156,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book provides easy access to the basic principles and methods for solving constrained and unconstrained convex optimization problems. Included are sections that cover: basic methods for solving constrained and unconstrained optimization problems with differentiable objective functions; convex sets and their properties; convex functions and their properties and generalizations; and basic principles of sub-differential calculus and convex programming problems. Convex Optimization provides detailed proofs for most of the results presented in the book and also includes many figures and exercises for a better understanding of the material. Exercises are given at the end of each chapter, with solutions and hints to selected exercises given at the end of the book. Undergraduate and graduate students, researchers in different disciplines, as well as practitioners will all benefit from this accessible approach to convex optimization methods.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 386
Veröffentlichungsjahr: 2021
Ähnliche
Table of Contents
Cover
Title Page
Copyright
Notations
Introduction
1 Optimization Problems with Differentiable Objective Functions
1.1. Basic concepts
1.2. Optimization problems with objective functions of one variable
1.3. Optimization problems with objective functions of several variables
1.4. Constrained optimization problems
1.5. Exercises
2 Convex Sets
2.1. Convex sets: basic definitions
2.2. Combinations of points and hulls of sets
2.3. Topological properties of convex sets
2.4. Theorems on separation planes and their applications
2.5. Systems of linear inequalities and equations
2.6. Extreme points of a convex set
2.7. Exercises
3 Convex Functions
3.1. Convex functions: basic definitions
3.2. Operations in the class of convex functions
3.3. Criteria of convexity of differentiable functions
3.4. Continuity and differentiability of convex functions
3.5. Convex minimization problem
3.6. Theorem on boundedness of Lebesgue set of a strongly convex function
3.7. Conjugate function
3.8. Basic properties of conjugate functions
3.9. Exercises
4 Generalizations of Convex Functions
4.1. Quasi-convex functions
4.2. Pseudo-convex functions
4.3. Logarithmically convex functions
4.4. Convexity in relation to order
4.5. Exercises
5 Sub-gradient and Sub-differential of Finite Convex Function
5.1. Concepts of sub-gradient and sub-differential
5.2. Properties of sub-differential of convex function
5.3. Sub-differential mapping
5.4. Calculus rules for sub-differentials
5.5. Systems of convex and linear inequalities
5.6. Exercises
6 Constrained Optimization Problems
6.1. Differential conditions of optimality
6.2. Sub-differential conditions of optimality
6.3. Exercises
6.4. Constrained optimization problems
6.5. Exercises
6.6. Dual problems in convex optimization
6.7. Exercises
Solutions, Answers and Hints
References
Index
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1.
Example 1.5
Figure 1.2.
Example 1.6
Chapter 2
Figure 2.1.
Convex set X
1
. Non-convex set X
2
Figure 2.2.
X
1
is a cone
.
X
2
is a convex cone
Figure 2.3.
Conjugate cones
Figure 2.4.
Affine set and linear subspace
Figure 2.5.
a) Convex hull. b) Conic hull
Figure 2.6.
a) Convex polyhedron. b) Polyhedral cone
Figure 2.7.
Unbounded closed convex set
Figure 2.8.
Projection of a point onto a set
Figure 2.9. Sets X1 and X2 are: a) properly separated; b) strongly separated; c)...
Figure 2.10.
a), c) Properly supporting hyperplanes; b) supporting hyperplane
Chapter 3
Figure 3.1.
Convex function
Figure 3.2.
Epigraph of convex function
Figure 3.3.
Epigraph of nonconvex function
Figure 3.4.
Separating linear function
Chapter 5
Figure 5.1.
Example 5.1
Guide
Cover
Table of Contents
Title Page
Copyright
Notations
Introduction
Begin Reading
Solutions, Answers and Hints
References
Index
End User License Agreement
Pages
v
iii
iv
ix
x
xi
xii
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
235
236
237
238
239
241
242
243
244
245
246
247
Series Editor Nikolaos Limnios
Convex Optimization
Introductory Course
Mikhail Moklyachuk
First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
27-37 St George’s Road
London SW19 4EU
UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2020
The rights of Mikhail Moklyachuk to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2020943973
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-683-8
Introduction
Convex analysis and optimization have an increasing impact on many areas of mathematics and applications including control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, statistics, and economics and finance. There are several fundamental books devoted to different aspects of convex analysis and optimization. Among them we can mention Optima and Equilibria: An Introduction to Nonlinear Analysis by Aubin (1998), Convex Analysis by Rockafellar (1970), Convex Analysis and Minimization Algorithms (in two volumes) by Hiriart-Urruty and Lemaréchal (1993) and its abridged version (2002), Convex Analysis and Nonlinear Optimization by Borwein and Lewis (2000), Convex Optimization by Boyd and Vandenberghe (2004), Convex Analysis and Optimization by Bertsekas et al. (2003), Convex Analysis and Extremal Problems by Pshenichnyj (1980), A Course in Optimization Methods by Sukharev et al. (2005), Convex Analysis: An Introductory Text by Van Tiel (1984), as well as other books listed in the bibliography (see Alekseev et al. (1984, 1987); Alekseev and Timokhov (1991); Clarke (1983); Hiriart-Urruty (1998); Ioffe and Tikhomirov (1979) and Nesterov (2004)).
This book provides easy access to the basic principles and methods for solving constrained and unconstrained convex optimization problems. Structurally, the book has been divided into the following parts: basic methods for solving constrained and unconstrained optimization problems with differentiable objective functions, convex sets and their properties, convex functions, their properties and generalizations, subgradients and subdifferentials, and basic principles and methods for solving constrained and unconstrained convex optimization problems. The first part of the book describes methods for finding the extremum of functions of one and many variables. Problems of constrained and unconstrained optimization (problems with restrictions of inequality and inequality types) are investigated. The necessary and sufficient conditions of the extremum, the Lagrange method, are described.
The second part is the most voluminous in terms of the amount of material presented. Properties of convex sets and convex functions directly related to extreme problems are described. The basic principles of subdifferential calculus are outlined. The third part is devoted to the problems of mathematical programming. The problem of convex programming is considered in detail. The Kuhn–Tucker theorem is proved and the economic interpretations of the Kuhn–Tucker vector are described.
We give detailed proofs for most of the results presented in the book and also include many figures and exercises for better understanding of the material. Finally, we present solutions and hints to selected exercises at the end of the book. Exercises are given at the end of each chapter while figures and examples are provided throughout the whole text. The list of references contains texts which are closely related to the topics considered in the book and may be helpful to the reader for advanced studies of convex analysis, its applications and further extensions. Since only elementary knowledge in linear algebra and basic calculus is required, this book can be used as a textbook for both undergraduate and graduate level courses in convex optimization and its applications. In fact, the author has used these lecture notes for teaching such classes at Kyiv National University. We hope that the book will make convex optimization methods more accessible to large groups of undergraduate and graduate students, researchers in different disciplines and practitioners. The idea was to prepare materials of lectures in accordance with the suggestion made by Einstein: “Everything should be made as simple as possible, but not simpler.”
