minimization problem are introduced and related, with emphasis on the quantitative aspects. These concepts are used to study the behavior of the solution sets of minimization problems for functions with connected sublevel sets, generalizing results of Attouch-Wets in the convex case. Applications to continuity properties of subdifferentials and to projection mappings are pointed out.We are grateful to M. Valadier for pointing out, during a lecture by the author in Montpellier in October 1990 presenting the main results of the present paper, that existence results in Section 2 of the present paper can be dissociated from estimates.     

Limiting subhessians, limiting subjets and their calculus

We study calculus rules for limiting subjets of order two. These subjets are obtained as limits of sequences of subjets, a subjet of a function at some point being the Taylor expansion of a twice differentiable function which minorizes and coincides with at . These calculus rules are deduced from approximate (or fuzzy) calculus rules for subjets of order two. In turn, these rules are consequences of delicate results of Crandall-Ishii-Lions. We point out the similarities and the differences with the case of first order limiting subdifferentials.