Markov decision processes: discrete stochastic dynamic programming
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, New Jersey
Wiley-Interscience
[2005]
|
| Schriftenreihe: | Wiley series in probability and statistics
Wiley-Interscience paperback series |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xvii, 649 Seiten Diagramme |
| ISBN: | 9780471727828 0471727822 |
Internformat
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| adam_text | Contents Preface 1. Introduction 1.1. The Sequential Decision Model, 1 1.2. Inventory Management, 3 1.3. Bus Engine Replacement, 4 1.4. Highway Pavement Maintenance, 5 1.5. Communication Models, 8 1.6. Mate Desertion in Cooper’s Hawks, 10 1.7. So Who’s Counting, 13 Historical Background, 15 2. Model Formulation 2.1. Problem Definition and Notation, 17 2.1.1. Decision Epochs and Periods, 17 2.1.2. State and Action Sets, 18 2.1.3. Rewards and Transition Probabilities, 19 2.1.4. Decision Rules, 21 2.1.5. Policies, 22 2.1.6. Induced Stochastic Processes, Conditional Probabilities, and Expectations, 22 2.2. A One-Period Markov Decision Problem, 25 2.3. Technical Considerations, 27 2.3.1. The Role of Model Assumptions, 28 2.3.2. The Borei Model, 28 Bibliographic Remarks, 30 Problems, 31 3. Examples 3.1. A Two-State Markov Decision Process, 33 3.2. Single-Product Stochastic Inventory Control, 37
CONTĽNTS viii 3.2.1. Model Formulation, 37 3.2.2. A Numerical Example, 41 3.3. Deterministic Dynamic Programs, 42 3.3.1. Problem Formulation, 42 3.3.2. Shortest Route and Critical Path Models, 43 3.3.3. Sequential Allocation Models, 45 3.3.4. Constrained Maximum Likelihood Estimation, 46 3.4. Optimal Stopping, 47 3.4.1. Problem Formulation, 47 3.4.2. Selling an Asset, 48 3.4.3. The Secretary Problem, 49 3.4.4. Exercising an Option, 50 3.5. Controlled Discrete-Time Dynamic Systems, 51 3.5.1. Model Formulation, 51 3.5.2. The Inventory Control Model Revisited, 53 3.5.3. Economic Growth Models, 55 3.5.4. Linear Quadratic Control, 56 3.6. Bandit Models, 57 3.6.1. Markov Decision Problem Formulation, 57 3.6.2. Applications, 58 3.6.3. Modifications, 61 3.7. Discrete-Time Queueing Systems, 62 3.7.1. Admission Control, 62 3.7.2. Service Rate Control, 64 Bibliographic Remarks, 66 Problems, 68 4. Finite-Horizon Markov Decision Processes 4.1. Optimality Criteria, 74 4.1.1. Some Preliminaries, 74 4.1.2. The Expected Total Reward Criteria, 78 4.1.3. Optimal Policies, 79 4.2. Finite-Horizon Policy Evaluation, 80 4.3. Optimality Equations and the Principle of Optimality, 83 4.4. Optimality of Deterministic Markov Policies, 88 4.5. Backward Induction, 92 4.6. Examples, 94 4.6.1. The Stochastic Inventory Model, 94 4.6.2. Routing Problems, 96 4.6.3. The Sequential Allocation Model, 98 4.6.4. The Secretary Problem, 100 4.7. Optimality of Monotone Policies, 102 4.7.1. Structured Policies, 103 4.7.2. Superadditive Functions, 103 74
CONTENTS 4.7.3. Optimality of Monotone Policies, 105 4.7.4. A Price Determination Model, 108 4.7.5. An Equipment Replacement Model, 109 4.7.6. Monotone Backward Induction, 111 Bibliographic Remarks, 112 Problems, 113 5. Infinite-Horizon Models: Foundations 5.1. 5.2. 5.3. 5.4. The Value of a Policy, 120 The Expected Total Reward Criterion, 123 The Expected Total Discounted Reward Criterion, 125 Optimality Criteria, 128 5.4.1. Criteria Based on Policy Value Functions, 128 5.4.2. Overtaking Optimality Criteria, 130 5.4.3. Discount Optimality Criteria, 133 5.5. Markov Policies, 134 5.6. Vector Notation for Markov Decision Processes, 137 Bibliographic Remarks, 138 Problems, 139 6. Discounted Markov Decision Problems 6.1. 6.2. Policy Evaluation, 143 Optimality Equations, 146 6.2.1. Motivation and Definitions, 146 6.2.2. Properties of Solutions of the Optimality Equations, 148 6.2.3. Solutions of the Optimality Equations, 149 6.2.4. Existence of Optimal Policies, 152 6.2.5. General State and Action Spaces, 157 6.3. Value Iteration and Its Variants, 158 6.3.1. Rates of Convergence, 159 6.3.2. Value Iteration, 160 6.3.3. Increasing the Efficiency of Value Iteration with Splitting Methods, 164 6.4. Policy Iteration, 174 6.4.1. The Algorithm 6.4.2. Finite State and Action Models, 176 6.4.3. Nonfinite Models, 177 6.4.4. Convergence Rates, 181 6.5. Modified Policy Iteration, 185 6.5.1. The Modified Policy Iteration Algorithm, 186 6.5.2. Convergence of Modified Policy Iteration, 188 6.5.3. Convergence Rates, 192 6.5.4. Variants of Modified Policy Iteration, 194 6.6. Spans, Bounds,
