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Fixed Point Theory, Variational Analysis, and Optimization [Kõva köide]

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Edited by , Edited by (Aligarh Muslim University, India, and King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia), Edited by (University of Tabuk, Saudia Arabia)
  • Formaat: Hardback, 368 pages, kõrgus x laius: 234x156 mm, kaal: 840 g, 9 Tables, black and white; 42 Illustrations, black and white
  • Ilmumisaeg: 03-Jun-2014
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1482222078
  • ISBN-13: 9781482222074
Teised raamatud teemal:
Fixed Point Theory, Variational Analysis, and OptimizationSuurem pilt
  • Formaat: Hardback, 368 pages, kõrgus x laius: 234x156 mm, kaal: 840 g, 9 Tables, black and white; 42 Illustrations, black and white
  • Ilmumisaeg: 03-Jun-2014
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1482222078
  • ISBN-13: 9781482222074
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781482222074.html
Märksõnad:
Fixed Point Theory, Variational Analysis, and Optimization not only covers three vital branches of nonlinear analysisfixed point theory, variational inequalities, and vector optimizationbut also explains the connections between them, enabling the study of a general form of variational inequality problems related to the optimality conditions involving differentiable or directionally differentiable functions. This essential reference supplies both an introduction to the field and a guideline to the literature, progressing from basic concepts to the latest developments. Packed with detailed proofs and bibliographies for further reading, the text:











Examines Mann-type iterations for nonlinear mappings on some classes of a metric space Outlines recent research in fixed point theory in modular function spaces Discusses key results on the existence of continuous approximations and selections for set-valued maps with an emphasis on the nonconvex case Contains definitions, properties, and characterizations of convex, quasiconvex, and pseudoconvex functions, and of their strict counterparts Discusses variational inequalities and variational-like inequalities and their applications Gives an introduction to multi-objective optimization and optimality conditions Explores multi-objective combinatorial optimization (MOCO) problems, or integer programs with multiple objectives

Fixed Point Theory, Variational Analysis, and Optimization is a beneficial resource for the research and study of nonlinear analysis, optimization theory, variational inequalities, and mathematical economics. It provides fundamental knowledge of directional derivatives and monotonicity required in understanding and solving variational inequality problems.

Arvustused

"There is a real need for this book. It is useful for people who work in areas of nonlinear analysis, optimization theory, variational inequalities, and mathematical economics." Nan-Jing Huang, Sichuan University, Chengdu, Peoples Republic of China

