Generalized Convexity and Vector Optimization (Nonconvex Optimization and Its Applications Book 90)

The present lecture note is dedicated to the study of the optimality conditions and the duality results for nonlinear vector optimization problems, in ?nite and in?nite dimensions. The problems include are nonlinear vector optimization problems, s- metric dual problems, continuous-time vector optimization problems, relationships between vector optimization and variational inequality problems. Nonlinear vector optimization problems arise in several contexts such as in the building and interpretation of economic models; the study of various technolo- cal processes; the development of optimal choices in ?nance; management science; production processes; transportation problems and statistical decisions, etc. In preparing this lecture note a special effort has been made to obtain a se- contained treatment of the subjects; so we hope that this may be a suitable source for a beginner in this fast growing area of research, a semester graduate course in nonlinear programing, and a good reference book. This book may be useful to theoretical economists, engineers, and applied researchers involved in this area of active research. The lecture note is divided into eight chapters: Chapter 1 brie?y deals with the notion of nonlinear programing problems with basic notations and preliminaries. Chapter 2 deals with various concepts of convex sets, convex functions, invex set, invex functions, quasiinvex functions, pseudoinvex functions, type I and generalized type I functions, V-invex functions, and univex functions Cover 1 Series: Nonconvex Optimization and Its Applications 2 Title: Generalized Convexity and Vector Optimization 4 Copyright 5 Preface 6 Contents 8 Chapter 1. Introduction 12 1.1 Nonlinear Symmetric Dual Pair of Programming Problems 14 1.2 Motivation 15 Chapter 2. Generalized Convex Functions 18 2.1 Convex and Generalized Convex Functions 19 2.2 Invex and Generalized Invex Functions 21 2.3 Type I and Related Functions 23 2.4 Univex and Related Functions 27 2.5 V-Invex and Related Functions 29 2.6 Further Generalized Convex Functions 33 Chapter 3. Generalized Type I and Related Functions 36 3.1 Generalized Type I Univex Functions 36 3.2 Nondifferentiable d–Type I and Related Functions 38 3.3 Continuous-Time Analogue of Generalized Type I Functions 39 3.4 Nondifferentiable Continuous-Time Analogue of Generalized Type I Functions 42 3.5 Generalized Convex Functions in Complex Spaces 43 3.6 Semilocally Connected Type I Functions 44 3.7 (Γ, ρ, σ , θ )-V-Type-I and Related n-Set Functions 47 3.8 Nondifferentiable d-V-Type-I and Related Functions 49 3.9 Nonsmooth Invex and Related Functions 51 3.10 Type I and Related Functions in Banach Spaces 52 Chapter 4. Optimality Conditions 56 4.1 Optimality Conditions for Vector Optimization Problems 56 4.2 Optimality Conditions for Nondifferentiable Vector Optimization Problems 59 4.3 Optimality Conditions for Minimax Fractional Programs 62 4.4 Optimality Conditions for Vector Optimization Problems on Banach Spaces 67 4.5 Optimality Conditions for Complex Minimax Programs on Complex Spaces 69 4.6 Optimality Conditions for Continuous-Time Optimization Problems 71 4.7 Optimality Conditions for Nondifferentiable Continuous-Time Optimization Problems 78 4.8 Optimality Conditions for Fractional Optimization Problems with Semilocally Type I Pre-invex Functions 84 4.9 Optimality Conditions for Vector Fractional Subset Optimization Problems 91 Chapter 5. Duality Theory 102 5.1 Mond–Weir Type Duality for Vector Optimization Problems 102 5.2 General Mond–Weir Type Duality for Vector Optimization Problems 105 5.3 Mond–Weir Duality for Nondifferentiable Vector Optimization Problems 107 5.4 General Mond–Weir Duality for Nondifferentiable Vector Optimization Problems 110 5.5 Mond–Weir Duality for Nondifferentiable Vector Optimization Problems with d−Univex Functions 112 5.6 General Mond–Weir Duality for