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Dynamic Programming

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Table of Contents

Introduction Welcome to Dynamic Programming! How to Read This Book SCIENCE Fundamentals Introduction Meta-Recipe Revisited Problem Formulation Decomposition of the Solution Set Principle of Conditional Optimization Conditional Problems Optimality Equation Solution Procedure Time Out: Direct Enumeration! Equivalent Conditional Problems Modified Problems The Role of a Decomposition Scheme Dynamic Programming Problem - Revisited Trivial Decomposition Scheme Summary and a Look Ahead Multistage Decision Model Introduction A Prototype Multistage Decision Model Problem vs Problem Formulation Policies Markovian Policies Remarks on the Notation Summary Bibliographic Notes Dynamic Programming - An Outline Introduction Preliminary Analysis Markovian Decomposition Scheme Optimality Equation Dynamic Programming Problems The Final State Model Principle of Optimality Summary Solution Methods Introduction Additive Functional Equations Truncated Functional Equations Nontruncated Functional Equations Summary Successive Approximation Methods Introduction Motivation Preliminaries Functional Equations of Type One Functional Equations of Type Two Truncation Method Stationary Models Truncation and Successive Approximation Summary Bibliographic Notes Optimal Policies Introduction Preliminary Analysis Truncated Functional Equations Nontruncated Functional Equations Successive Approximation in the Policy Space Summary Bibliographic Notes The Curse of Dimensionality Introduction Motivation Discrete Problems Special Cases Complete Enumeration Conclusions The Rest Is Mathematics and Experience Introduction Choice of Model Dynamic Programming Models Forward Decomposition Models Practice What You Preach! Computational Schemes Applications Dynamic Programming Software Summary ART Refinements Introduction Weak-Markovian Condition Markovian Formulations Decomposition Schemes Sequential Decision Models Example Shortest Path Model The Art of Dynamic Programming Modeling Summary Bibliographic Notes The State Introduction Preliminary Analysis Mathematically Speaking Decomposition Revisited Infeasible States and Decisions State Aggregation Nodes as States Multistage vs Sequential Models Models vs Functional Equations Easy Problems Modeling Tips Concluding Remarks Summary Parametric Schemes Introduction Background and Motivation Fractional Programming Scheme C-Programming Scheme Lagrange Multiplier Scheme Summary Bibliographic Notes The Principle of Optimality Introduction Bellman's Principle of Optimality Prevailing Interpretation Variations on a Theme Criticism So What Is Amiss? The Final State Model Revisited Bellman's Treatment of Dynamic Programming Summary Post Script: Pontryagin's Maximum Principle Forward Decomposition Introduction Function Decomposition Initial Problem Separable Objective Functions Revisited Modified Problems Revisited Backward Conditional Problems Revisited Markovian Condition Revisited Forward Functional Equation Impact on the State Space Anomaly Pathologic Cases Summary and Conclusions Bibliographic Notes Push! Introduction The Pull Method The Push Method Monotone Accumulated Return Processes Dijkstra's Algorithm Summary Bibliographic Notes EPILOGUE What Then Is Dynamic Programming? Review Non-Optimization Problems An Abstract Dynamic Programming Model Examples The Towers of Hanoi Problem Optimization-Free Dynamic Programming Concluding Remarks Appendix A: Contraction Mapping Appendix B: Fractional Programming Appendix C: Composite Concave Programming Appendix D: The Principle of Optimality in Stochastic Processes Appendix E: The Corridor Method Bibliography Index

About the Author

Moshe Sniedovich is a Principal Fellow (Associate) in the Department of Mathematics and Statistics at the University of Melbourne in Australia. Dr. Sniedovich has worked at the Israel Ministry of Agriculture, University of Arizona, Princeton University, IBM TJ Watson Research Center, and South Africa National Research Institute for Mathematical Sciences. He earned his B.Sc. from Technion and his Ph.D. from the University of Arizona.

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