Verfügbarkeit: Fundamentals of convex analysis

Fundamentals of convex analysis:

Gespeichert in:

Bibliographische Detailangaben
Hauptverfasser:	Hiriart-Urruty, Jean-Baptiste 1949- (VerfasserIn), Lemaréchal, Claude 1944- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Berlin [u.a.] Springer 2004
Ausgabe:	Corr. 2. print.
Schriftenreihe:	Grundlehren Text editions
Schlagworte:	Konvexe Analysis Minimierung
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	X, 259 S. graph. Darst.
ISBN:	3540422056 9783540422051

Internformat

MARC


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245	1	0	\|a Fundamentals of convex analysis \|c Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal
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Datensatz im Suchindex

_version_	1804135033326272512
adam_text	CONTENTS PREFACE ****************************************************** V 0. INTRODUCTION: NOTATION, ELEMENTARY RESULTS ***************** 1 1 SOME FACTS ABOUT LOWER AND UPPER BOUNDS . . . . . . . . . . . . . . . . . . . 1 2 THE SET OF EXTENDED REAL NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 LINEAR AND BILINEAR ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 DIFFERENTIATION IN A EUCLIDEAN SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 SET-VALUED ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 RECALLS ON CONVEX FUNCTIONS OF THE REAL VARIABLE . . . . . . . . . . . . . . . 14 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 A. CONVEX SETS ********************************************** 19 1 GENERALITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.1 DEFINITION AND FIRST EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2 CONVEXITY-PRESERVING OPERATIONS ON SETS . . . . . . . . . . . . . . . . 22 1.3 CONVEX COMBINATIONS AND CONVEX HULLS. . . . . . . . . . . . . . . . . 26 1.4 CLOSED CONVEX SETS AND HULLS . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 CONVEX SETS ATTACHED TO A CONVEX SET . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 THE RELATIVE INTERIOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 THE ASYMPTOTIC CONE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 EXTREME POINTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4 EXPOSED FACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3 PROJECTION ONTO CLOSED CONVEX SETS . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1 THE PROJECTION OPERATOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 PROJECTION ONTO A CLOSED CONVEX CONE . . . . . . . . . . . . . . . . . . 49 4 SEPARATION AND APPLICATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 SEPARATION BETWEEN CONVEX SETS . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 FIRST CONSEQUENCES OF THE SEPARATION PROPERTIES . . . . . . . . . . 54 * EXISTENCE OF SUPPORTING HYPERPLANES . . . . . . . . . . . . . . . . . . 54 * OUTER DESCRIPTION OF CLOSED CONVEX SETS . . . . . . . . . . . . . . 55 * PROOF OF MINKOWSKIS THEOREM . . . . . . . . . . . . . . . . . . . . . . . 57 BIPOLAR OF A CONVEX CONE . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3 THE LEMMA OF MINKOWSKI-FARKAS . . . . . . . . . . . . . . . . . . . . . . 58 5 CONICAL APPROXIMATIONS OF CONVEX SETS . . . . . . . . . . . . . . . . . . . . . . . 62 VIII CONTENTS 5.1 CONVENIENT DEFINITIONS OF TANGENT CONES . . . . . . . . . . . . . . . . 62 5.2 THE TANGENT AND NORMAL CONES TO A CONVEX SET . . . . . . . . . . 65 5.3 SOME PROPERTIES OF TANGENT AND NORMAL CONES . . . . . . . . . . . 67 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 B. CONVEX FUNCTIONS ******************************************* 73 1 BASIC DEFINITIONS AND EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 1.1 THE DEFINITIONS OF A CONVEX FUNCTION . . . . . . . . . . . . . . . . . . . 73 1.2 SPECIAL CONVEX FUNCTIONS: AFFINITY AND CLOSEDNESS. . . . . . . . 76 * LINEAR AND AFFINE FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 77 * CLOSED CONVEX FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 * OUTER CONSTRUCTION OF CLOSED CONVEX FUNCTIONS . . . . . . . . . 80 1.3 FIRST EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2 FUNCTIONAL OPERATIONS PRESERVING CONVEXITY . . . . . . . . . . . . . . . . . . . . 87 2.1 OPERATIONS PRESERVING CLOSEDNESS . . . . . . . . . . . . . . . . . . . . . . 87 2.2 DILATIONS AND PERSPECTIVES OF A FUNCTION . . . . . . . . . . . . . . . . . 