**ISBN:** 1-886529-42-6, 978-1-886529-42-7

**Publication:** April 2013, 256 pages, hardcover

**Price:** $65.00

Table of Contents and Preface, Overview Slides

A research monograph providing a synthesis of old research on the foundations of dynamic programming, with the modern theory of approximate dynamic programming and new research on semicontractive models.

It aims at a unified and economical development of the core theory and algorithms of total cost sequential decision problems, based on the strong connections of the subject with fixed point theory. The analysis focuses on the abstract mapping that underlies dynamic programming and defines the mathematical character of the associated problem. The discussion centers on two fundamental properties that this mapping may have: monotonicity and (weighted sup-norm) contraction. It turns out that the nature of the analytical and algorithmic DP theory is determined primarily by the presence or absence of these two properties, and the rest of the problem's structure is largely inconsequential. New research is focused on two areas: 1) The ramifications of these properties in the context of algorithms for approximate dynamic programming, and 2) The new class of semicontractive models, exemplified by stochastic shortest path problems, where some but not all policies are contractive.

The book is an excellent supplement to several of our books: Dynamic Programming and Optimal Control (Athena Scientific, 2012), and Neuro-Dynamic Programming (Athena Scientific, 1996).

**From the review by Panos Pardalos (Optimization Methods and Software):**

"The mathematical development in this book is elegant and mathematically rigorous, relying on the power of abstraction to focus on the fundamentals. The monograph provides for the first time a comprehensive synthesis of the field, while presenting much new research, some of which is relevant to currently very active fields such as approximate dynamic programming. Many examples are sprinkled through the book, illustrating the unifying power of the theory and applying it to specific types of problems, such as discounted, stochastic shortest path, semi-Markov, minimax, sequential games, multiplicative, and risk-sensitive models. The book also includes end-of-chapter exercises (with complete solutions), which supplement the text with examples, counterexamples, and theoretical extensions."

"Like several other books by Bertsekas, this book is well-written, and well-suited for self-study. It can be used as a supplement to graduate dynamic programming classes. I highly recommend this book for everyone interested in dynamic optimization."

**From the review by Eugene Feinberg at Amazon.com:**

"Dimitri Bertsekas is also the author of "Dynamic Programming and Optimal Control," Athena Scientific, 2007, a comprehensive text in which most of the dynamic programming concepts and applications are presented in a way interesting and available to a large spectrum of readers from undergraduate students in business and engineering to researches in the field. He wrote many other wonderful books on dynamic programming, data networks, probability, numerical methods, and optimization. His books are well written, informative, and easy to read. This book carries all these merits."

"Historical notes, sources, and exercises are also included at the end of each chapter, and solutions to all exercises are provided. This, and the lucid exposition, makes this book ideal both for self-study and as a supplement for graduate-level courses/texts in dynamic programming or reinforcement learning. In addition, readers interested in pursuing research in dynamic programming can find new research directions mentioned in the introduction of the book."

**Among its features, the book:**

develops the algorithmic foundations of approximate dynamic programming within an abstract unifying framework

discusses algorithms for approximate dynamic programming within a broadly applicable setting

develops the connections of dynamic programming with fixed point theory

contains significant new research

Dimitri P. Bertsekas is McAfee Professor of Engineering at the Massachusetts Institute of Technology and a member of the prestigious United States National Academy of Engineering. He is the recipient of the 2001 A. R. Raggazini ACC education award and the 2009 INFORMS expository writing award. He has also received 2014 ACC Richard E. Bellman Control Heritage Award for "contributions to the foundations of deterministic and stochastic optimization-based methods in systems and control," the 2014 Khachiyan Prize for Life-Time Accomplishments in Optimization, and the SIAM/MOS 2015 George B. Dantzig Prize.

The material listed below can be freely downloaded, reproduced, and distributed.

- D. P. Bertsekas, "Robust Shortest Path Planning and Semicontractive Dynamic Programming", Lab. for Information and Decision Systems Report LIDS-P-2915, MIT, Jan. 2015.
- D. P. Bertsekas and H. Yu, "Stochastic Shortest Path Problems Under Weak Conditions", Lab. for Information and Decision Systems Report LIDS-P-2909, MIT, March 2015; to appear in Math. of OR.
- D. P. Bertsekas, "Value and Policy Iteration in Deterministic Optimal Control and Adaptive Dynamic Programming", Lab. for Information and Decision Systems Report LIDS-P-3174, MIT, May 2015 (revised Sept. 2015); to appear in IEEE Transactions on Neural Networks and Learning Systems.
- An updated version of Chapter 4 and a new Appendix B, of the author's Dynamic Programming book, Vol. II, which incorporate recent research on a variety of undiscounted problem abstract DP topics; (Related Lecture Slides).

**The following four papers have a strong connection to the book, and amplify on the analysis and the range of applications of the semicontractive models of Chapters 3 and 4:**

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