Stopping Criteria, and Relative Value Iteration, 195
CONTHNTS x 6 6.1. 6.6.2. The Span Seminorm, 195 Bounds on the Value of a Discounted Markov Decision Processes, 199 6.6.3. Stopping Criteria, 201 6.6.4. Value Iteration and Relative Value Iteration, 203 6.7. Action Elimination Procedures, 206 6.7.1. Identification of Nonoptimal Actions, 206 6.7.2. Action Elimination Procedures, 208 6.7.3. Modified Policy Iteration with Action Elimination and an Improved Stopping Criterion, 213 6.7.4. Numerical Performance of Modified Policy Iteration with Action Elimination, 216 6.8. Convergence of Policies, Turnpikes and Planning Horizons, 218 6.9. Linear Programming, 223 6.9.1. Model Formation, 223 6.9.2. Basic Solutions and Stationary Policies, 224 6.9.3. Optimal Solutions and Optimal Policies, 227 6.9.4. An Example, 229 6.10. Countable-State Models, 231 6.10.1. Unbounded Rewards, 231 6.10.2. Finite-State Approximations to Countable-State Discounted Models, 239 6.10.3. Bounds for Approximations, 245 6.10.4. An Equipment Replacement Model, 248 6.11. The Optimality of Structured Policies, 255 6.11.1. A General Framework, 255 6.11.2. Optimal Monotone Policies, 258 6.11.3. Continuous and Measurable Optimal Policies, 260 Bibliographic Remarks, 263 Problems, 266 7. The Expected Total-Reward Criterion 7.1. Model Classification and General Results, 277 7.1.1. Existence of the Expected Total Reward, 278 7.1.2. The Optimality Equation, 280 7.1.3. Identification of Optimal Policies, 282 7.1.4. Existence of Optimal Policies, 283 7.2. Positive Bounded Models, 284 7.2.1. The Optimality Equation, 285 7.2.2. Identification of Optimal Policies, 288 7.2.3.
Existence of Optimal Policies, 289 7.2.4. Value Iteration, 294 7.2.5. Policy Iteration, 295 7.2.6. Modified Policy Iteration, 298 7.2.7. Linear Programming, 299 7.2.8. Optimal Stopping, 303 277
CONTENTS xi 7.3. Negative Models, 309 7.3.1. The Optimality Equation, 309 7.3.2. Identification and Existence of Optimal Policies, 311 7.3.3. Value Iteration, 313 7.3.4. Policy Iteration, 316 7.3.5. Modified Policy Iteration, 318 7.3.6. Linear Programming, 319 7.3.7. Optimal Parking, 319 7.4. Comparison of Positive and Negative Models, 324 Bibliographic Remarks, 325 Problems, 326 8. Average Reward and Related Criteria 8.1. Optimality Criteria, 332 8.1.1. The Average Reward of a Fixed Policy, 332 8.1.2. Average Optimality Criteria, 333 8.2. Markov Reward Processes and Evaluation Equations, 336 8.2.1. The Gain and Bias, 337 8.2.2. The Laurent Series Expansion, 341 8.2.3. Evaluation Equations, 343 8.3. Classification of Markov Decision Processes, 348 8.3.1. Classification Schemes, 348 8.3.2. Classifying a Markov Decision Process, 350 8.3.3. Model Classification and the Average Reward Criterion, 351 8.4. The Average Reward Optimality Equation— Unichain Models, 353 8.4.1. The Optimality Equation, 354 8.4.2. Existence of Solutions to the Optimality Equation, 358 8.4.3. Identification and Existence of Optimal Policies, 360 8.4.4. Models with Compact Action Sets, 361 8.5. Value Iteration in Unichain Models, 364 8.5.1. The Value Iteration Algorithm, 364 8.5.2. Convergence of Value Iteration, 366 8.5.3. Bounds on the Gain, 370 8.5.4. An Aperiodicity Transformation, 371 8.5.5. Relative Value Iteration, 373 8.5.6. Action Elimination, 373 8.6. Policy Iteration in Unichain Models, 377 8.6.1. The Algorithm, 378 8.6.2. Convergence of Policy Iteration for Recurrent Models, 379 8.6.3.
