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Duality for Nonconvex Approximation and Optimization 2006 ed. [Kõva köide]

  • Formaat: Hardback, 356 pages, kõrgus x laius: 235x155 mm, kaal: 1560 g, 17 Illustrations, black and white; XX, 356 p. 17 illus., 1 Hardback
  • Sari: CMS Books in Mathematics
  • Ilmumisaeg: 16-Feb-2006
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 0387283943
  • ISBN-13: 9780387283944
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  • Formaat: Hardback, 356 pages, kõrgus x laius: 235x155 mm, kaal: 1560 g, 17 Illustrations, black and white; XX, 356 p. 17 illus., 1 Hardback
  • Sari: CMS Books in Mathematics
  • Ilmumisaeg: 16-Feb-2006
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 0387283943
  • ISBN-13: 9780387283944
The theory of convex optimization has been constantly developing over the past 30 years.  Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems.  This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity.  This manuscript will be of great interest for experts in this and related fields.

Arvustused

From the reviews:









"Being the first monograph devoted to nonconvex duality, this book is going to become a fundamental source for researchers in the field. An important feature of the book is that it is also accessible to nonspecialists, since, in spite of dealing with a rather specialized topic, it is essentially self-contained. this monograph is a very useful addition to the existing literature on optimization and approximation and is undoubtedly going to constitute a major reference on nonconvex duality." (Juan-Enrique Martinez-Legaz, Mathematical Reviews, Issue 2006 k)



"This monograph, being the first book of this kind in the literature, covers a wide range of optimization and approximation problems. It provides an excellent overview over the literature and, moreover, it contains a lot of new results and new proofs of known results. The results and the choice of the classes of problems are well motivated. The monograph is appropriate for graduate students and advanced readers." (Andreas Löhne, Mathematical Methods of Operations Research, Vol. 66, 2007)



"In this monograph the author presents some approaches to duality in nonconvex approximation in normed linear spaces and to duality in nonconvex global optimization in locally convex spaces. It is my belief that the monograph under review will become a fundamental reference on nonconvex duality for researchers in the field, and, although the topics are very specialized, the monograph is also accessible to nonspecialists . is strongly recommended to researchers, postgraduate and graduate students interested in nonconvex duality theory." (Fabián Flores Bazán, Zentralblatt MATH, Vol. 1119 (21), 2007)



"This is a nice addition to the literature on nonconvex optimization in locally convex spaces, devoted primarily to nonconvex duality. Most of the material appears for the first time in book form and examples are abundant. The style is friendly. I stronglyrecommend this book to graduate students studying nonconvex optimization theory." (Constantin P. Niculescu, Revue Roumaine de Mathématique Pures et Appliquées, Vol. LII (5), 2007)

List of Figures xi
Preface xiii
1 Preliminaries
1(84)
1.1 Some preliminaries from convex analysis
1(26)
1.2 Some preliminaries from abstract convex analysis
27(12)
1.3 Duality for best approximation by elements of convex sets
39(7)
1.4 Duality for convex and quasi-convex infimization
46(39)
1.4.1 Unperturbational theory
47(24)
1.4.2 Perturbational theory
71(14)
2 Worst Approximation
85(16)
2.1 The deviation of a set from an element
86(7)
2.2 Characterizations and existence of farthest points
93(8)
3 Duality for Quasi-convex Supremization
101(36)
3.1 Some hyperplane theorems of surrogate duality
103(5)
3.2 Unconstrained surrogate dual problems for quasi-convex supremization
108(13)
3.3 Constrained surrogate dual problems for quasi-convex supremization
121(6)
3.4 Lagrangian duality for convex supremization
127(4)
3.4.1 Unperturbational theory
127(2)
3.4.2 Perturbational theory
129(2)
3.5 Duality for quasi-convex supremization over structured primal constraint sets
131(6)
4 Optimal Solutions for Quasi-convex Maximization
137(16)
4.1 Maximum points of quasi-convex functions
137(7)
4.2 Maximum points of continuous convex functions
144(5)
4.3 Some basic subdifferential characterizations of maximum points
149(4)
5 Reverse Convex Best Approximation
153(16)
5.1 The distance to the complement of a convex set
154(7)
5.2 Characterizations and existence of elements of best approximation in complements of convex sets
161(8)
6 Unperturbational Duality for Reverse Convex Infimization
169(34)
6.1 Some hyperplane theorems of surrogate duality
171(4)
6.2 Unconstrained surrogate dual problems for reverse convex infimization
175(9)
6.3 Constrained surrogate dual problems for reverse convex infimization
184(5)
6.4 Unperturbational Lagrangian duality for reverse convex infimization
189(1)
6.5 Duality for infimization over structured primal reverse convex constraint sets
190(13)
6.5.1 Systems
190(8)
6.5.2 Inequality constraints
198(5)
7 Optimal Solutions for Reverse Convex Infimization
203(10)
7.1 Minimum points of functions on reverse convex subsets of locally convex spaces
203(6)
7.2 Subdifferential characterizations of minimum points of functions on reverse convex sets
209(4)
8 Duality for D.C. Optimization Problems
213(46)
8.1 Unperturbational duality for unconstrained d.c. infimization
213(8)
8.2 Minimum points of d.c. functions
221(4)
8.3 Duality for d.c. infimization with a d.c. inequality constraint
225(7)
8.4 Duality for d.c. infimization with finitely many d.c. inequality constraints
232(12)
8.5 Perturbational theory
244(3)
8.6 Duality for optimization problems involving maximum operators
247(12)
8.6.1 Duality via conjugations of type Lau
248(4)
8.6.2 Duality via Fenchel conjugations
252(7)
9 Duality for Optimization in the Framework of Abstract Convexity
259(20)
9.1 Additional preliminaries from abstract convex analysis
259(8)
9.2 Surrogate duality for abstract quasi-convex supremization, using polarities ΔG: 2x —> 2W and ΔG: 2X —> 2WxR
267(3)
9.3 Constrained surrogate duality for abstract quasi-convex supremization, using families of subsets of X
270(1)
9.4 Surrogate duality for abstract reverse convex infimization, using polarities ΔG: 2X —> 2W and ΔG: 2X —> 2WxR
271(2)
9.5 Constrained surrogate duality for abstract reverse convex infimization, using families of subsets of X
273(2)
9.6 Duality for unconstrained abstract d.c. infimization
275(4)
10 Notes and Remarks 279(50)
References 329(18)
Index 347