Preface xi
List of Figures
xv
List of Tables
xvii
Contributors xix
I Fixed Point Theory
1(136)
1 Common Fixed Points in Convex Metric Spaces
3(42)
Abdul Rahim Khan
Hafiz Fukhar-ud-din
1.1 Introduction
3(1)
1.2 Preliminaries
4(11)
1.3 Ishikawa Iterative Scheme
15(9)
1.4 Multistep Iterative Scheme
24(8)
1.5 One-Step Implicit Iterative Scheme
32(13)
Bibliography
39(6)
2 Fixed Points of Nonlinear Semigroups in Modular Function Spaces
45(32)
B. A. Bin Dehaish
M. A. Khamsi
2.1 Introduction
45(1)
2.2 Basic Definitions and Properties
46(7)
2.3 Some Geometric Properties of Modular Function Spaces
53(6)
2.4 Some Fixed-Point Theorems in Modular Spaces
59(2)
2.5 Semigroups in Modular Function Spaces
61(3)
2.6 Fixed Points of Semigroup of Mappings
64(13)
Bibliography
71(6)
3 Approximation and Selection Methods for Set-Valued Maps and Fixed Point Theory
77(60)
Hichem Ben-El-Mechaiekh
3.1 Introduction
78(2)
3.2 Approximative Neighborhood Retracts, Extensors, and Space Approximation
80(17)
3.2.1 Approximative Neighborhood Retracts and Extensors
80(4)
3.2.2 Contractibility and Connectedness
84(1)
3.2.2.1 Contractible Spaces
84(1)
3.2.2.2 Proximal Connectedness
85(1)
3.2.3 Convexity Structures
86(4)
3.2.4 Space Approximation
90(1)
3.2.4.1 The Property A(K; P) for Spaces
90(2)
3.2.4.2 Domination of Domain
92(3)
3.2.4.3 Domination, Extension, and Approximation
95(2)
3.3 Set-Valued Maps, Continuous Selections, and Approximations
97(25)
3.3.1 Semicontinuity Concepts
98(1)
3.3.2 USC Approachable Maps and Their Properties
99(1)
3.3.2.1 Conservation of Approachability
100(6)
3.3.2.2 Homotopy Approximation, Domination of Domain, and Approachability
106(2)
3.3.3 Examples of A--Maps
108(5)
3.3.4 Continuous Selections for LSC Maps
113(1)
3.3.4.1 Michael Selections
114(2)
3.3.4.2 A Hybrid Continuous Approximation-Selection Property
116(1)
3.3.4.3 More on Continuous Selections for Non-Convex Maps
116(5)
3.3.4.4 Non-Expansive Selections
121(1)
3.4 Fixed Point and Coincidence Theorems
122(15)
3.4.1 Generalizations of the Himmelberg Theorem to the Non-Convex Setting
122(1)
3.4.1.1 Preservation of the FPP from P to A(K; P)
123(3)
3.4.1.2 A Leray-Schauder Alternative for Approachable Maps
126(1)
3.4.2 Coincidence Theorems
127(4)
Bibliography
131(6)
II Convex Analysis and Variational Analysis
137(110)
4 Convexity, Generalized Convexity, and Applications
139(32)
N. Hadjisavvas
4.1 Introduction
139(1)
4.2 Preliminaries
140(1)
4.3 Convex Functions
141(7)
4.4 Quasiconvex Functions
148(9)
4.5 Pseudoconvex Functions
157(4)
4.6 On the Minima of Generalized Convex Functions
161(2)
4.7 Applications
163(3)
4.7.1 Sufficiency of the KKT Conditions
163(1)
4.7.2 Applications in Economics
164(2)
4.8 Further Reading
166(5)
Bibliography
167(4)
5 New Developments in Quasiconvex Optimization
171(36)
D. Aussel
5.1 Introduction
171(3)
5.2 Notations
174(2)
5.3 The Class of Quasiconvex Functions
176(8)
5.3.1 Continuity Properties of Quasiconvex Functions
181(1)
5.3.2 Differentiability Properties of Quasiconvex Functions
182(1)
5.3.3 Associated Monotonicities
183(1)
5.4 Normal Operator: A Natural Tool for Quasiconvex Functions
184(12)
5.4.1 The Semistrictly Quasiconvex Case
185(3)
5.4.2 The Adjusted Sublevel Set and Adjusted Normal Operator
188(1)
5.4.2.1 Adjusted Normal Operator: Definitions
188(3)
5.4.2.2 Some Properties of the Adjusted Normal Operator
191(5)
5.5 Optimality Conditions for Quasiconvex Programming
196(3)
5.6 Stampacchia Variational Inequalities
199(4)
5.6.1 Existence Results: The Finite Dimensions Case
199(2)
5.6.2 Existence Results: The Infinite Dimensional Case
201(2)
5.7 Existence Result for Quasiconvex Programming
203(4)
Bibliography
204(3)
6 An Introduction to Variational-like Inequalities
207(40)
Qamrul Hasan Ansari
6.1 Introduction
207(1)
6.2 Formulations of Variational-like Inequalities
