Fundamentals of Convex Analysis

by Lemarechal, Claude; Chenciner, A.; Chern, S. S.; Eckmann, B.; Harpe, P. De LA; Hirzebruch, F.

• ISBN13: 9783540422051
• ISBN10: 3540422056
• Format: Hardcover
• Copyright: 2001-09-01
• Publisher: Springer Verlag

Summary

This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306), which presented an introduction to the basic concepts in convex analysis and a study of convex minimization problems. The "backbone" of both volumes was extracted, some material deleted that was deemed too advanced for an introduction, or too closely related to numerical algorithms. Some exercises were included and finally the index has been considerably enriched. The main motivation of the authors was to "light the entrance" of the monument Convex Analysis. This book is not a reference book to be kept on the shelf by experts who already know the building and can find their way through it; it is far more a book for the purpose of learning and teaching.

Table of Contents

Preface	v
Introduction: Notation, Elementary Results	1	(18)
Some Facts About Lower and Upper Bounds	1	(4)
The Set of Extended Real Numbers	5	(1)
Linear and Bilinear Algebra	6	(3)
Differentiation in a Euclidean Space	9	(3)
Set-Valued Analysis	12	(2)
Recalls on Convex Functions of the Real Variable	14	(5)
Exercises	16	(3)
Convex Sets	19	(54)
Generalities	19	(14)
Definition and First Examples	19	(3)
Convexity-Preserving Operations on Sets	22	(4)
Convex Combinations and Convex Hulls	26	(5)
Closed Convex Sets and Hulls	31	(2)
Convex Sets Attached to a Convex Set	33	(13)
The Relative Interior	33	(6)
The Asymptotic Cone	39	(2)
Extreme Points	41	(2)
Exposed Faces	43	(3)
Projection onto Closed Convex Sets	46	(5)
The Projection Operator	46	(3)
Projection onto a Closed Convex Cone	49	(2)
Separation and Applications	51	(11)
Separation Between Convex Sets	51	(3)
First Consequences of the Separation Properties	54	(1)
Existence of Supporting Hyperplanes	54	(1)
Outer Description of Closed Convex Sets	55	(2)
Proof of Minkowski's Theorem	57	(1)
Bipolar of a Convex Cone	57	(1)
The Lemma of Minkowski-Farkas	58	(4)
Conical Approximations of Convex Sets	62	(11)
Convenient Definitions of Tangent Cones	62	(3)
The Tangent and Normal Cones to a Convex Set	65	(3)
Some Properties of Tangent and Normal Cones	68	(2)
Exercises	70	(3)
Convex Functions	73	(48)
Basic Definitions and Examples	73	(14)
The Definitions of a Convex Function	73	(3)
Special Convex Functions: Affinity and Closedness	76	(1)
Linear and Affine Functions	77	(1)
Closed Convex Functions	78	(2)
Outer Construction of Closed Convex Functions	80	(2)
First Examples	82	(5)
Functional Operations Preserving Convexity	87	(15)
Operations Preserving Closedness	87	(2)
Dilations and Perspectives of a Function	89	(3)
Infimal Convolution	92	(4)
Image of a Function Under a Linear Mapping	96	(2)
Convex Hull and Closed Convex Hull of a Function	98	(4)
Local and Global Behaviour of a Convex Function	102	(8)
Continuity Properties	102	(4)
Behaviour at Infinity	106	(4)
First-and Second-Order Differentiation	110	(11)
Differentiable Convex Functions	110	(4)
Nondifferentiable Convex Functions	114	(1)
Second-Order Differentiation	115	(2)
Exercises	117	(4)
Sublinearity and Support Functions	121	(42)
Sublinear Functions	123	(11)
Definitions and First Properties	123	(4)
Some Examples	127	(4)
The Convex Cone of All Closed Sublinear Functions	131	(3)
The Support Function of a Nonempty Set	134	(9)
Definitions, Interpretations	134	(2)
Basic Properties	136	(4)
Examples	140	(3)
Correspondence Between Convex Sets and Sublinear Functions	143	(20)
The Fundamental Correspondence	143	(3)
Example: Norms and Their Duals, Polarity	146	(5)
Calculus with Support Functions	151	(7)
Example: Support Functions of Closed Convex Polyhedra	158	(3)
Exercises	161	(2)
Subdifferentials of Finite Convex Functions	163	(46)
The Subdifferential: Definitions and Interpretations	164	(9)
First Definition: Directional Derivatives	164	(3)
Second Definition: Minorization by Affine Functions	167	(2)
Geometric Constructions and Interpretations	169	(4)
Local Properties of the Subdifferential	173	(7)
First-Order Developments	173	(4)
Minimality Conditions	177	(1)
Mean-Value Theorems	178	(2)
First Examples	180	(3)
Calculus Rules with Subdifferentials	183	(11)
Positive Combinations of Functions	183	(1)
Pre-Composition with an Affine Mapping	184	(1)
Post-Composition with an Increasing Convex Function of Several Variables	185	(3)
Supremum of Convex Functions	188	(3)
Image of a Function Under a Linear Mapping	191	(3)
Further Examples	194	(5)
Largest Eigenvalue of a Symmetric Matrix	194	(2)
Nested Optimization	196	(2)
Best Approximation of a Continuous Function on a Compact Interval	198	(1)
The Subdifferential as a Multifunction	199	(10)
Monotonicity Properties of the Subdifferential	199	(2)
Continuity Properties of the Subdifferential	201	(3)
Subdifferentials and Limits of Subgradients	204	(1)
Exercises	205	(4)
Conjugacy in Convex Analysis	209	(36)
The Convex Conjugate of a Function	211	(11)
Definition and First Examples	211	(3)
Interpretations	214	(2)
First Properties	216	(1)
Elementary Calculus Rules	216	(2)
The Biconjugate of a Function	218	(1)
Conjugacy and Coercivity	219	(1)
Subdifferentials of Extended-Valued Functions	220	(2)
Calculus Rules on the Conjugacy Operation	222	(11)
Image of a Function Under a Linear Mapping	222	(2)
Pre-Composition with an Affine Mapping	224	(3)
Sum of Two Functions	227	(2)
Infima and Suprema	229	(2)
Post-Composition with an Increasing Convex Function	231	(2)
Various Examples	233	(4)
The Cramer Transformation	234	(1)
The Conjugate of Convex Partially Quadratic Functions	234	(1)
Polyhedral Functions	235	(2)
Differentiability of a Conjugate Function	237	(8)
First-Order Differentiability	238	(2)
Lipschitz Continuity of the Gradient Mapping	240	(1)
Exercises	241	(4)
Bibliographical Comments	245	(4)
The Founding Fathers of the Discipline	249	(2)
References	251	(2)
Index	253