Nondifferentiable Vector Optimization Problems with d−Univex Functions 116 5.7 Mond–Weir Duality for Nondifferentiable Vector Optimization Problems with d–Type-I Univex Functions 118 5.8 General Mond–Weir Duality for Nondifferentiable Vector Optimization Problems with d–Type-I Univex Functions 122 5.9 First Duality Model for Fractional Minimax Programs 125 5.10 Second Duality Model for Fractional Minimax Programs 128 5.11 Third Duality Model for Fractional Minimax Programs 130 5.12 Mond–Weir Duality for Nondifferentiable Vector Optimization Problems 132 5.13 Duality for Vector Optimization Problems on Banach Spaces 137 5.14 First Dual Model for Complex Minimax Programs 139 5.15 Second Dual Model for Complex Minimax Programs 143 5.16 Mond–Weir Duality for Continuous–Time Vector Optimization Problems 146 5.17 General Mond–Weir Duality for Continuous–Time Vector Optimization Problems 149 5.18 Duality for Nondifferentiable Continuous–Time Optimization Problems 152 5.19 Duality for Vector Control Problems 155 5.20 Duality for Vector Fractional Subset Optimization Problems 163 Chapter 6. Second and Higher Order Duality 176 6.1 Second Order Duality for Nonlinear Optimization Problems 176 6.2 Second Order Duality for Minimax Programs 180 6.3 Second Order Duality for Nondifferentiable Minimax Programs 187 6.4 Higher Order Duality for Nonlinear Optimization Problems 192 6.5 Mond–Weir Higher Order Duality for Nonlinear Optimization Problems 195 6.6 General Mond–Weir Higher Order Duality for Nonlinear Optimization Problems 199 6.7 Mangasarian Type Higher Order Duality for Nondifferentiable Optimization Problems 200 6.8 Mond–Weir Type Higher Order Duality for Nondifferentiable Optimization Problems 204 6.9 General Mond–Weir Type Higher Order Duality for Nondifferentiable Optimization Problems 206 Chapter 7. Symmetric Duality 210 7.1 Higher Order Symmetric Duality 210 7.2 Mond–Weir Type Higher Order Symmetric Duality 214 7.3 Self Duality 220 7.4 Higher Order Vector Nondifferentiable Symmetric Duality 221 7.5 Minimax Mixed Integer Optimization Problems 225 7.6 Mixed Symmetric Duality in Nondifferentiable Vector Optimization Problems 226 7.7 Mond–Weir Type Mixed Symmetric First and Second Order Duality in Nondifferentiable Optimization Problems 235 7.8 Second Order Mixed Symmetric Duality in Nondifferentiable Vector Optimization Problems 243 7.9 Symmetric Duality for a Class of Nondifferentiable Vector Fractional Variational problems 252 Chapter 8. Vector Variational-like Inequality Problems 266 8.1 Relationships Between Vector Variational-Like Inequalities and Vector Optimization Problems 266 8.2 On Relationships Between Vector Variational Inequalities and Vector Optimization Problems with Pseudo-Univexity 271 8.3 Relationship Between Vector Variational-Like Inequalities and Nondifferentiable Vector Optimization Problems 275 8.4 Characterization of Generalized Univex functions 280 8.5 Characterization of Nondifferentiable Generalized Invex Functions 286 References 292 Index 304 Content: Introduction and Motivation -- Generalized Convex Functions -- Type I and Related Functions -- Optimality Conditions -- Duality Theory -- Second and Higher Order Duality -- Symmetric Duality -- Vector Variational-like Inequality Problems. Abstract: This book discusses the Kuhn-Tucker Optimality, Karush-Kuhn-Tucker Necessary and Sufficient Optimality Conditions in presence of various types of generalized convexity assumptions. It details the present state of knowledge on research done in this area. Read more... Discusses the Kuhn-Tucker Optimality, Karush-Kuhn-Tucker Necessary and Sufficient Optimality Conditions in presence of various types of generalized convexity assumptions. This book also discusses Wolfe-type Duality, Mond-Weir type Duality, and Mixed type Duality for Multiobjective optimization problems such as Nonlinear programming problems. Shashi Kant Mishra, Shou-yang Wang, Kin Keung Lai. Includes Bibliographical References (p. 281-292) And Index.