Generalized Convexity and Vector Optimization

by Mishra, Shashi Kant; Wang, Shou-yang; Lai, Kin Keung

• ISBN13: 9783540856702
• ISBN10: 3540856706
• Format: Hardcover
• Copyright: 2009-01-01
• Publisher: Springer Verlag

Summary

The present book discusses the Kuhn-Tucker Optimality, Karush-Kuhn-Tucker Necessary and Sufficient Optimality Conditions in presence of various types of generalized convexity assumptions. Wolfe-type Duality, Mond-Weir type Duality, Mixed type Duality for Multiobjective optimization problems such as Nonlinear programming problems, Fractional programming problems, Nonsmooth programming problems, Nondifferentiable programming problems, Variational and Control problems under various types of generalized convexity assumptions.

Table of Contents

Introduction	p. 1
Nonlinear Symmetric Dual Pair of Programming Problems	p. 3
Motivation	p. 4
Generalized Convex Functions	p. 7
Convex and Generalized Convex Functions	p. 8
Invex and Generalized Invex Functions	p. 10
Type I and Related Functions	p. 12
Univex and Related Functions	p. 16
V-Invex and Related Functions	p. 18
Further Generalized Convex Functions	p. 22
Generalized Type I and Related Functions	p. 25
Generalized Type I Univex Functions	p. 25
Nondifferentiable d-Type I and Related Functions	p. 27
Continuous-Time Analogue of Generalized Type I Functions	p. 28
Nondifferentiable Continuous-Time Analogue of Generalized Type I Functions	p. 31
Generalized Convex Functions in Complex Spaces	p. 32
Semilocally Connected Type I Functions	p. 33
(&Ghe;, ¿, ¿, ¿)-V-Type-I and Related n-Set Functions	p. 36
Nondifferentiable d-V-Type-I and Related Functions	p. 38
Nonsmooth Invex and Related Functions	p. 40
Type I and Related Functions in Banach Spaces	p. 41
Optimality Conditions	p. 45
Optimality Conditions for Vector Optimization Problems	p. 45
Optimality Conditions for Nondifferentiable Vector Optimization Problems	p. 48
Optimality Conditions for Minimax Fractional Programs	p. 51
Optimality Conditions for Vector Optimization Problems on Branch Spaces	p. 56
Optimality Conditions for Complex Minimax Programs on Complex Spaces	p. 58
Optimality Conditions for Continuous-Time Optimization Problems	p. 60
Optimality Conditions for Nondifferentiable Continuous-Time Optimization Problems	p. 67
Optimality Conditions for Fractional Optimization Problems with Semilocally Type I Pre-invex Functions	p. 73
Optimality Conditions for Vector Fractional Subset Optimization Problems	p. 80
Duality Theory	p. 91
Mond-Weir Type Duality for Vector Optimization Problems	p. 91
General Mond-Weir Type Duality for Vector Optimization Problems	p. 94
Mond-Weir Duality for Nondifferentiable Vector Optimization Problems	p. 96
General Mond-Weir Duality for Nondifferentiable Vector Optimization Problems	p. 99
Mond-Weir Duality for Nondifferentiable Vector Optimization Problems with d-Univex Functions	p. 101
General Mond-Weir Duality for Nondifferentiable Vector Optimization Problems with d-Univex Functions	p. 105
Mond-Weir Duality for Nondifferentiable Vector Optimization Problems with d-Type-I Univex Functions	p. 107
General Mond-Weir Duality for Nondifferentiable Vector Optimization Problems with d-Type-I Univex Functions	p. 111
First Duality Model for Fractional Minimax Programs	p. 114
Second Duality Model for Fractional Minimax Programs	p. 117
Third Duality Model for Fractional Minimax Programs	p. 119
Mond-Weir Duality for Nondifferentiable Vector Optimization Problems	p. 121
Duality for Vector Optimization Problems on Banach Spaces	p. 126
First Dual Model for Complex Minimax Programs	p. 128
Second Dual Model for Complex Minimax Programs	p. 132
Mond-Weir Duality for Continuous-Time Vector Optimization Problems	p. 135
General Mond-Weir Duality for Continuous-Time Vector Optimization Problems	p. 138
Duality for Nondifferentiable Continuous-Time Optimization Problems	p. 141
Duality for Vector Control Problems	p. 144
Duality for Vector Fractional Subset Optimization Problems	p. 152
Second and Higher Order Duality	p. 165
Second Order Duality for Nonlinear Optimization Problems	p. 165
Second Order Duality for Minimax Programs	p. 169
Second Order Duality for Nondifferentiable Minimax Programs	p. 176
Higher Order Duality for Nonlinear Optimization Problems	p. 181
Mond-Weir Higher Order Duality for Nonlinear Optimization Problems	p. 184
General Mond-Weir Higher Order Duality for Nonlinear Optimization Problems	p. 188
Mangasarian Type Higher Order Duality for Nondifferentiable Optimization Problems	p. 189
Mond-Weir Type Higher Order Duality for Nondifferentiable Optimization Problems	p. 193
General Mond-Weir Type Higher Order Duality for Nondifferentiable Optimization Problems	p. 195
Symmetric Duality	p. 199
Higher Order Symmetric Duality	p. 199
Mond-Weir Type Higher Order Symmetric Duality	p. 203
Self Duality	p. 209
Higher Order Vector Nondifferentiable Symmetric Duality	p. 210
Minimax Mixed Integer Optimization Problems	p. 214
Mixed Symmetric Duality in Nondifferentiable Vector Optimization Problems	p. 215
Mond-Weir Type Mixed Symmetric First and Second Order Duality in Nondifferentiable Optimization Problems	p. 224
Second Order Mixed Symmetric Duality in Nondifferentiable Vector Optimization Problems	p. 232
Symmetric Duality for a Class of Nondifferentiable Vector Fractional Variational Problems	p. 241
Vector Variational-like Inequality Problems	p. 255
Relationships Between Vector Variational-Like Inequalities and Vector Optimization Problems	p. 255
On Relationships Between Vector Variational Inequalities and Vector Optimization Problems with Pseudo-Univexity	p. 260
Relationship Between Vector Variational-Like Inequalities and Nondifferentiable Vector Optimization Problems	p. 264
Characterization of Generalized Univex functions	p. 269
Characterization of Nondifferentiable Generalized Invex Functions	p. 275
References	p. 281
Index	p. 293
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