89 2.3 INFIMAL CONVOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.4 IMAGE OF A FUNCTION UNDER A LINEAR MAPPING . . . . . . . . . . . . 96 2.5 CONVEX HULL AND CLOSED CONVEX HULL OF A FUNCTION . . . . . . . 98 3 LOCAL AND GLOBAL BEHAVIOUR OF A CONVEX FUNCTION . . . . . . . . . . . . . . . 102 3.1 CONTINUITY PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.2 BEHAVIOUR AT INFINITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4 FIRST- AND SECOND-ORDER DIFFERENTIATION . . . . . . . . . . . . . . . . . . . . . . . . 110 4.1 DIFFERENTIABLE CONVEX FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . 110 4.2 NONDIFFERENTIABLE CONVEX FUNCTIONS . . . . . . . . . . . . . . . . . . . . 113 4.3 SECOND-ORDER DIFFERENTIATION . . . . . . . . . . . . . . . . . . . . . . . . . . 114 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 C. SUBLINEARITY AND SUPPORT FUNCTIONS ************************** 121 1 SUBLINEAR FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 1.1 DEFINITIONS AND FIRST PROPERTIES. . . . . . . . . . . . . . . . . . . . . . . . . 123 1.2 SOME EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 1.3 THE CONVEX CONE OF ALL CLOSED SUBLINEAR FUNCTIONS . . . . . . 131 2 THE SUPPORT FUNCTION OF A NONEMPTY SET . . . . . . . . . . . . . . . . . . . . . . 134 2.1 DEFINITIONS, INTERPRETATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.2 BASIC PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.3 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3 CORRESPONDENCE BETWEEN CONVEX SETS AND SUBLINEAR FUNCTIONS . . . . 143 3.1 THE FUNDAMENTAL CORRESPONDENCE . . . . . . . . . . . . . . . . . . . . . . 143 3.2 EXAMPLE: NORMS AND THEIR DUALS, POLARITY . . . . . . . . . . . . . . . 146 3.3 CALCULUS WITH SUPPORT FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . 151 3.4 EXAMPLE: SUPPORT FUNCTIONS OF CLOSED CONVEX POLYHEDRA . . 158 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 CONTENTS IX D. SUBDIFFERENTIALS OF FINITE CONVEX FUNCTIONS ***************** 163 1 THE SUBDIFFERENTIAL: DEFINITIONS AND INTERPRETATIONS . . . . . . . . . . . . . . 164 1.1 FIRST DEFINITION: DIRECTIONAL DERIVATIVES . . . . . . . . . . . . . . . . . 164 1.2 SECOND DEFINITION: MINORIZATION BY AFFINE FUNCTIONS . . . . . . 167 1.3 GEOMETRIC CONSTRUCTIONS AND INTERPRETATIONS . . . . . . . . . . . . . 169 2 LOCAL PROPERTIES OF THE SUBDIFFERENTIAL . . . . . . . . . . . . . . . . . . . . . . . . . 173 2.1 FIRST-ORDER DEVELOPMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 2.2 MINIMALITY CONDITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 2.3 MEAN-VALUE THEOREMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3 FIRST EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4 CALCULUS RULES WITH SUBDIFFERENTIALS . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.1 POSITIVE COMBINATIONS OF FUNCTIONS . . . . . . . . . . . . . . . . . . . . . 183 4.2 PRE-COMPOSITION WITH AN AFFINE MAPPING . . . . . . . . . . . . . . . . 184 4.3 POST-COMPOSITION WITH AN INCREASING CONVEX FUNCTION OF SEVERAL VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.4 SUPREMUM OF CONVEX FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . 187 4.5 IMAGE OF A FUNCTION UNDER A LINEAR MAPPING . . . . . . . . . . . . 191 5 FURTHER EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5.1 LARGEST EIGENVALUE OF A SYMMETRIC MATRIX . . . . . . . . . . . . . . . 193 5.2 NESTED OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.3 BEST APPROXIMATION OF A CONTINUOUS FUNCTION ON A COM- PACT INTERVAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6 THE SUBDIFFERENTIAL AS A MULTIFUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.1 MONOTONICITY PROPERTIES OF THE SUBDIFFERENTIAL . . . . . . . . . . . . 198 6.2 CONTINUITY PROPERTIES OF THE SUBDIFFERENTIAL . . . . . . . . . . . . . . 200 6.3 SUBDIFFERENTIALS AND LIMITS OF SUBGRADIENTS . . . . . . . . . . . . . . 203 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 E. CONJUGACY IN CONVEX ANALYSIS ******************************* 209 1 THE CONVEX CONJUGATE OF A FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 211 1.1 DEFINITION AND FIRST EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . 211 1.2 INTERPRETATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 1.3 FIRST PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 * ELEMENTARY CALCULUS RULES . . . . . . . . . . . . . . . . . . . . . . . . . . 