# Luận văn Some qualitative problems in optimization

Contents

Half-title page i

Honor Statement ii

Acknowledgements iii

Table of contents v

Notations viii

Introduction 1

Chapter 1. Preliminaries 6

1.1 Notations and basic concepts . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Some basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Some known results concerning convex infinite problems . . . . . . . . 14

Chapter 2. Optimality conditions, Lagrange duality, and stability of

convex infinite problems 16

2.1 Introduction. 16

2.2 Qualification/Constraint qualification conditions . . . . . . . . . . . . . 17

vi

2.2.1 Relation between generalized Slater’s conditions and (FM) con-dition . . . . . . . . . . . . . . . . 18

2.2.2 Relation between Slater and (FM) conditions in semi-infinite pro-gramming . . . . . . . . . . . . . . . 19

2.3 New version of Farkas’ lemma . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Duality. 29

2.6 Stability . 34

Chapter 3. Characterizations of solution sets of convex infinite problems

and extensions 37

3.1 Introduction. 37

3.2 Characterizations of solution sets of convex infinite programs . . . . . 39

3.2.1 Characterizations of solution set via Lagrange multipliers . . . . 39

3.2.2 Characterizations of solution set via subdifferential of Lagrangian

function . . . . . . . . . . . . . . . 41

3.2.3 Characterizations of solution set via minimizing sequence . . . . 45

3.2.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Characterizations of solution sets of semi-convex programs . . . . . . . 48

3.3.1 Some basic results concerning semiconvex function . . . . . . . . 49

3.3.2 Characterizations of solution sets of semiconvex programs . . . . 52

3.4 Characterization of solution sets of linear fractional programs . . . . . . 57

Chapter 4.ε- Optimality and ε-Lagrangian duality for convex infinite

problems 61

4.1 Introduction. 61

vii

4.2 Approximate optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Necessary and sufficient conditions forε-solutions . . . . . . . . 63

4.2.2 Special case: Cone-constrained convex programs . . . . . . . . . 68

4.3 ε-Lagrangian duality and ε-saddle points . . . . . . . . . . . . . . . . . 69

4.4 Some more approximate results concerning Lagrangian function of (P) . 73

Chapter 5.ε-Optimality andε-Lagrangian duality for non-convex infinite

problems 76

5.1 Introduction. 76

5.2 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Generalized KKT Conditions up toε . 81

5.4 Quasi Saddle-Points andε-Lagrangian Duality . . . . . . . . . . . . . . 88

Main results and open problems 94

The papers related to the thesis 96

Index 106

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