The vector valued maximin:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston u.a.
Acad. Press
1994
|
| Series: | Mathematics in science and engineering
193 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 404 S. graph. Darst. |
| ISBN: | 0127799508 |
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MARC
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| 245 | 1 | 0 | |a The vector valued maximin |c V. I. Zhukovskiy ; M. E. Salukvadze |
| 246 | 1 | 3 | |a The vector-valued maximin |
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Record in the Search Index
| _version_ | 1804123407356264448 |
|---|---|
| adam_text | Contents
Preface xi
Notation xv
Abstract xvii
Chapter 1 Quasimotions and their Properties
1. Reference Information 1
1.1. Control System 1
1.2. Motion Properties 4
1.3. Kononenko s Counter Example 6
2. Piecewise Continuous Stepwise Quasimotion 8
2.1. Subbotin s Counter Example 8
2.2. Stepwise Quasimotion 11
2.3. Properties of Quasimotions 17
2.4. Completeness of a Quasimotion Bunch 20
3. The Alternative and the Saddle Point 25
3.1. Useful Information 25
3.2. Proof of the Alternative 27
3.3. Minimax, Maximin, and Saddle Point 29
3.4. Saddle Point Properties of To 33
4. Corollaries of the Alternative 38
4.1. Additional Proposition 38
4.2. Corollaries 41
4.3. A Property of the Linear Quadratic Problem 42
Chapter 2 Slater Optimality
1. Slater Maximal Strategy 49
1.1. Formalization of Multicriterial Problem 49
1.2. Definition of Slater Maximal Strategy 51
V
vi Contents
1.3 Properties of Slater Maximal Strategy 53
1.4. Stability 57
1.5. Properties of Inheritance and Rejection 61
2. Sufficient Conditions 63
2.1. Conditions 63
2.2. Corollaries of Theorem 2.1 65
2.3. A Universal Slater Maximal Strategy 66
3. Structure in the Case of Slater Optimality 70
3.1. Description of Structure 70
3.2. Structure 71
3.3. Comparison With Another Definition of Problem (1.1) Solution 74
3.4. Example. Multicriterial Dynamic Problem Without
Slater Maximal Strategy 77
Chapter 3 Pareto Optimality
1. Pareto Optimal Strategy 81
1.1. Definition and Geometric Interpretation 81
1.2. Properties of the Pareto Maximal Strategy 83
1.3. Stability 84
2. Relations Between the Sets iTp and fs 90
2.1. Preliminary Remarks 90
2.2. Quasiconcave Multicriterial Problems 90
2.3. Remark 92
3. Structure in the Case of Pareto Optimality 93
3.1. Description 93
3.2. Structure 94
3.3. Existence 96
3.4. Specific Features of Pareto Maximal Strategies 96
4. Sufficient Conditions 100
4.1. Auxiliary Propositions 100
4.2. Sufficient Conditions 105
4.3. Corollaries 107
5. A Linear Quadratic Multicriterial Problem 110
5.1. Problem Statement 110
5.2. An Auxiliary Optimal Control Problem 111
5.3. Formal Procedure for Obtaining V 113
5.4. Theoretical Basis of Algorithm 115
5.5. Exact Solution of System (5.22), (5.23) 118
6. Comparison to Pl Optimality 120
6.1. Definition 120
6.2. Properties of Pi Maximal Strategy 121
6.3. Uniqueness of Values of the Goal Functional Vector 124
6.4. Relationship Between f v and tri 125
Contents vii
Chapter 4 Geoffrion Optimality
1. Geoffrion Maximal Strategy 131
1.1. Definition 131
1.2. Properties 134
1.3. Structure 137
1.4. External and Dynamic Stability 142
2. Necessary and Sufficient Conditions 145
2.1. Auxiliary Propositions 145
2.2. Necessary Conditions 148
2.3. Sufficient Conditions 151
3. A optimality 153
3.1. Compactness of the Set Xs 153
3.2. Formalization of the Relation A and its Properties 155
3.3. A optimal Strategies of Problem (1.1) in Chapter 2 158
Chapter 5 Vector Valued Saddle Points
1. Definition 169
1.1. Why Saddle Points in Differential Games? 169
1.2. Formalization of Vector Valued Saddle Points and Some of Their Properties 172
1.3. Geometric Interpretation 181
2. Properties of Saddle Points 186
2.1. Existence of /1 saddle Point 186
2.2. Dynamic Stability 191
2.3. Compactness 194
3. Invariance of Vector Valued Saddle Points 202
3.1. Affine Transformations 202
3.2. Addition of Criteria 204
3.3. Use of Increasing Functions 207
4. Sufficient Conditions 210
4.1 Examples of Functions Increasing over and over on R 210
4.2. Slater Saddle Points 212
4.3. Pareto Saddle Points 216
4.4. Geoffrion Saddle Points 218
4.5. A Linear Quadratic Game 220
4.6. /1 saddle Points 227
4.7. Specific Features of Vector Valued Saddle Points 228
Chapter 6 Vector Valued Guarantees
1. Vector Valued Maximin and Minimax 237
1.1. Formalization of Vector Valued Maximin 237
Viii Contents
1.2. Relationship to Scalar Maximin 243
1.3. Geometric Interpretation 245
1.4. Why is the Maximin Better than the Vector Valued Saddle Point? 249
2. Existence of Vector Valued Maximins 251
2.1. Existence of Slater Maximin 252
2.2. Topological Properties of Slater Maximin 257
2.3. Stability of the Set of Slater Maximins 259
2.4. Relation Between Maximins and Vector Valued Saddle Points 262
2.5. Vector Valued Maximin as Solution of a Differential Game 266
3. Pareto e maximin 270
3.1. Formalization of c saddle Points in Static Games 270
3.2. Properties of f. saddle Points of the Game (3.1) 275
3.3. Existence of a Pareto s maximin in the Differential Game (1.1) 277
Chapter 7 The Competition Problem
