Vector Optimization (eBook)

Preface 8
Contents 12
Part I Convex Analysis 18
Chapter 1 Linear Spaces 20
1.1 Linear Spaces and Convex Sets 20
1.2 Partially Ordered Linear Spaces 29
1.3 Topological Linear Spaces 38
1.4 Some Examples 49
Chapter 2 Maps on Linear Spaces 54
2.1 Convex Maps 54
2.2 Differentiable Maps 62
Notes 76
Chapter 3 Some Fundamental Theorems 77
3.1 Zorn’s Lemma and the Hahn-Banach Theorem 77
3.2 Separation Theorems 87
3.3 A James Theorem 97
3.4 Two Krein-Rutman Theorems 103
3.5 Contingent Cones and a Lyusternik Theorem 106
Notes 115
Part II Theory of Vector Optimization 117
Chapter 4 Optimality Notions 119
Notes 129
Chapter 5 Scalarization 131
5.1 Necessary Conditions for Optimal Elements of a Set 131
5.2 Sufficient Conditions for Optimal Elements of a Set 145
5.3 Parametric Approximation Problems 155
Notes 163
Chapter 6 Existence Theorems 165
Notes 175
Chapter 7 Generalized Lagrange Multiplier Rule 177
7.1 Necessary Conditions for Minimal and Weakly Minimal Elements 177
7.2 Sufficient Conditions for Minimal and Weakly Minimal Elements 190
7.2.1 Generalized Quasiconvex Maps 190
7.2.2 Sufficiency of the Generalized Multiplier Rule 197
Notes 203
Chapter 8 Duality 205
8.1 A General Duality Principle 205
8.2 Duality Theorems for Abstract Optimization Problems 208
8.3 Specialization to Abstract Linear Optimization Problems 216
Notes 223
Part III Mathematical Applications 224
Chapter 9 Vector Approximation 226
9.1 Introduction 226
9.2 Simultaneous Approximation 228
9.3 Generalized Kolmogorov Condition 231
9.4 Nonlinear Chebyshev Vector Approximation 233
9.5 Linear Chebyshev Vector Approximation 241
9.5.1 Duality Results 242
9.5.2 An Alternation Theorem 248
Notes 256
Chapter 10 Cooperative n Player Differential Games 258
10.1 Basic Remarks on the Cooperation Concept 258
10.2 A Maximum Principle 260
10.2.1 Necessary Conditions for Optimal and Weakly Optimal Controls 262
10.2.2 Sufficient Conditions for Optimal and Weakly Optimal Controls 274
10.3 A Special Cooperative n Player Differential Game 285
Notes 292
Part IV Engineering Applications 294
Chapter 11 Theoretical Basics of Multiobjective Optimization 296
11.1 Basic Concepts 296
11.2 Special Scalarization Results 306
11.2.1 Weighted Sum Approach 307
11.2.2 Weighted Chebyshev Norm Approach 319
11.2.3 Special Scalar Problems 322
Notes 326
Chapter 12 Numerical Methods 329
12.1 Modified Polak Method 329
12.2 Eichfelder-Polak Method 335
12.3 Interactive Methods 339
12.3.1 Modified STEM Method 340
12.3.2 Method of Reference Point Approximation 344
12.4 Method for Discrete Problems 357
Notes 362
Chapter 13 Multiobjective Design Problems 364
13.1 Design of Antennas 365
13.2 Design of FDDI ComputerNetworks 372
13.2.1 A Cooperative Game 373
13.2.2 Minimization of Mean Waiting Times 375
13.2.3 Numerical Results 378
13.3 Fluidized Reactor-Heater System 380
13.3.1 Simplification of the Constraints 382
13.3.2 Numerical Results 384
13.4 A Cross-Current Multistage Extraction Process 386
13.5 Field Design of a Magnetic Resonance System 389
Notes 393
Part V Extensions to Set Optimization 395
Chapter 14 Basic Concepts and Results of Set Optimization 397
Notes 403
Chapter 15 Contingent Epiderivatives 405
15.1 Contingent Derivatives and Contingent Epiderivatives 405
15.2 Properties of Contingent Epiderivatives 409
15.3 Contingent Epiderivatives of Real-Valued Functions 413
15.4 Generalized Contingent Epiderivatives 417
Notes 421
Chapter 16 Subdifferential 422
16.1 Concept of Subdifferential 422
16.2 Properties of the Subdifferential 424
16.3 Weak Subgradients 428
Notes 432
Chapter 17 Optimality Conditions 433
17.1 Optimality Conditions with Contingent Epiderivatives 433
17.2 Optimality Conditions with Subgradients 438
17.3 Optimality Conditions with Weak Subgradients 439
17.4 Generalized Lagrange Multiplier Rule 441
17.4.1 A Necessary Optimality Condition 442
17.4.2 A Sufficient Optimality Condition 451
Notes 456
Bibliography 458
List of Symbols 484
Index 486