Convergence of Policy Iteration for Unichain Models, 380 8.7. Modified Policy Iteration in Unichain Models, 385 8.7.1. The Modified Policy Iteration Algorithm, 386 331
CONTF.NTS xii 8.7.2. Convergence of the Algorithm, 387 8.7.3. Numerical Comparison of Algorithms, 388 8.8. Linear Programming in Unichain Models, 391 8.8.1. Linear Programming for Recurrent Models, 392 8.8.2. Linear Programming for Unichain Models, 395 8.9. State Action Frequencies, Constrained Models and Models with Variance Criteria, 398 8.9.1. Limiting Average State Action Frequencies, 398 8.9.2. Models with Constraints, 404 8.9.3. Variance Criteria, 408 8.10. Countable-State Models, 412 8.10.1. Counterexamples, 413 8.10.2. Existence Results, 414 8.10.3. A Communications Model, 421 8.10.4. A Replacement Model, 424 8.11. The Optimality of Structured Policies, 425 8.11.1. General Theory, 425 8.11.2. Optimal Monotone Policies, 426 Bibliographic Remarks, 429 Problems, 433 9. The Average Reward Criterion-Multichain and Communicating Models 9.1. Average Reward Optimality Equations: Multichain Models, 442 9.1.1. Multichain Optimality Equations, 443 9.1.2. Property of Solutions of the Optimality Equations, 445 9.1.3. Existence of Solutions to the Optimality Equations, 448 9.1.4. Identification and Existence of Optimal Policies, 450 9.2. Policy Iteration for Multichain Models, 451 9.2.1. The Algorithm, 452 9.2.2. An Example, 454 9.2.3. Convergence of the Policy Iteration in Multichain Models, 455 9.2.4. Behavior of the Iterates of Policy Iteration, 458 9.3. Linear Programming in Multichain Models, 462 9.3.1. Dual Feasible Solutions and Randomized Decision Rules, 463 9.3.2. Basic Feasible Solutions and Deterministic Decision Rules, 467 9.3.3. Optimal Solutions and Policies, 468
9.4. Value Iteration, 472 9.4.1. Convergence of un - ng*, 472 9.4.2. Convergence of Value Iteration, 476 9.5. Communicating Models, 478 9.5.1. Policy Iteration, 478 9.5.2. Linear Programming, 482 9.5.3. Value Iteration, 483 441
CONTENTS Bibliographic Remarks, 484 Problems, 487 10. Sensitive Discount Optimality 10.1. Existence of Optimal Policies, 493 10.1.1. Definitions, 494 10.1.2. Blackwell Optimality, 495 10.1.3. Stationary «-Discount Optimal Policies, 497 10.2. Optimality Equations, 501 10.2.1. Derivation of Sensitive Discount Optimality Equations, 501 10.2.2. Lexicographic Ordering, 503 10.2.3. Properties of Solutions of the Sensitive Optimality Equations, 505 10.3. Policy Iteration, 511 10.3.1. The Algorithm, 511 10.3.2. An Example, 513 10.3.3. Convergence of the Algorithm, 515 10.3.4. Finding Blackwell Optimal Policies, 517 10.4. The Expected Total-Reward Criterion Revisited, 519 10.4.1. Relationship of the Bias and Expected Total Reward, 519 10.4.2. Optimality Equations and Policy Iteration, 520 10.4.3. Finding Average Overtaking Optimal Policies, 525 Bibliographic Remarks, 526 Problems, 528 11. Continuous-Time Models 11.1. Model Formulation, 531 11.1.1. Probabilistic Structure, 531 11.1.2. Rewards or Costs, 533 11.1.3. Decision Rules and Policies, 534 11.1.4. Induced Stochastic Processes, 534 11.2. Applications, 536 11.2.1. A Two-State Semi-Markov Decision Process, 536 11.2.2. Admission Control for a G/M/l Queueing System, 537 11.2.3. Service Rate Control in an M/G/l Queueing System, 539 11.3. Discounted Models, 540 11.3.1. Model Formulation, 540 11.3.2. Policy Evaluation, 540 11.3.3. The Optimality Equation and Its Properties, 545 11.3.4. Algorithms, 546 11.3.5. Unbounded Rewards, 547