208(4)
6.3 Variational-like Inequalities and Optimization Problems
212(6)
6.3.1 Invexity
212(2)
6.3.2 Relations between Variational-like Inequalities and an Optimization Problem
214(4)
6.4 Existence Theory
218(7)
6.5 Solution Methods
225(13)
6.5.1 Auxiliary Principle Method
226(5)
6.5.2 Proximal Method
231(7)
6.6 Appendix
238(9)
Bibliography
240(7)
III Vector Optimization
247(96)
7 Vector Optimization: Basic Concepts and Solution Methods
249(58)
Augusta Ratiu
7.1 Introduction
250(1)
7.2 Mathematical Backgrounds
251(9)
7.2.1 Partial Orders
252(5)
7.2.2 Increasing Sequences
257(1)
7.2.3 Monotone Functions
258(1)
7.2.4 Biggest Weakly Monotone Functions
259(1)
7.3 Pareto Maximality
260(8)
7.3.1 Maximality with Respect to Extended Orders
262(1)
7.3.2 Maximality of Sections
263(1)
7.3.3 Proper Maximality and Weak Maximality
263(3)
7.3.4 Maximal Points of Free Disposal Hulls
266(2)
7.4 Existence
268(5)
7.4.1 The Main Theorems
268(1)
7.4.2 Generalization to Order-Complete Sets
269(2)
7.4.3 Existence via Monotone Functions
271(2)
7.5 Vector Optimization Problems
273(4)
7.5.1 Scalarization
274(3)
7.6 Optimality Conditions
277(5)
7.6.1 Differentiable Problems
277(2)
7.6.2 Lipschitz Continuous Problems
279(2)
7.6.3 Concave Problems
281(1)
7.7 Solution Methods
282(25)
7.7.1 Weighting Method
282(10)
7.7.2 Constraint Method
292(10)
7.7.3 Outer Approximation Method
302(3)
Bibliography
305(2)
8 Multi-objective Combinatorial Optimization
307(36)
Matthias Ehrgott
Xavier Gandibleux
8.1 Introduction
307(1)
8.2 Definitions and Properties
308(5)
8.3 Two Easy Problems: Multi-objective Shortest Path and Spanning Tree
313(2)
8.4 Nice Problems: The Two-Phase Method
315(5)
8.4.1 The Two-Phase Method for Two Objectives
315(4)
8.4.2 The Two-Phase Method for Three Objectives
319(1)
8.5 Difficult Problems: Scalarization and Branch and Bound
320(7)
8.5.1 Scalarization
321(3)
8.5.2 Multi-objective Branch and Bound
324(3)
8.6 Challenging Problems: Metaheuristics
327(6)
8.7 Conclusion
333(10)
Bibliography
334(9)
Index 343
Saleh Abdullah R. Al-Mezel is a full professor of mathematics at King Abdulaziz University, Jeddah, Saudi Arabia and the vice president for academic affairs at the University of Tabuk, Saudi Arabia. He holds a B.Sc from King Abdulaziz University; an M.Phil from Swansea University, Wales; and a Ph.D from Cardiff University, Wales. He possesses over ten years of teaching experience and has participated in several sponsored research projects. His publications span numerous books and international journals.

Falleh Rajallah M. Al-Solamy is a professor of mathematics at King Abdulaziz University, Jeddah, Saudi Arabia and the vice president for graduate studies and scientific research at the University of Tabuk, Saudi Arabia. He holds a B.Sc from King Abdulaziz University and a Ph.D from Swansea University, Wales. A member of several academic societies, he possesses over 7 years of academic and administrative experience. He has completed 30 research projects on differential geometry and its applications, participated in over 14 international conferences, and published more than 60 refereed papers.

Qamrul Hasan Ansari is a professor of mathematics at Aligarh Muslim University, India, from which he also received his M.Phil and Ph.D. He has co/edited, co/authored, and/or contributed to 8 scholarly books. He serves as associate editor of the Journal of Optimization Theory and Applications and the Fixed Point Theory and Applications, and has guest-edited special issues of several other journals. He has more than 150 research papers published in world-class journals and his work has been cited in over 1,400 ISI journals. His fields of specialization and/or interest include nonlinear analysis, optimization, convex analysis, and set-valued analysis.