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This 4th edition is a major revision of Vol. II of the leading two-volume dynamic programming textbook by Bertsekas, and contains a substantial amount of new material, as well asa reorganization of old material. The length has increased by more than 60% from the third edition, andmost of the old material has been restructured and/or revised. Volume II now numbers more than 700 pages and is larger in size than Vol. I. It can arguably be viewed as a new book!Approximate DP has become the central focal point of Vol. II, and occupies more than half of the book (the last two chapters, and large parts of Chapters 1-3). Thus one may also view Vol. II as a followup of the author's 1996 book ``Neuro-Dynamic Programming" (coauthored with John Tsitsiklis). The present book focuses to a great extenton new research that became available after 1996. On the other hand, the textbook style of the book has been preserved, and some material has been explained at an intuitive or informal level, while
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