216 * THE BICONJUGATE OF A FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . 218 * CONJUGACY AND COERCIVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 1.4 SUBDIFFERENTIALS OF EXTENDED-VALUED FUNCTIONS . . . . . . . . . . . 220 2 CALCULUS RULES ON THE CONJUGACY OPERATION . . . . . . . . . . . . . . . . . . . . 222 2.1 IMAGE OF A FUNCTION UNDER A LINEAR MAPPING . . . . . . . . . . . . 222 2.2 PRE-COMPOSITION WITH AN AFFINE MAPPING . . . . . . . . . . . . . . . . 224 2.3 SUM OF TWO FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 2.4 INFIMA AND SUPREMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 2.5 POST-COMPOSITION WITH AN INCREASING CONVEX FUNCTION . . . . . 231 3 VARIOUS EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3.1 THE CRAMER TRANSFORMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 X CONTENTS 3.2 THE CONJUGATE OF CONVEX PARTIALLY QUADRATIC FUNCTIONS . . . . 234 3.3 POLYHEDRAL FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 4 DIFFERENTIABILITY OF A CONJUGATE FUNCTION . . . . . . . . . . . . . . . . . . . . . . . 237 4.1 FIRST-ORDER DIFFERENTIABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 4.2 LIPSCHITZ CONTINUITY OF THE GRADIENT MAPPING . . . . . . . . . . . . 240 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 BIBLIOGRAPHICAL COMMENTS ************************************** 245 THE FOUNDING FATHERS OF THE DISCIPLINE ************************* 249 REFERENCES ************************************************* 249 INDEX ******************************************************** 256
adam_txt	CONTENTS PREFACE ****************************************************** V 0. INTRODUCTION: NOTATION, ELEMENTARY RESULTS ***************** 1 1 SOME FACTS ABOUT LOWER AND UPPER BOUNDS . . . . . . . . . . . . . . . . . . . 1 2 THE SET OF EXTENDED REAL NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 LINEAR AND BILINEAR ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 DIFFERENTIATION IN A EUCLIDEAN SPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 SET-VALUED ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 RECALLS ON CONVEX FUNCTIONS OF THE REAL VARIABLE . . . . . . . . . . . . . . . 14 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 A. CONVEX SETS ********************************************** 19 1 GENERALITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.1 DEFINITION AND FIRST EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2 CONVEXITY-PRESERVING OPERATIONS ON SETS . . . . . . . . . . . . . . . . 22 1.3 CONVEX COMBINATIONS AND CONVEX HULLS. . . . . . . . . . . . . . . . . 26 1.4 CLOSED CONVEX SETS AND HULLS . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 CONVEX SETS ATTACHED TO A CONVEX SET . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 THE RELATIVE INTERIOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 THE ASYMPTOTIC CONE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 EXTREME POINTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4 EXPOSED FACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3 PROJECTION ONTO CLOSED CONVEX SETS . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1 THE PROJECTION OPERATOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 PROJECTION ONTO A CLOSED CONVEX CONE . . . . . . . . . . . . . . . . . . 49 4 SEPARATION AND APPLICATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 SEPARATION BETWEEN CONVEX SETS . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 FIRST CONSEQUENCES OF THE SEPARATION PROPERTIES . . . . . . . . . . 54 * EXISTENCE OF SUPPORTING HYPERPLANES . . . . . . . . . . . . . . . . . . 54 * OUTER DESCRIPTION OF CLOSED CONVEX SETS . . . . . . . . . . . . . . 55 * PROOF OF MINKOWSKIS THEOREM . . . . . . . . . . . . . . . . . . . . . . . 57 BIPOLAR OF A CONVEX CONE . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3 THE LEMMA OF MINKOWSKI-FARKAS . . . . . . . . . . . . . . . . . . . . . . 58 5 CONICAL APPROXIMATIONS OF CONVEX SETS . . . . . . . . . . . . . . . . . . . . . . . 62 VIII CONTENTS 5.1 CONVENIENT DEFINITIONS OF TANGENT CONES . . . . . . . . . . . . . . . . 62 5.2 THE TANGENT AND NORMAL CONES TO A CONVEX SET . . . . . . . . . . 65 5.3 SOME PROPERTIES OF TANGENT AND NORMAL CONES . . . . . . . . . . . 