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E-raamat: Generalized Convexity and Vector Optimization

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Generalized Convexity and Vector OptimizationSuurem pilt
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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.

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.



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.

Arvustused

Aus den Rezensionen: "... Das Buch ist eine sehr stringente und anspruchsvolle Aneinanderreihung von Definitionen, Lemmata, Theoremen und Korollaren sowie von deren Beweisen ... Das vorliegende Werk ist ... ein umfassendes und in seiner Tiefe beeindruckendes Fachbuch, es richtet sich ... an Spezialisten auf dem Gebiet der Optimierung ... Das Buch uberzeugt durch seine gute und ubersichtliche Form, seine konsequente detaillierter Behandlung vieler Problemklassen sowie den mathematischen Umfang der behandelten Probleme ..." (in: Rho - Mathematik Verein der Uni Rostock, Mai 2009)

1 Introduction 1
1.1 Nonlinear Symmetric Dual Pair of Programming Problems
3
1.2 Motivation
4
2 Generalized Convex Functions 7
2.1 Convex and Generalized Convex Functions
8
2.2 Invex and Generalized Invex Functions
10
2.3 Type I and Related Functions
12
2.4 Univex and Related Functions
16
2.5 V-Invex and Related Functions
18
2.6 Further Generalized Convex Functions
22
3 Generalized Type I and Related Functions 25
3.1 Generalized Type I Univex Functions
25
3.2 Nondifferentiable d–Type I and Related Functions
27
3.3 Continuous-Time Analogue of Generalized Type I Functions
28
3.4 Nondifferentiable Continuous-Time Analogue of Generalized Type I Functions
31
3.5 Generalized Convex Functions in Complex Spaces
32
3.6 Semilocally Connected Type I Functions
33
3.7 ( ,ρ,σ,θ)-V-Type-I and Related n-Set Functions
36
3.8 Nondifferentiable d-V-Type-I and Related Functions
38
3.9 Nonsmooth Invex and Related Functions
40
3.10 Type I and Related Functions in Banach Spaces
41
4 Optimality Conditions 45
4.1 Optimality Conditions for Vector Optimization Problems
45
4.2 Optimality Conditions for Nondifferentiable Vector Optimization Problems
48
4.3 Optimality Conditions for Minimax Fractional Programs
51
4.4 Optimality Conditions for Vector Optimization Problems on Banach Spaces
56
4.5 Optimality Conditions for Complex Minimax Programs on Complex Spaces
58
4.6 Optimality Conditions for Continuous-Time Optimization Problems
60
4.7 Optimality Conditions for Nondifferentiable Continuous-Time Optimization Problems
67
4.8 Optimality Conditions for Fractional Optimization Problems with Semilocally Type I Pre-invex Functions
73
4.9 Optimality Conditions for Vector Fractional Subset Optimization Problems
80
5 Duality Theory 91
5.1 Mond—Weir Type Duality for Vector Optimization Problems
91
5.2 General Mond—Weir Type Duality for Vector Optimization Problems
94
5.3 Mond—Weir Duality for Nondifferentiable Vector Optimization Problems
96
5.4 General Mond—Weir Duality for Nondifferentiable Vector Optimization Problems
99
5.5 Mond—Weir Duality for Nondifferentiable Vector Optimization Problems with d—Univex Functions
101
5.6 General Mond—Weir Duality for Nondifferentiable Vector Optimization Problems with d—Univex Functions
105
5.7 Mond—Weir Duality for Nondifferentiable Vector Optimization Problems with d—Type-I Univex Functions
107
5.8 General Mond—Weir Duality for Nondifferentiable Vector Optimization Problems with d—Type-I Univex Functions
111
5.9 First Duality Model for Fractional Minimax Programs
114
5.10 Second Duality Model for Fractional Minimax Programs
117
5.11 Third Duality Model for Fractional Minimax Programs
119
5.12 Mond—Weir Duality for Nondifferentiable Vector Optimization Problems
121
5.13 Duality for Vector Optimization Problems on Banach Spaces
126
5.14 First Dual Model for Complex Minimax Programs
128
5.15 Second Dual Model for Complex Minimax Programs
132
5.16 Mond—Weir Duality for Continuous—Time Vector Optimization Problems
135
5.17 General Mond—Weir Duality for Continuous—Time Vector Optimization Problems
138
5.18 Duality for Nondifferentiable Continuous—Time Optimization Problems
141
5.19 Duality for Vector Control Problems
144
5.20 Duality for Vector Fractional Subset Optimization Problems
152
6 Second and Higher Order Duality 165
6.1 Second Order Duality for Nonlinear Optimization Problems
165
6.2 Second Order Duality for Minimax Programs
169
6.3 Second Order Duality for Nondifferentiable Minimax Programs
176
6.4 Higher Order Duality for Nonlinear Optimization Problems
181
6.5 Mond–Weir Higher Order Duality for Nonlinear Optimization Problems
184
6.6 General Mond–Weir Higher Order Duality for Nonlinear Optimization Problems
188
6.7 Mangasarian Type Higher Order Duality for Nondifferentiable Optimization Problems
189
6.8 Mond–Weir Type Higher Order Duality for Nondifferentiable Optimization Problems
193
6.9 General Mond–Weir Type Higher Order Duality for Nondifferentiable Optimization Problems
195
7 Symmetric Duality 199
7.1 Higher Order Symmetric Duality
199
7.2 Mond–Weir Type Higher Order Symmetric Duality
203
7.3 Self Duality
209
7.4 Higher Order Vector Nondifferentiable Symmetric Duality
210
7.5 Minimax Mixed Integer Optimization Problems
214
7.6 Mixed Symmetric Duality in Nondifferentiable Vector Optimization Problems
215
7.7 Mond–Weir Type Mixed Symmetric First and Second Order Duality in Nondifferentiable Optimization Problems
224
7.8 Second Order Mixed Symmetric Duality in Nondifferentiable Vector Optimization Problems
232
7.9 Symmetric Duality for a Class of Nondifferentiable Vector Fractional Variational problems
241
8 Vector Variational-like Inequality Problems 255
8.1 Relationships Between Vector Variational-Like Inequalities and Vector Optimization Problems
255
8.2 On Relationships Between Vector Variational Inequalities and Vector Optimization Problems with Pseudo-Univexity
260
8.3 Relationship Between Vector Variational-Like Inequalities and Nondifferentiable Vector Optimization Problems
264
8.4 Characterization of Generalized Univex functions
269
8.5 Characterization of Nondifferentiable Generalized Invex Functions
275
References 281
Index 293
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