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 Название: Abstract Dynamic Programming, 2nd Edition
Автор: Dimitri P. Bertsekas
Издательство: Athena Scientific
Год: 2018
Страниц: 360
Язык: английский
Формат: pdf (true), djvu
Размер: 10.1 MB
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.
Dynamic programming (DP for short) is the principal method for analysis of a large and diverse class of sequential decision problems. Examples are deterministic and stochastic optimal control problems with a continuous state space, Markov and semi-Markov decision problems with a discrete state space, minimax problems, and sequential zero-sum games. While the nature of these problems may vary widely, their underlying structures turn out to be very similar. In all cases there is an underlying mapping that depends on an associated controlled dynamic system and corresponding cost per stage. This mapping, the DP operator, provides a compact “mathematical signature” of the problem. It defines the cost function of policies and the optimal cost function, and it provides a convenient shorthand notation for algorithmic description and analysis.
More importantly, the structure of the DP operator defines the mathematical character of the associated problem. The purpose of this book is to provide an analysis of this structure, centering on two fundamental properties: 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 one or both of these two properties, and the rest of the problem’s structure is largely inconsequential.
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