Mathematics of optimization: smooth and nonsmooth case
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier
2004
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| Edition: | 1. ed. |
| Subjects: | |
| Online Access: | Table of contents Publisher description Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XV, 598 S. graph. Darst. 24 cm |
| ISBN: | 0444505504 |
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| adam_text | Titel: Mathematics of optimization
Autor: Giorgi, Giorgio
Jahr: 2004
Mathematics of Optimization:
Smooth and Nonsmooth Case
G. Giorgi
Faculty ofEconomics
University ofPavia
Pavia, Italy
A. Guerraggio
Faculty ofEconomics
Insubria University
Varese, Italy
J. Thierfelder
Institute of Mathematics
Technical University Ilmenau
Ilmenau, Germany
Amsterdam - Boston - Heidelberg - London - New York - Oxford
Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo
CONTENTS
PREFACE vii
CHAPTER I. INTRODUCTION 1
1.1 Optimization Problems 1
1.2 Basic Mathematical Preliminaries and Notations 11
References to Chapter I 20
CHAPTER II. CONVEX SETS, CONVEX AND GENERALIZED
CONVEX FUNCTIONS 23
2.1 Convex Sets and Their Main Properties 23
2.2 Separation Theorems 34
2.3 Some Particular Convex Sets. Convex Cones 47
2.4 Theorems of the Alternative for Linear Systems 59
2.5 Convex Functions 70
2.6 Directional Derivatives and Subgradients of Convex
Functions 94
2.7 Conjugate Functions 102
2.8 Extrema of Convex Functions 109
2.9 Systems of Convex Functions and Nonlinear Theorems
of the Alternative 111
2.10 Generalized Convex Functions 118
2.11 Relationships Between the Various Classes of Generalized
Convex Functions. Properties in Optimization Problems 140
2.12 Generalized Monotonicity and Generalized Convexity 149
2.13 Comparison Between Convex and Generalized Convex
Functions 153
2.14 Generalized Convexity at a Point 156
XIV
2.15 Convexity, Pseudoconvexity and Quasiconvexity of
Composite Functions 161
2.16 Convexity, Pseudoconvexity and Quasiconvexity of
Quadratic Functions 168
2.17 Other Types of Generalized Convex Functions 173
References to Chapter II 186
CHAPTER III. SMOOTH OPTIMIZATION PROBLEMS.
SADDLE POINT CONDITIONS 201
3.1 Introduction 201
3.2 Unconstrained Extremum Problems and Extremum
Problems with a Set Constraint 202
3.3 Equality Constrained Extremum Problems 209
3.4 Local Cone Approximations of Sets 221
3.5 Necessary Optimality Conditions for Problem (P)
where the Optimal Point is Interior to X 248
3.6 Necessary Optimality Conditions for Problems (Pe )
and (P); The Case of a Set Constraint 258
3.7 Again on Constraint Qualifications 270
3.8 Necessary Optimality Conditions for (Pi) 277
3.9 Sufficient First-Order Optimality Conditions for
(P) and (Pi) 287
3.10 Second-Order Optimality Conditions 303
3.11 Linearization Properties of a Nonlinear Programming
Problem 317
3.12 Some Specific Cases 321
3.13 Extensions to Topological Spaces 328
3.14 Optimality Criteria of the Saddle Point Type 332
References to Chapter III 342
CHAPTER IV. NONSMOOTH OPTIMIZATION PROBLEMS 359
4.1 Preliminary Remarks 359
4.2 Differentiability 360
XV
4.3 Directional Derivatives and Subdifferentials for
Convex Functions 371
4.4 Generalized Directional Derivatives 379
4.5 Generalized Gradient Mappings 392
4.6 Abstract Cone Approximations of Sets and
Relating Differentiability Notions 396
4.7 Special iC -Directional Derivative 414
4.8 Generalized Optimality Conditions 433
References to Chapter IV 452
CHAPTERV. DUALITY 459
5.1 Preliminary Remarks 459