67 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 B. CONVEX FUNCTIONS ******************************************* 73 1 BASIC DEFINITIONS AND EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 1.1 THE DEFINITIONS OF A CONVEX FUNCTION . . . . . . . . . . . . . . . . . . . 73 1.2 SPECIAL CONVEX FUNCTIONS: AFFINITY AND CLOSEDNESS. . . . . . . . 76 * LINEAR AND AFFINE FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 77 * CLOSED CONVEX FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 * OUTER CONSTRUCTION OF CLOSED CONVEX FUNCTIONS . . . . . . . . . 80 1.3 FIRST EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2 FUNCTIONAL OPERATIONS PRESERVING CONVEXITY . . . . . . . . . . . . . . . . . . . . 87 2.1 OPERATIONS PRESERVING CLOSEDNESS . . . . . . . . . . . . . . . . . . . . . . 87 2.2 DILATIONS AND PERSPECTIVES OF A FUNCTION . . . . . . . . . . . . . . . . . 89 2.3 INFIMAL CONVOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.4 IMAGE OF A FUNCTION UNDER A LINEAR MAPPING . . . . . . . . . . . . 96 2.5 CONVEX HULL AND CLOSED CONVEX HULL OF A FUNCTION . . . . . . . 98 3 LOCAL AND GLOBAL BEHAVIOUR OF A CONVEX FUNCTION . . . . . . . . . . . . . . . 102 3.1 CONTINUITY PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.2 BEHAVIOUR AT INFINITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4 FIRST- AND SECOND-ORDER DIFFERENTIATION . . . . . . . . . . . . . . . . . . . . . . . . 110 4.1 DIFFERENTIABLE CONVEX FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . 110 4.2 NONDIFFERENTIABLE CONVEX FUNCTIONS . . . . . . . . . . . . . . . . . . . . 113 4.3 SECOND-ORDER DIFFERENTIATION . . . . . . . . . . . . . . . . . . . . . . . . . . 114 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 C. SUBLINEARITY AND SUPPORT FUNCTIONS ************************** 121 1 SUBLINEAR FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 1.1 DEFINITIONS AND FIRST PROPERTIES. . . . . . . . . . . . . . . . . . . . . . . . . 123 1.2 SOME EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 1.3 THE CONVEX CONE OF ALL CLOSED SUBLINEAR FUNCTIONS . . . . . . 131 2 THE SUPPORT FUNCTION OF A NONEMPTY SET . . . . . . . . . . . . . . . . . . . . . . 134 2.1 DEFINITIONS, INTERPRETATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.2 BASIC PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.3 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3 CORRESPONDENCE BETWEEN CONVEX SETS AND SUBLINEAR FUNCTIONS . . . . 143 3.1 THE FUNDAMENTAL CORRESPONDENCE . . . . . . . . . . . . . . . . . . . . . . 143 3.2 EXAMPLE: NORMS AND THEIR DUALS, POLARITY . . . . . . . . . . . . . . . 146 3.3 CALCULUS WITH SUPPORT FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . 151 3.4 EXAMPLE: SUPPORT FUNCTIONS OF CLOSED CONVEX POLYHEDRA . . 158 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 CONTENTS IX D. SUBDIFFERENTIALS OF FINITE CONVEX FUNCTIONS ***************** 163 1 THE SUBDIFFERENTIAL: DEFINITIONS AND INTERPRETATIONS . . . . . . . . . . . . . . 164 1.1 FIRST DEFINITION: DIRECTIONAL DERIVATIVES . . . . . . . . . . . . . . . . . 164 1.2 SECOND DEFINITION: MINORIZATION BY AFFINE FUNCTIONS . . . . . . 167 1.3 GEOMETRIC CONSTRUCTIONS AND INTERPRETATIONS . . . . . . . . . . . . . 169 2 LOCAL PROPERTIES OF THE SUBDIFFERENTIAL . . . . . . . . . . . . . . . . . . . . . . . . . 173 2.1 FIRST-ORDER DEVELOPMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 2.2 MINIMALITY CONDITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 2.3 MEAN-VALUE THEOREMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3 FIRST EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4 CALCULUS RULES WITH SUBDIFFERENTIALS . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.1 POSITIVE COMBINATIONS OF FUNCTIONS . . . . . . . . . . . . . . . . . . . . . 183 4.2 PRE-COMPOSITION WITH AN AFFINE MAPPING . . . . . . . . . . . . . . . . 184 4.3 POST-COMPOSITION WITH AN INCREASING CONVEX FUNCTION OF SEVERAL VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.4 SUPREMUM OF CONVEX FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . 187 4.5 IMAGE OF A FUNCTION UNDER A LINEAR MAPPING . . . . . . . . . . . . 191 5 FURTHER EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5.1 LARGEST EIGENVALUE OF A SYMMETRIC MATRIX . . . . . . . . . . . . . . . 