Fixed Point Theory, Variational Analysis, and Optimization
Fixed Point Theory, Variational Analysis, and Optimization
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DOI：
10.1201/b17068
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发表时间：
2014-06
期刊：
影响因子：
--
通讯作者：
S. Al-Mezel;F. Al-Solamy;Q. Ansari
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文献类型：
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作者：
S. Al-Mezel;F. Al-Solamy;Q. Ansari
前言图表一览表不动点理论凸度量空间中的公共不动点Abdul Rahim Khan和Hafiz Ehhar-ud-din引论石川迭代格式多步迭代格式一步隐式迭代格式参考书目模函数空间B中非线性半群的不动点。A. Bin Dehaish和M. A. Khamsi引言基本定义和性质模函数空间的一些几何性质模空间中的一些不动点定理模函数空间中的半群映射半群的不动点参考书目集值映射的近似和选择方法以及不动点理论Hichem Ben-El-Mechaiekh引言近似邻域收缩，扩张，与空间逼近逼近邻域收缩与扩张可收缩性与连通性可收缩空间邻近连通性凸性结构空间逼近域控制，扩张，与逼近集值映射、连续选择与逼近半连续性概念USC可逼近映射及其性质可逼近性守恒同伦逼近、Domain控制A-映射的可逼近性LSC映射的连续选择Michael选择一个混合的连续近似选择性质更多关于非凸映射的连续选择扩张选择不动点和重合定理Himmelberg定理到非凸集的推广FPP从P到A（K P）的保持可逼近映射的Leray-Schauder替代重合定理参考书目II。凸分析与变分分析凸性、广义凸性及其应用。Hadjisavvas引论凸函数拟凸函数伪凸函数关于广义凸函数极小值的应用KKT条件的充分性在经济学中的应用进一步阅读参考书目拟凸优化的新发展D. Aussel引言记法拟凸函数类拟凸函数的连续性拟凸函数的可微性拟凸函数的相关单调性正规算子：拟凸函数的自然工具半严格拟凸情形调整子水平集和调整正规算子调整正规算子：拟凸规划的调整正规算子最优性条件Stampacchia变分不等式存在性结果如下：有限维情形的存在性结果：拟凸规划的无限维情形的存在性结果参考书目类变分不等式简介Qamrul Hasan Ansari类变分不等式的介绍类变分不等式的公式类变分不等式与优化问题类变分不等式与优化问题之间的不变凸关系存在性理论求解方法辅助原理方法近似方法附录参考书目III.矢量优化矢量优化：基本概念和求解方法Dinh The Luc and Augusta Ratiu引言数学背景偏序递增序列单调函数最大弱单调函数Pareto极大值关于扩阶的极大值截面极大值真极大值和弱极大值自由处理壳极大点的存在性主要定理推广到序-完备集通过单调函数的存在向量优化问题标量化最优性条件可微问题Lipschitz连续问题凹问题求解方法加权方法约束方法外逼近方法参考书目目标组合优化Matthias Ehrgott和Xavier Gandibleux介绍定义和性质两个简单的问题：多目标最短路径和生成树好的问题：两阶段法两目标两阶段法三目标两阶段法标量化与分支定界标量化多目标分支定界问题元分析
Preface List of Figures List of Tables Contributors I. Fixed Point Theory Common Fixed Points in Convex Metric Spaces Abdul Rahim Khan and Hafiz Fukhar-ud-din Introduction Preliminaries Ishikawa Iterative Scheme Multistep Iterative Scheme One-Step Implicit Iterative Scheme Bibliography Fixed Points of Nonlinear Semigroups in Modular Function Spaces B. A. Bin Dehaish and M. A. Khamsi Introduction Basic Definitions and Properties Some Geometric Properties of Modular Function Spaces Some Fixed-Point Theorems in Modular Spaces Semigroups in Modular Function Spaces Fixed Points of Semigroup of Mappings Bibliography Approximation and Selection Methods for Set-Valued Maps and Fixed Point Theory Hichem Ben-El-Mechaiekh Introduction Approximative Neighborhood Retracts, Extensors, and Space Approximation Approximative Neighborhood Retracts and Extensors Contractibility and Connectedness Contractible Spaces Proximal Connectedness Convexity Structures Space Approximation The Property A(K P) for Spaces Domination of Domain Domination, Extension, and Approximation Set-Valued Maps, Continuous Selections, and Approximations Semicontinuity Concepts USC Approachable Maps and Their Properties Conservation of Approachability Homotopy Approximation, Domination of Domain, and Approachability Examples of A-Maps Continuous Selections for LSC Maps Michael Selections A Hybrid Continuous Approximation-Selection Property More on Continuous Selections for Non-Convex Maps Non-Expansive Selections Fixed Point and Coincidence Theorems Generalizations of the Himmelberg Theorem to the Non-Convex Setting Preservation of the FPP from P to A(K P) A Leray-Schauder Alternative for Approachable Maps Coincidence Theorems Bibliography II. Convex Analysis and Variational Analysis Convexity, Generalized Convexity, and Applications N. Hadjisavvas Introduction Preliminaries Convex Functions Quasiconvex Functions Pseudoconvex Functions On the Minima of Generalized Convex Functions Applications Sufficiency of the KKT Conditions Applications in Economics Further Reading Bibliography New Developments in Quasiconvex Optimization D. Aussel Introduction Notations The Class of Quasiconvex Functions Continuity Properties of Quasiconvex Functions Differentiability Properties of Quasiconvex Functions Associated Monotonicities Normal Operator: A Natural Tool for Quasiconvex Functions The Semistrictly Quasiconvex Case The Adjusted Sublevel Set and Adjusted Normal Operator Adjusted Normal Operator: Definitions Some Properties of the Adjusted Normal Operator Optimality Conditions for Quasiconvex Programming Stampacchia Variational Inequalities Existence Results: The Finite Dimensions Case Existence Results: The Infinite Dimensional Case Existence Result for Quasiconvex Programming Bibliography An Introduction to Variational-Like Inequalities Qamrul Hasan Ansari Introduction Formulations of Variational-Like Inequalities Variational-Like Inequalities and Optimization Problems Invexity Relations between Variational-Like Inequalities and an Optimization Problem Existence Theory Solution Methods Auxiliary Principle Method Proximal Method Appendix Bibliography III. Vector Optimization Vector Optimization: Basic Concepts and Solution Methods Dinh The Luc and Augusta Ratiu Introduction Mathematical Backgrounds Partial Orders Increasing Sequences Monotone Functions Biggest Weakly Monotone Functions Pareto Maximality Maximality with Respect to Extended Orders Maximality of Sections Proper Maximality and Weak Maximality Maximal Points of Free Disposal Hulls Existence The Main Theorems Generalization to Order-Complete Sets Existence via Monotone Functions Vector Optimization Problems Scalarization Optimality Conditions Differentiable Problems Lipschitz Continuous Problems Concave Problems Solution Methods Weighting Method Constraint Method Outer Approximation Method Bibliography Multi-Objective Combinatorial Optimization Matthias Ehrgott and Xavier Gandibleux Introduction Definitions and Properties Two Easy Problems: Multi-Objective Shortest Path and Spanning Tree Nice Problems: The Two-Phase Method The Two-Phase Method for Two Objectives The Two-Phase Method for Three Objectives Difficult Problems: Scalarization and Branch and Bound Scalarization Multi-Objective Branch and Bound Challenging Problems: Metaheuristics Conclusion Bibliography Index