Fixed point theory, variational analysis, and optimization

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Fixed Point Theory, Variational Analysis, and Optimization not only covers three vital branches of nonlinear analysis--fixed point theory, variational inequalities, and vector optimization--but 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.

Author Notes

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.

Abdul Rahim Khan and Hafiz Fukhar-ud-dinB. A. Bin Dehaish and M. A. KhamsiHichem Ben-El-MechaiekhN. HadjisavvasD. AusselQamrul Hasan AnsariDinh The Luc and Augusta RatuiMatthias Ehrgott and Xavier Gandibleux

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

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