SOME QUALITATIVE PROBILEMS IN OPTIMIZATION TA QUANG SON Trang nhan đề Lời cam đoan Lời cảm ơn Mục lục Danh mục các ký hiệu Lời giới thiệu Chương_1: Preliminaries Chương_2: Optimality conditions, Lagrange duality, and stability for convex infinite problems Chương_3: Characterizations of solutions sets of convex infinite problems and extensions Chương 4: ε- Optimality and ε- Lagrangian duality for conver infinite problems Chương_5: ε- Optimality and ε- Lagrangian duality for non -conver infinite problems Kết luận và hướng phát triển 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) condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 Relation between Slater and (FM) conditions in semi-infinite programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. "- Optimality and "-Lagrangian duality for convex infinite problems 61 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 vii 4.2 Approximate optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.1 Necessary and sufficient conditions for "-solutions . . . . . . . . 63 4.2.2 Special case: Cone-constrained convex programs . . . . . . . . . 68 4.3 "-Lagrangian duality and "-saddle points . . . . . . . . . . . . . . . . . 69 4.4 Some more approximate results concerning Lagrangian function of (P) . 73 Chapter 5. "-Optimality and "-Lagrangian duality for non-convex infinite problems 76 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Generalized KKT Conditions up to " . . . . . . . . . . . . . . . . . . . 81 5.4 Quasi Saddle-Points and "-Lagrangian Duality . . . . . . . . . . . . . . 88 Main results and open problems 94 The papers related to the thesis 96 Index 106 Danh mục các công trình của tác giả Phụ lục Tài liệu tham khảoSOME QUALITATIVE PROBILEMS IN OPTIMIZATION TA QUANG SON Trang nhan đề Lời cam đoan Lời cảm ơn Mục lục Danh mục các ký hiệu Lời giới thiệu Chương_1: Preliminaries Chương_2: Optimality conditions, Lagrange duality, and stability for convex infinite problems Chương_3: Characterizations of solutions sets of convex infinite problems and extensions Chương 4: ε- Optimality and ε- Lagrangian duality for conver infinite problems Chương_5: ε- Optimality and ε- Lagrangian duality for non -conver infinite problems Kết luận và hướng phát triển 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) condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 Relation between Slater and (FM) conditions in semi-infinite programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. "- Optimality and "-Lagrangian duality for convex infinite problems 61 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 vii 4.2 Approximate optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.1 Necessary and sufficient conditions for "-solutions . . . . . . . . 63 4.2.2 Special case: Cone-constrained convex programs . . . . . . . . . 68 4.3 "-Lagrangian duality and "-saddle points . . . . . . . . . . . . . . . . . 69 4.4 Some more approximate results concerning Lagrangian function of (P) . 73 Chapter 5. "-Optimality and "-Lagrangian duality for non-convex infinite problems 76 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Generalized KKT Conditions up to " . . . . . . . . . . . . . . . . . . . 81 5.4 Quasi Saddle-Points and "-Lagrangian Duality . . . . . . . . . . . . . . 88 Main results and open problems 94 The papers related to the thesis 96 Index 106 Danh mục các công trình của tác giả Phụ lục Tài liệu tham khảo