Preface to the First Edition xv

Acknowledgments xvii

1 The Challenges of Dynamic Programming 1

1.1 A Dynamic Programming Example: A Shortest Path Problem, 2

1.2 The Three Curses of Dimensionality, 3

1.3 Some Real Applications, 6

1.4 Problem Classes, 11

1.5 The Many Dialects of Dynamic Programming, 15

1.6 What Is New in This Book?, 17

1.7 Pedagogy, 19

1.8 Bibliographic Notes, 22

2 Some Illustrative Models 25

2.1 Deterministic Problems, 26

2.2 Stochastic Problems, 31

2.3 Information Acquisition Problems, 47

2.4 A Simple Modeling Framework for Dynamic Programs, 50

2.5 Bibliographic Notes, 54

Problems, 54

3 Introduction to Markov Decision Processes 57

3.1 The Optimality Equations, 58

3.2 Finite Horizon Problems, 65

3.3 Infinite Horizon Problems, 66

3.4 Value Iteration, 68

3.5 Policy Iteration, 74

3.6 Hybrid Value-Policy Iteration, 75

3.7 Average Reward Dynamic Programming, 76

3.8 The Linear Programming Method for Dynamic Programs, 77

3.9 Monotone Policies*, 78

3.10 Why Does It Work?**, 84

3.11 Bibliographic Notes, 103

Problems, 103

4 Introduction to Approximate Dynamic Programming 111

4.1 The Three Curses of Dimensionality (Revisited), 112

4.2 The Basic Idea, 114

4.3 Q-Learning and SARSA, 122

4.4 Real-Time Dynamic Programming, 126

4.5 Approximate Value Iteration, 127

4.6 The Post-Decision State Variable, 129

4.7 Low-Dimensional Representations of Value Functions, 144

4.8 So Just What Is Approximate Dynamic Programming?, 146

4.9 Experimental Issues, 149

4.10 But Does It Work?, 155

4.11 Bibliographic Notes, 156

Problems, 158

5 Modeling Dynamic Programs 167

5.1 Notational Style, 169

5.2 Modeling Time, 170

5.3 Modeling Resources, 174

5.4 The States of Our System, 178

5.5 Modeling Decisions, 187

5.6 The Exogenous Information Process, 189

5.7 The Transition Function, 198

5.8 The Objective Function, 206

5.9 A Measure-Theoretic View of Information**, 211

5.10 Bibliographic Notes, 213

Problems, 214

6 Policies 221

6.1 Myopic Policies, 224

6.2 Lookahead Policies, 224

6.3 Policy Function Approximations, 232

6.4 Value Function Approximations, 235

6.5 Hybrid Strategies, 239

6.6 Randomized Policies, 242

6.7 How to Choose a Policy?, 244

6.8 Bibliographic Notes, 247

Problems, 247

7 Policy Search 249

7.1 Background, 250

7.2 Gradient Search, 253

7.3 Direct Policy Search for Finite Alternatives, 256

7.4 The Knowledge Gradient Algorithm for Discrete Alternatives, 262

7.5 Simulation Optimization, 270

7.6 Why Does It Work?**, 274

7.7 Bibliographic Notes, 285

Problems, 286

8 Approximating Value Functions 289

8.1 Lookup Tables and Aggregation, 290

8.2 Parametric Models, 304

8.3 Regression Variations, 314

8.4 Nonparametric Models, 316

8.5 Approximations and the Curse of Dimensionality, 325

8.6 Why Does It Work?**, 328

8.7 Bibliographic Notes, 333

Problems, 334

9 Learning Value Function Approximations 337

9.1 Sampling the Value of a Policy, 337

9.2 Stochastic Approximation Methods, 347

9.3 Recursive Least Squares for Linear Models, 349

9.4 Temporal Difference Learning with a Linear Model, 356

9.5 Bellman's Equation Using a Linear Model, 358

9.6 Analysis of TD(0), LSTD, and LSPE Using a Single State, 364

9.7 Gradient-Based Methods for Approximate Value Iteration*, 366

9.8 Least Squares Temporal Differencing with Kernel Regression*, 371

9.9 Value Function Approximations Based on Bayesian Learning*, 373

9.10 Why Does It Work*, 376

9.11 Bibliographic Notes, 379

Problems, 381

10 Optimizing While Learning 383

10.1 Overview of Algorithmic Strategies, 385

10.2 Approximate Value Iteration and Q-Learning Using Lookup Tables, 386

10.3 Statistical Bias in the Max Operator, 397

10.4 Approximate Value Iteration and Q-Learning Using Linear Models, 400

10.5 Approximate Policy Iteration, 402

10.6 The Actor-Critic Paradigm, 408

10.7 Policy Gradient Methods, 410

10.8 The Linear Programming Method Using Basis Functions, 411

10.9 Approximate Policy Iteration Using Kernel Regression*, 413

10.10 Finite Horizon Approximations for Steady-State Applications, 415

10.11 Bibliographic Notes, 416

Problems, 418

11 Adaptive Estimation and Stepsizes 419

11.1 Learning Algorithms and Stepsizes, 420

11.2 Deterministic Stepsize Recipes, 425

11.3 Stochastic Stepsizes, 433

11.4 Optimal Stepsizes for Nonstationary Time Series, 437

11.5 Optimal Stepsizes for Approximate Value Iteration, 447

11.6 Convergence, 449

11.7 Guidelines for Choosing Stepsize Formulas, 451

11.8 Bibliographic Notes, 452

Problems, 453

12 Exploration Versus Exploitation 457

12.1 A Learning Exercise: The Nomadic Trucker, 457

12.2 An Introduction to Learning, 460

12.3 Heuristic Learning Policies, 464

12.4 Gittins Indexes for Online Learning, 470

12.5 The Knowledge Gradient Policy, 477

12.6 Learning with a Physical State, 482

12.7 Bibliographic Notes, 492

Problems, 493

13 Value Function Approximations for Resource Allocation Problems 497

13.1 Value Functions versus Gradients, 498

13.2 Linear Approximations, 499

13.3 Piecewise-Linear Approximations, 501

13.4 Solving a Resource Allocation Problem Using Piecewise-Linear Functions, 505

13.5 The SHAPE Algorithm, 509

13.6 Regression Methods, 513

13.7 Cutting Planes*, 516

13.8 Why Does It Work?**, 528

13.9 Bibliographic Notes, 535

Problems, 536

14 Dynamic Resource Allocation Problems 541

14.1 An Asset Acquisition Problem, 541

14.2 The Blood Management Problem, 547

14.3 A Portfolio Optimization Problem, 557

14.4 A General Resource Allocation Problem, 560

14.5 A Fleet Management Problem, 573

14.6 A Driver Management Problem, 580

14.7 Bibliographic Notes, 585

Problems, 586

15 Implementation Challenges 593

15.1 Will ADP Work for Your Problem?, 593

15.2 Designing an ADP Algorithm for Complex Problems, 594

15.3 Debugging an ADP Algorithm, 596

15.4 Practical Issues, 597

15.5 Modeling Your Problem, 602

15.6 Online versus Offline Models, 604

15.7 If It Works, Patent It!, 606

Bibliography 607

Index 623