Implicit Functions and Solution Mappings

A View from Variational Analysis

verfasst von: Asen L. Dontchev, R. Tyrrell Rockafellar

Verlag: Springer New York

Buchreihe : Springer Monographs in Mathematics

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

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Über dieses Buch

Setting up equations and solving them has long been so important that, in popular imagination, it has virtually come to describe what mathematical analysis and its applications are all about. A central issue in the subject is whether the solution to an equation involving parameters may be viewed as a function of those parameters, andif so,what propertiesthat functionmighthave.Thisisaddressedbytheclassical theory of implicit functions, which began with single real variables and progressed through multiple variables to equations in in?nite dimensions, such as equations associated with integral and differential operators. A major aim of the book is to lay out that celebrated theory in a broader way than usual, bringing to light many of its lesser known variants, for instance where standard assumptions of differentiability are relaxed. However, another major aim is to explain how the same constellation of ideas, when articulated in a suitably expanded framework, can deal successfully with many other problems than just solving equations.

Inhaltsverzeichnis

Frontmatter

1. Functions Defined Implicitly by Equations

Abstract

The idea of solving an equation f(p,x) = 0 for x as a function of p, say x = s(p), plays a huge role in classical analysis and its applications. The function obtained in this way is said to be defined implicitly by the equation. The closely related idea of solving an equation f(x) = y for x as a function of y concerns the inversion of f. The circumstances in which an implicit function or an inverse function exists and has properties like differentiability have long been studied. Still, there are features which are not widely appreciated and variants which are essential to seeing how the subject might be extended beyond solving only equations. For one thing, properties other than differentiability, such as Lipschitz continuity, can come in. But fundamental expansions in concept, away from thinking just about functions, can serve in interesting ways as well.

As a starter, consider for real variables x and y the extent to which the equation x ² = y can be solved for x as a function of y. This concerns the inversion of the function f(x) = x ² in Figure 1.1 below, as depicted through the reflection that interchanges the x and y axes. The reflection of the graph is not the graph of a function, but some parts of it may have that character. For instance, a function is obtained from a neighborhood of the point B, but not from one of the point A, no matter how small.

Asen L. Dontchev, R. Tyrrell Rockafellar

2. Implicit Function Theorems for Variational Problems

Abstract

Solutions mappings in the classical setting of the implicit function theorem concern problems in the form of parameterized equations. The concept can go far beyond that, however. In any situation where some kind of problem in x depends on a parameter p, there is the mapping S that assigns to each p the corresponding set of solutions x. The same questions then arise about the extent to which a localization of S around a pair \((\bar{p},\bar{x})\) in its graph yields a function s which might be continuous or differentiable, and so forth.

This chapter moves into that much wider territory in replacing equation-solving problems by more complicated problems termed “generalized equations.” Such problems arise variationally in constrained optimization, models of equilibrium, and many other areas. An important feature, in contrast to ordinary equations, is that functions obtained implicitly from their solution mappings typically lack differentiability, but often exhibit Lipschitz continuity and sometimes combine that with the existence of one-sided directional derivatives.

Asen L. Dontchev, R. Tyrrell Rockafellar

3. Regularity properties of set-valued solution mappings

Abstract

In the concept of a solution mapping for a problem dependent on parameters, whether formulated with equations or something broader like variational inequalities, we have always had to face the possibility that solutions might not exist, or might not be unique when they do exist. This goes all the way back to the setting of the classical implicit function theorem. In letting S(p) denote the set of all x satisfying f(p, x) = 0, where f is a given function from \({\mathbb{R}}^{d} \times {\mathbb{R}}^{n}\) to \({\mathbb{R}}^{m}\), we cannot expect to be defining a function S from \({\mathbb{R}}^{d}\) to \({\mathbb{R}}^{n}\), even when m = n. In general, we only get a set-valued mapping S. However, this mapping S could have a single-valued localization s with properties of continuity or differentiability. The study of such localizations, as “subfunctions” within a set-valued mapping, has been our focus so far, but now we open up to a wider view.

Asen L. Dontchev, R. Tyrrell Rockafellar

4. Regularity Properties Through Generalized Derivatives

Abstract

In the wide-ranging generalizations we have been developing of the inverse function theorem and implicit function theorem, we have followed the idea that conclusions about a solution mapping, concerning the Aubin property, say, or the existence of a single-valued localization, can be drawn by confirming that some auxiliary solution mapping, obtained from a kind of approximation, has the property in question. In the classical framework, we can appeal to a condition like the invertibility of a Jacobian matrix and thus tie in with standard calculus. Now, though, we are far away in another world where even a concept of differentiability seems to be lacking. However, substitutes for classical differentiability can very well be introduced and put to work. In this chapter we show the way to that and explain numerous consequences.

First, graphical differentiation of a set-valued mapping is defined through the variational geometry of the mapping’s graph. A characterization of the Aubin property is derived and applied to the case of a solution mapping. Strong metric subregularity is characterized next. Applications are made to parameterized constraint systems and special features of solution mappings for variational inequalities. There is a review then of some other derivative concepts and the associated inverse function theorems of Clarke and Kummer. Finally, alternative results using coderivatives are described.

Asen L. Dontchev, R. Tyrrell Rockafellar

5. Regularity in infinite dimensions

Abstract

The theme of this chapter has origins in the early days of functional analysis and the Banach open mapping theorem, which concerns continuous linear mappings from one Banach space to another. The graphs of such mappings are subspaces of the product of the two Banach spaces, but remarkably much of the classical theory extends to set-valued mappings whose graphs are convex sets or cones instead of subspaces. Openness connects up then with metric regularity and interiority conditions on domains and ranges, as seen in the Robinson–Ursescu theorem. Infinite-dimensional inverse function theorems and implicit function theorems due to Lyusternik, Graves, and Bartle and Graves can be derived and extended. Banach spaces can even be replaced to some degree by more general metric spaces.

Asen L. Dontchev, R. Tyrrell Rockafellar

Chapter 6. Applications in Numerical Variational Analysis

Abstract

The classical implicit function theorem finds a wide range of applications in numerical analysis. For instance, it helps in deriving error estimates for approximations to differential equations and is often relied on in establishing the convergence of algorithms. Can the generalizations of the classical theory to which we have devoted so much of this book have comparable applications in the numerical treatment of nonclassical problems for generalized equations and beyond? In this chapter we provide positive answers in several directions.

We begin with a topic at the core of numerical work, the “conditioning” of a problem and how it extends to concepts like metric regularity. We also explain how the conditioning of a feasibility problem, like solving a system of inequalities, can be understood. Next we take up a general iterative scheme for solving generalized equations under metric regularity, obtaining convergence by means of our earlier basic results. As particular cases, we get various modes of convergence of the age-old procedure known as Newton’s method in several guises, and of the much more recently introduced proximal point algorithm. We go a step further with Newton’s method by showing that the mapping which assigns to an instance of a parameter the set of all sequences generated by the method obeys, in a Banach space of sequences, the implicit function theorem paradigm in the same pattern as the solution mapping for the underlying generalized equation. Approximations of quadratic optimization problems in Hilbert spaces are then studied. Finally, we apply our methodology to discrete approximations in optimal control.

Asen L. Dontchev, R. Tyrrell Rockafellar

Backmatter

Titel: Implicit Functions and Solution Mappings
verfasst von: Asen L. Dontchev
R. Tyrrell Rockafellar
Verlag: Springer New York
Electronic ISBN: 978-0-387-87821-8
Print ISBN: 978-0-387-87820-1
DOI: https://doi.org/10.1007/978-0-387-87821-8