CONTENTS xiv 11.4. Average Reward Models, 548 11.4.1. Model Formulation, 548 11.4.2. Policy Evaluation, 550 11.4.3. Optimality Equations, 554 11.4.4. Algorithms, 558 11.4.5. Countable-State Models, 559 11.5. Continuous-Time Markov Decision Processes, 560 11.5.1. Continuous-Time Markov Chains, 561 11.5.2. The Discounted Model, 563 11.5.3. The Average Reward Model, 567 11.5.4. Queueing Admission Control, 568 Bibliographic Remarks, 573 Problems, 574 Afterword 579 Notation 581 Appendix A. Markov Chains 587 A.l. A.2. A.3. A.4. A.5. Basic Definitions, 587 Classification of States, 588 Classifying the States of a Finite Markov Chain, 589 The Limiting Matrix, 591 Matrix Decomposition, the Drazin Inverse and the Deviation Matrix, 594 A.6. The Laurent Series Expansion of the Resolvent, 599 A.7. A. A. Markov, 600 Appendix B. Semicontinuous Functions Appendix C. Normed Linear Spaces C.l. C.2. C.3. C.4. 602 605 Linear Spaces, 605 Eigenvalues and Eigenvectors, 607 Series Expansions of Inverses, 607 Continuity of Inverses and Products, 609 Appendix D. Linear Programming 610 Bibliography Index 643
|
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| indexdate | 2024-07-09T20:04:01Z |
| institution | BVB |
| isbn | 9780471727828 0471727822 |
| language | English |
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| publishDate | 2005 |
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| spelling | Puterman, Martin L. 1945- Verfasser (DE-588)170238296 aut Markov decision processes discrete stochastic dynamic programming Martin L. Puterman, University of British Columbia Hoboken, New Jersey Wiley-Interscience [2005] xvii, 649 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Wiley-Interscience paperback series Dynamic programming Markov processes Statistical decision Markov-Entscheidungsprozess (DE-588)4168927-6 gnd rswk-swf Markov-Entscheidungsprozess (DE-588)4168927-6 s DE-604 Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013022843&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Puterman, Martin L. 1945- Markov decision processes discrete stochastic dynamic programming Dynamic programming Markov processes Statistical decision Markov-Entscheidungsprozess (DE-588)4168927-6 gnd |
| subject_GND | (DE-588)4168927-6 |
| title | Markov decision processes discrete stochastic dynamic programming |
| title_auth | Markov decision processes discrete stochastic dynamic programming |
| title_exact_search | Markov decision processes discrete stochastic dynamic programming |
| title_full | Markov decision processes discrete stochastic dynamic programming Martin L. Puterman, University of British Columbia |
| title_fullStr | Markov decision processes discrete stochastic dynamic programming Martin L. Puterman, University of British Columbia |
| title_full_unstemmed | Markov decision processes discrete stochastic dynamic programming Martin L. Puterman, University of British Columbia |
| title_short | Markov decision processes |
| title_sort | markov decision processes discrete stochastic dynamic programming |
| title_sub | discrete stochastic dynamic programming |
| topic | Dynamic programming Markov processes Statistical decision Markov-Entscheidungsprozess (DE-588)4168927-6 gnd |
| topic_facet | Dynamic programming Markov processes Statistical decision Markov-Entscheidungsprozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013022843&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT putermanmartinl markovdecisionprocessesdiscretestochasticdynamicprogramming |