1. Mathematical Model of Competition 281
1.1. A Model of a Special System 281
1.2. A Model of Competition 284
1.3. Optimal Decision Making in Competition Problems 285
1.4. A Geometric Interpretation of the ZS Solution 286
1.5. Procedure for the Construction of a ZS Solution 290
2. A Game With Separable Payoff Function 291
2.1. Problem Statement 29
2.2. Vector Valued Saddle Points 293
2.3. Vector Valued Maximin and Minimax 300
2.4. Existence of ZS Solutions With N = 2 305
3. The ZS Solution in the Competition Problem 311
3.1. A Game With Mirror Payoff Function (Saddle Points) 311
3.2. Vector Valued Maximin and Minimax of the Game (3.1) 315
3.3. Obtaining ZS Solutions in the Competition Problem 319
4. Model of Competing Research Activities 322
4.1. Single Firm Model 322
4.2. Game Theoretic Model of Competition 324
4.3. Pareto Optimal Controls 326
4.4. Decision Making in the Competition Model 330
Chapter 8 A Pursuit Game With Noise
1. Statement of Problem 335
1.1. Cooperative Pursuit Game with Negligible Noise 335
1.2. Recognition of Noise (Uncertainties) in Problem (1.3) 337
Contents ix
2. Pursuit of Two Target Points, one of which with Uncertain Location 339
2.1. Problem Statement and Essential Results 339
2.2. Pareto Minimaxes 342
2.3. Pareto Saddle Points 344
3. Pursuit of Two Target Points 347
3.1. The Problem 348
3.2. Auxiliary Information 351
3.3. Solution of the Pursuit Game 356
4. A Three Criterial Pursuit Problem 359
4.1. Problem Statement 359
4.2. An Auxiliary Proposition 362
4.3. Pareto Saddle Points of the Game (4.5) 364
Appendix 1 Concepts from Topology 371
A 1.1. The Topological Space 371
A1.2. Metric Spaces 372
A 1.3. Convergence (in the Topological Sense) 373
A1.4. The Space {2x}m 375
Appendix 2 Upper Semicontinuous Multivalent Mappings 377
Appendix 3 Auxiliary Propositions from the Theory of
Multicriterial Problems 379
Appendix 4 Vector Valued Maximins in
Static Problems 383
A4.1. Slater Maximin 383
A4.2. Other Notions of Vector Valued Maximins 385
References 389
Author Index 397
Subject Index 401
|
| any_adam_object | 1 |
| author | Zukovskij, Vladislav I. Salukvadze, Mindija E. |
| author_facet | Zukovskij, Vladislav I. Salukvadze, Mindija E. |
| author_role | aut aut |
| author_sort | Zukovskij, Vladislav I. |
| author_variant | v i z vi viz m e s me mes |
| building | Verbundindex |
| bvnumber | BV009051381 |
| classification_rvk | SK 860 |
| ctrlnum | (OCoLC)468640101 (DE-599)BVBBV009051381 |
| dewey-full | 519.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.3 |
| dewey-search | 519.3 |
| dewey-sort | 3519.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV009051381 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:29:17Z |
| institution | BVB |
| isbn | 0127799508 |
| language | English |
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| physical | XVII, 404 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Acad. Press |
| record_format | marc |
| series | Mathematics in science and engineering |
| series2 | Mathematics in science and engineering |
| spelling | Zukovskij, Vladislav I. Verfasser aut The vector valued maximin V. I. Zhukovskiy ; M. E. Salukvadze The vector-valued maximin Boston u.a. Acad. Press 1994 XVII, 404 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics in science and engineering 193 Jeux différentiels ram Optimisation mathématique ram Optimierung (DE-588)4043664-0 gnd rswk-swf Vektor (DE-588)4202708-1 gnd rswk-swf Differentialspiel (DE-588)4012253-0 gnd rswk-swf Differentialspiel (DE-588)4012253-0 s Optimierung (DE-588)4043664-0 s DE-604 Vektor (DE-588)4202708-1 s 1\p DE-604 Salukvadze, Mindija E. Verfasser aut Mathematics in science and engineering 193 (DE-604)BV000001196 193 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005991064&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Zukovskij, Vladislav I. Salukvadze, Mindija E. The vector valued maximin Mathematics in science and engineering Jeux différentiels ram Optimisation mathématique ram Optimierung (DE-588)4043664-0 gnd Vektor (DE-588)4202708-1 gnd Differentialspiel (DE-588)4012253-0 gnd |
| subject_GND | (DE-588)4043664-0 (DE-588)4202708-1 (DE-588)4012253-0 |
| title | The vector valued maximin |
| title_alt | The vector-valued maximin |
| title_auth | The vector valued maximin |
| title_exact_search | The vector valued maximin |
| title_full | The vector valued maximin V. I. Zhukovskiy ; M. E. Salukvadze |
| title_fullStr | The vector valued maximin V. I. Zhukovskiy ; M. E. Salukvadze |
| title_full_unstemmed | The vector valued maximin V. I. Zhukovskiy ; M. E. Salukvadze |
| title_short | The vector valued maximin |
| title_sort | the vector valued maximin |
| topic | Jeux différentiels ram Optimisation mathématique ram Optimierung (DE-588)4043664-0 gnd Vektor (DE-588)4202708-1 gnd Differentialspiel (DE-588)4012253-0 gnd |
| topic_facet | Jeux différentiels Optimisation mathématique Optimierung Vektor Differentialspiel |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005991064&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000001196 |
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