193 5.2 NESTED OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.3 BEST APPROXIMATION OF A CONTINUOUS FUNCTION ON A COM- PACT INTERVAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6 THE SUBDIFFERENTIAL AS A MULTIFUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.1 MONOTONICITY PROPERTIES OF THE SUBDIFFERENTIAL . . . . . . . . . . . . 198 6.2 CONTINUITY PROPERTIES OF THE SUBDIFFERENTIAL . . . . . . . . . . . . . . 200 6.3 SUBDIFFERENTIALS AND LIMITS OF SUBGRADIENTS . . . . . . . . . . . . . . 203 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 E. CONJUGACY IN CONVEX ANALYSIS ******************************* 209 1 THE CONVEX CONJUGATE OF A FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 211 1.1 DEFINITION AND FIRST EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . 211 1.2 INTERPRETATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 1.3 FIRST PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 * ELEMENTARY CALCULUS RULES . . . . . . . . . . . . . . . . . . . . . . . . . . 216 * THE BICONJUGATE OF A FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . 218 * CONJUGACY AND COERCIVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 1.4 SUBDIFFERENTIALS OF EXTENDED-VALUED FUNCTIONS . . . . . . . . . . . 220 2 CALCULUS RULES ON THE CONJUGACY OPERATION . . . . . . . . . . . . . . . . . . . . 222 2.1 IMAGE OF A FUNCTION UNDER A LINEAR MAPPING . . . . . . . . . . . . 222 2.2 PRE-COMPOSITION WITH AN AFFINE MAPPING . . . . . . . . . . . . . . . . 224 2.3 SUM OF TWO FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 2.4 INFIMA AND SUPREMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 2.5 POST-COMPOSITION WITH AN INCREASING CONVEX FUNCTION . . . . . 231 3 VARIOUS EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3.1 THE CRAMER TRANSFORMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 X CONTENTS 3.2 THE CONJUGATE OF CONVEX PARTIALLY QUADRATIC FUNCTIONS . . . . 234 3.3 POLYHEDRAL FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 4 DIFFERENTIABILITY OF A CONJUGATE FUNCTION . . . . . . . . . . . . . . . . . . . . . . . 237 4.1 FIRST-ORDER DIFFERENTIABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 4.2 LIPSCHITZ CONTINUITY OF THE GRADIENT MAPPING . . . . . . . . . . . . 240 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 BIBLIOGRAPHICAL COMMENTS ************************************** 245 THE FOUNDING FATHERS OF THE DISCIPLINE ************************* 249 REFERENCES ************************************************* 249 INDEX ******************************************************** 256
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spelling	Hiriart-Urruty, Jean-Baptiste 1949- Verfasser (DE-588)128857102 aut Fundamentals of convex analysis Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal Corr. 2. print. Berlin [u.a.] Springer 2004 X, 259 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren Text editions Konvexe Analysis Minimierung Minimierung (DE-588)4251074-0 gnd rswk-swf Konvexe Analysis (DE-588)4138566-4 gnd rswk-swf Konvexe Analysis (DE-588)4138566-4 s DE-604 Minimierung (DE-588)4251074-0 s Lemaréchal, Claude 1944- Verfasser (DE-588)128857137 aut SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014580774&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Hiriart-Urruty, Jean-Baptiste 1949- Lemaréchal, Claude 1944- Fundamentals of convex analysis Konvexe Analysis Minimierung Minimierung (DE-588)4251074-0 gnd Konvexe Analysis (DE-588)4138566-4 gnd
subject_GND	(DE-588)4251074-0 (DE-588)4138566-4
title	Fundamentals of convex analysis
title_auth	Fundamentals of convex analysis
title_exact_search	Fundamentals of convex analysis
title_exact_search_txtP	Fundamentals of convex analysis
title_full	Fundamentals of convex analysis Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal
title_fullStr	Fundamentals of convex analysis Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal
title_full_unstemmed	Fundamentals of convex analysis Jean-Baptiste Hiriart-Urruty ; Claude Lemaréchal
title_short	Fundamentals of convex analysis
title_sort	fundamentals of convex analysis
topic	Konvexe Analysis Minimierung Minimierung (DE-588)4251074-0 gnd Konvexe Analysis (DE-588)4138566-4 gnd
topic_facet	Konvexe Analysis Minimierung
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014580774&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT hiriarturrutyjeanbaptiste fundamentalsofconvexanalysis AT lemarechalclaude fundamentalsofconvexanalysis

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