Convex Analysis (Princeton Landmarks in Mathematics and Physics): (PMS-28) (Princeton Mathematical Series, 28) - Softcover

Rockafellar, Ralph Tyrell

Softcover

ISBN 10: 0691015864 ISBN 13: 9780691015866

Verlag: Princeton University Press, 1997

Inhaltsangabe

Available for the first time in paperback, R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions. This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading.

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R. Tyrrell Rockafellar

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Convex Analysis

By R. Tyrrell Rockafellar

PRINCETON UNIVERSITY PRESS

Copyright © 1970 Princeton University Press
All rights reserved.

ISBN: 978-0-691-01586-6

Contents

Preface, vii, 
Introductory Remarks: a Guide for the Reader, xi, 
PART I: BASIC CONCEPTS, 
§1. Affine Sets, 3, 
§2. Convex Sets and Cones, 10, 
§3. The Algebra of Convex Sets, 16, 
§4. Convex Functions, 23, 
§5. Functional Operations, 32, 
PART II: TOPOLOGICAL PROPERTIES, 
§6. Relative Interiors of Convex Sets, 43, 
§7. Closures of Convex Functions, 51, 
§8. Recession Cones and Unboundedness, 60, 
§9. Some Closedness Criteria, 72, 
§10. Continuity of Convex Functions, 82, 
PART III: DUALITY CORRESPONDENCES, 
§11. Separation Theorems, 95, 
§12. Conjugates of Convex Functions, 102, 
§13. Support Functions, 112, 
§14. Polars of Convex Sets, 121, 
§15. Polars of Convex Functions, 128, 
§16. Dual Operations, 140, 
PART IV: REPRESENTATION AND INEQUALITIES, 
§17. Caratheodory's Theorem, 153, 
§18. Extreme Points and Faces of Convex Sets, 162, 
§19. Polyhedral Convex Sets and Functions, 170, 
§20. Some Applications of Polyhedral Convexity, 179, 
§21. Helly's Theorem and Systems of Inequalities, 185, 
§22. Linear Inequalities, 198, 
PART V: DIFFERENTIAL THEORY, 
§23. Directional Derivatives and Subgradients, 213, 
§24. Differential Continuity and Monotonicity, 227, 
§25. Differentiability of Convex Functions, 241, 
§26. The Legendre Transformation, 251, 
PART VI: CONSTRAINED EXTREMUM PROBLEMS, 
§27. The Minimum of a Convex Function, 263, 
§28. Ordinary Convex Programs and Lagrange Multipliers, 273, 
§29. Bifunctions and Generalized Convex Programs, 291, 
§30. Adjoint Bifunctions and Dual Programs, 307, 
§31. Fenchel's Duality Theorem, 327, 
§32. The Maximum of a Convex Function, 342, 
PART VII: SADDLE-FUNCTIONS AND MINIMAX THEORY, 
§33. Saddle-Functions, 349, 
§34. Closures and Equivalence Classes, 359, 
§35. Continuity and Differentiability of Saddle-functions, 370, 
§36. Minimax Problems, 379, 
§37. Conjugate Saddle-functions and Minimax Theorems, 388, 
PART VIII: CONVEX ALGEBRA, 
§38. The Algebra of Bifunctions, 401, 
§39. Convex Processes, 413, 
Comments and References, 425, 
Bibliography, 433, 
Index, 447, 

CHAPTER 1

SECTION 1

Affine Sets

Throughout this book, R denotes the real number system, and Rn is the usual vector space of real n-tuples x = ([xi]1 ..., [xi]n). Everything takes place in Rn unless otherwise specified. The inner product of two vectors x and x* in Rn is expressed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The same symbol A is used to denote an m X n real matrix A and the corresponding linear transformation x -> Ax from Rn to Rn. The transpose matrix and the corresponding adjoint linear transformation from Rn to Rn are denoted by A*, so that one has the identity

= .

(In a symbol denoting a vector, * has no operational significance; all vectors are to be regarded as column vectors for purposes of matrix multiplication. Vector symbols involving * are used from time to time merely to bring out the familiar duality between vectors considered as points and vectors considered as the coefficient n-tuples of linear functions.) The end of a proof is signalled by [parallel].

If x and y are different points in Rn, the set of points of the form

(1 - λ)x + λy = x + λ(y - x), λ [member of] R,

is called the line through x and y. A subset M of Rn is called an affine set if (1 - λ)x + λy [member of] M for every x [member of] M, y [member of] M and A [member of] R. (Synonyms for "affine set" used by other authors are "affine manifold," "affine variety," "linear variety" or "flat.")

The empty set Ø and the space Rn itself are extreme examples of affine sets. Also covered by the definition is the case where M consists of a solitary point. In general, an affine set has to contain, along with any two different points, the entire line through those points. The intuitive picture is that of an endless uncurved structure, like a line or a plane in space.

The formal geometry of affine sets may be developed from the theorems of linear algebra about subspaces of Rn. The exact correspondence between affine sets and subspaces is described in the two theorems which follow.

Theorem 1.1. The subspaces of Rn are the affine sets which contain the origin.

Proof. Every subspace contains 0 and, being closed under addition and scalar multiplication, is in particular an affine set.

Conversely, suppose M is an affine set containing 0. For any x [member of] M and λ [member of] R, we have

λx = (1 - λ)0 + λx [member of] M,

so M is closed under scalar multiplication. Now, if x [member of] M and y [member of] M, we have

1/2 (x + y) = 1/2x + (1 - 1/2)y [member of] M,

and hence

x + y = 2(|1/2 + y)) [member of] M.

Thus M is also closed under addition and is a subspace. [parallel]|

For M [subset] Rn and a [member of] Rn, the translate of M by a is defined to be the set

M + a=-{x + a\x [member of] M}.

A translate of an affine set is another affine set, as is easily verified.

An affine set M is said to be parallel to an affine set L if M = L + a for some a. Evidently "M is parallel to L" is an equivalence relation on the collection of affine subsets of Rn. Note that this definition of parallelism is more restrictive than the everyday one, in that it does not include the idea of a line being parallel to a plane. One has to speak of a line which is parallel to another line within a given plane, and so forth.

Theorem 1.2. Each non-empty affine set M is parallel to a unique subspace L. This L is given by

L = M - M = {x - y\ x [member of] M, y [member of] M}.

Proof. Let us show first that M cannot be parallel to two different subspaces. Subspaces L1 and L2 parallel to M would be parallel to each other, so that L2 = L1 + a for some a. Since 0 [member of] L2, we would then have - a [member of L1, and hence a [member of] L1. But then L1 [contains] + a = L2. By a similar argument L2L1, so L1 = L2. This establishes the uniqueness. Now observe that, for any y [member of] M, M - y = M + (-y) is a translate of M containing 0. By Theorem 1.1 and what we have just proved, this affine set must be the unique subspace L parallel to M. Since L = M - y no matter which y [member of] M is chosen, we actually have L = M - M. [parallel]|

The dimension of a non-empty affine set is defined as the dimension of the subspace parallel to it. (The dimension of Ø is -1 by convention.) Naturally,...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 1997
• Sprache: Englisch
• ISBN 10: 0691015864
• ISBN 13: 9780691015866
• Einband: Taschenbuch
• Anzahl der Seiten: 470
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar