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2013 | Buch

Metric and Topological Tools Kapitel lesen Erstes Kapitel lesen

Calculus Without Derivatives

verfasst von: Jean-Paul Penot

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Calculus Without Derivatives expounds the foundations and recent advances in nonsmooth analysis, a powerful compound of mathematical tools that obviates the usual smoothness assumptions. This textbook also provides significant tools and methods towards applications, in particular optimization problems. Whereas most books on this subject focus on a particular theory, this text takes a general approach including all main theories.

In order to be self-contained, the book includes three chapters of preliminary material, each of which can be used as an independent course if needed. The first chapter deals with metric properties, variational principles, decrease principles, methods of error bounds, calmness and metric regularity. The second one presents the classical tools of differential calculus and includes a section about the calculus of variations. The third contains a clear exposition of convex analysis.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Metric and Topological Tools
Abstract
We devote this opening chapter to some preliminary material dealing with sets, set-valued maps, convergences, estimates, and well-posedness.
Jean-Paul Penot
Chapter 2. Elements of Differential Calculus
Abstract
Differential calculus is at the core of several sciences and techniques. Our world would not be the same without it: astronomy, electromagnetism, mechanics, optimization, thermodynamics, among others, use it as a fundamental tool.
Jean-Paul Penot
Chapter 3. Elements of Convex Analysis
Abstract
The class of convex functions is an important class that enjoys striking and useful properties. A homogenization procedure makes it possible to reduce this class to the subclass of sublinear functions. This subclass is next to the family of linear functions in terms of simplicity: the epigraph of a sublinear function is a convex cone, a notion almost as simple and useful as the notion of linear subspace. These two facts explain the rigidity of the class, and its importance.
Jean-Paul Penot
Chapter 4. Elementary and Viscosity Subdifferentials
Abstract
We devote the present chapter to some fundamental notions of nonsmooth analysis upon which some other constructions can be built. Their main features are easy consequences of the definitions. Normal cones have already been considered in connection with optimality conditions. Here we present their links with subdifferentials for nonconvex, nonsmooth functions. When possible, we mention the corresponding notions of tangent cones and directional derivatives; then one gets a full picture of four related objects that can be considered the four pillars of nonsmooth analysis, or even the six pillars if one considers graphical derivatives and coderivatives of multimaps. In the present framework, in contrast to the convex objects defined in Chap. 5, the passages from directional derivatives and tangent cones to subdifferentials and normal cones respectively are one-way routes, because the first notions are nonconvex, while a dual object exhibits convexity properties. On the other hand, the passages from analytical notions to geometrical notions and the reverse passages are multiple and useful. These connections are part of the attractiveness of nonsmooth analysis.
Jean-Paul Penot
Chapter 5. Circa-Subdifferentials, Clarke Subdifferentials
Abstract
We devote the present chapter to one of the most famous attempts to generalize the concept of derivative. When limited to the class of locally Lipschitzian functions, it is of simple use, a fact that explains its success. The general case requires a more sophisticated approach. We choose a geometrical route to it involving the concept of normal cone. It makes easy the proofs of calculus rules. In fact, in this theory, a complete primal–dual picture is available: besides a normal cone concept, one has a notion of tangent cone to a set, and besides a subdifferential for a function one has a notion of directional derivative. Moreover, inherent convexity properties ensure a full duality between these notions. Furthermore, the geometrical notions are related to the analytical notions in the same way as those that have been obtained for elementary subdifferentials. These facts represent great theoretical and practical advantages.
Jean-Paul Penot
Chapter 6. Limiting Subdifferentials
Abstract
The fuzzy character of the rules devised in Chap. 4 incites us to pass to the limit. Such a process is simple enough in finite-dimensional Banach spaces. However, since a number of problems are set in functional spaces, one is led to examine what can be done in infinite-dimensional spaces. It appears that the situation may depend on the nature of the space. For that reason, we mainly limit the study of this chapter to the framework of Asplund spaces (Sects. 6.1–6.5). As seen in Chap. 4, all the rules concerning Fréchet subdifferentials in Fréchet smooth Banach spaces are valid in the framework of Asplund spaces. Passing to the limit gives a particularly simple and striking character to the rules concerning sums and compositions. However, in Sect. 6.6, we give some attention to a limiting procedure involving directional subdifferentials. Such a construction is particularly adapted to the wide class of weakly compactly generated (WCG) spaces that encompasses separable spaces and reflexive spaces.
Jean-Paul Penot
Chapter 7. Graded Subdifferentials, Ioffe Subdifferentials
Abstract
In this last chapter we present an approach valid in every Banach space. The key idea, due to A.D. Ioffe, that yields such a universal theory consists in reducing the study to a convenient class of linear subspaces. Initially, Ioffe used the class of finite-dimensional subspaces of X [512, 513, 515, 516]; then he turned to the class of closed separable subspaces, which has some convenient permanence properties [527]. Since such an approach presents some analogy with the notion of inductive limit of topological linear spaces, one could call the obtained subdifferentials inductive subdifferentials rather than geometric subdifferentials. However, we adopt a different terminology that is evocative of this restriction process, and we keep Ioffe’s notation ∂ G .
Jean-Paul Penot
Backmatter
Metadaten
Titel
Calculus Without Derivatives
verfasst von
Jean-Paul Penot
Copyright-Jahr
2013
Verlag
Springer New York
Electronic ISBN
978-1-4614-4538-8
Print ISBN
978-1-4614-4537-1
DOI
https://doi.org/10.1007/978-1-4614-4538-8

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