5.2 Duality in Linear Optimization 460
5.3 Duality in Convex Optimization (Wolfe Duality) 463
5.4 Lagrange Duality 469
5.5 Perturbed Optimization Problems 486
References to Chapter V 498
CHAPTER VI. VECTOR OPTIMIZATION 503
6.1 Vector Optimization Problems 503
6.2 Conical Preference Orders 505
6.3 Optimality (or Efficiency) Notions 509
6.4 Proper Efficiency 512
6.5 Theorems of Existence 533
6.6 Optimality Conditions 546
6.7 Scalarization 554
6.8 The Nondifferentiable Case 567
References to Chapter VI 581
SUBJECT INDEX 593
|
| adam_txt |
Titel: Mathematics of optimization
Autor: Giorgi, Giorgio
Jahr: 2004
Mathematics of Optimization:
Smooth and Nonsmooth Case
G. Giorgi
Faculty ofEconomics
University ofPavia
Pavia, Italy
A. Guerraggio
Faculty ofEconomics
Insubria University
Varese, Italy
J. Thierfelder
Institute of Mathematics
Technical University Ilmenau
Ilmenau, Germany
Amsterdam - Boston - Heidelberg - London - New York - Oxford
Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo
CONTENTS
PREFACE vii
CHAPTER I. INTRODUCTION 1
1.1 Optimization Problems 1
1.2 Basic Mathematical Preliminaries and Notations 11
References to Chapter I 20
CHAPTER II. CONVEX SETS, CONVEX AND GENERALIZED
CONVEX FUNCTIONS 23
2.1 Convex Sets and Their Main Properties 23
2.2 Separation Theorems 34
2.3 Some Particular Convex Sets. Convex Cones 47
2.4 Theorems of the Alternative for Linear Systems 59
2.5 Convex Functions 70
2.6 Directional Derivatives and Subgradients of Convex
Functions 94
2.7 Conjugate Functions 102
2.8 Extrema of Convex Functions 109
2.9 Systems of Convex Functions and Nonlinear Theorems
of the Alternative 111
2.10 Generalized Convex Functions 118
2.11 Relationships Between the Various Classes of Generalized
Convex Functions. Properties in Optimization Problems 140
2.12 Generalized Monotonicity and Generalized Convexity 149
2.13 Comparison Between Convex and Generalized Convex
Functions 153
2.14 Generalized Convexity at a Point 156
XIV
2.15 Convexity, Pseudoconvexity and Quasiconvexity of
Composite Functions 161
2.16 Convexity, Pseudoconvexity and Quasiconvexity of
Quadratic Functions 168
2.17 Other Types of Generalized Convex Functions 173
References to Chapter II 186
CHAPTER III. SMOOTH OPTIMIZATION PROBLEMS.
SADDLE POINT CONDITIONS 201
3.1 Introduction 201
3.2 Unconstrained Extremum Problems and Extremum
Problems with a Set Constraint 202
3.3 Equality Constrained Extremum Problems 209
3.4 Local Cone Approximations of Sets 221
3.5 Necessary Optimality Conditions for Problem (P)
where the Optimal Point is Interior to X 248
3.6 Necessary Optimality Conditions for Problems (Pe )
and (P); The Case of a Set Constraint 258
3.7 Again on Constraint Qualifications 270
3.8 Necessary Optimality Conditions for (Pi) 277
3.9 Sufficient First-Order Optimality Conditions for
(P) and (Pi) 287
3.10 Second-Order Optimality Conditions 303
3.11 Linearization Properties of a Nonlinear Programming
Problem 317
3.12 Some Specific Cases 321
3.13 Extensions to Topological Spaces 328
3.14 Optimality Criteria of the Saddle Point Type 332
References to Chapter III 342
CHAPTER IV. NONSMOOTH OPTIMIZATION PROBLEMS 359
4.1 Preliminary Remarks 359
4.2 Differentiability 360
XV
4.3 Directional Derivatives and Subdifferentials for
Convex Functions 371
4.4 Generalized Directional Derivatives 379
4.5 Generalized Gradient Mappings 392
4.6 Abstract Cone Approximations of Sets and
Relating Differentiability Notions 396
4.7 Special iC'-Directional Derivative 414
4.8 Generalized Optimality Conditions 433
References to Chapter IV 452
CHAPTERV. DUALITY 459
5.1 Preliminary Remarks 459
5.2 Duality in Linear Optimization 460
5.3 Duality in Convex Optimization (Wolfe Duality) 463
5.4 Lagrange Duality 469
5.5 Perturbed Optimization Problems 486
References to Chapter V 498
CHAPTER VI. VECTOR OPTIMIZATION 503
6.1 Vector Optimization Problems 503
6.2 Conical Preference Orders 505
6.3 Optimality (or Efficiency) Notions 509
6.4 Proper Efficiency 512
6.5 Theorems of Existence 533
6.6 Optimality Conditions 546
6.7 Scalarization 554
6.8 The Nondifferentiable Case 567
References to Chapter VI 581
SUBJECT INDEX 593 |
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| spelling | Giorgi, Giorgio Verfasser aut Mathematics of optimization smooth and nonsmooth case G. Giorgi, A. Guerraggio, J. Thierfelder 1. ed. Amsterdam [u.a.] Elsevier 2004 XV, 598 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Otimização matemática larpcal Programação matemática larpcal Mathematical optimization Nonlinear programming Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf Nichtkonvexe Optimierung (DE-588)4309215-9 gnd rswk-swf Nichtlineare Optimierung (DE-588)4128192-5 s Nichtkonvexe Optimierung (DE-588)4309215-9 s DE-604 Guerraggio, Angelo 1948- Verfasser (DE-588)136405614 aut Thierfelder, Jörg 1955- Verfasser (DE-588)121071987 aut http://www.loc.gov/catdir/toc/els051/2004040362.html Table of contents http://www.loc.gov/catdir/description/els051/2004040362.html Publisher description HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016163111&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Giorgi, Giorgio Guerraggio, Angelo 1948- Thierfelder, Jörg 1955- Mathematics of optimization smooth and nonsmooth case Otimização matemática larpcal Programação matemática larpcal Mathematical optimization Nonlinear programming Nichtlineare Optimierung (DE-588)4128192-5 gnd Nichtkonvexe Optimierung (DE-588)4309215-9 gnd |
| subject_GND | (DE-588)4128192-5 (DE-588)4309215-9 |
| title | Mathematics of optimization smooth and nonsmooth case |
| title_auth | Mathematics of optimization smooth and nonsmooth case |
| title_exact_search | Mathematics of optimization smooth and nonsmooth case |
| title_exact_search_txtP | Mathematics of optimization smooth and nonsmooth case |
| title_full | Mathematics of optimization smooth and nonsmooth case G. Giorgi, A. Guerraggio, J. Thierfelder |
| title_fullStr | Mathematics of optimization smooth and nonsmooth case G. Giorgi, A. Guerraggio, J. Thierfelder |
| title_full_unstemmed | Mathematics of optimization smooth and nonsmooth case G. Giorgi, A. Guerraggio, J. Thierfelder |
| title_short | Mathematics of optimization |
| title_sort | mathematics of optimization smooth and nonsmooth case |
| title_sub | smooth and nonsmooth case |
| topic | Otimização matemática larpcal Programação matemática larpcal Mathematical optimization Nonlinear programming Nichtlineare Optimierung (DE-588)4128192-5 gnd Nichtkonvexe Optimierung (DE-588)4309215-9 gnd |
| topic_facet | Otimização matemática Programação matemática Mathematical optimization Nonlinear programming Nichtlineare Optimierung Nichtkonvexe Optimierung |
| url | http://www.loc.gov/catdir/toc/els051/2004040362.html http://www.loc.gov/catdir/description/els051/2004040362.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016163111&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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