First, any optimization problem has some objective: minimizing travel time, minimizing cost, maximizing profits, maximizing utility, etc. To understand the Bellman equation, several underlying concepts must be understood. Analytical concepts in dynamic programming Alternatively, it has been shown that if the cost function of the multi-stage optimization problem satisfies a "backward separable" structure, then the appropriate Bellman equation can be found without state augmentation. However, the resulting augmented-state multi-stage optimization problem has a higher dimensional state space than the original multi-stage optimization problem - an issue that can potentially render the augmented problem intractable due to the “ curse of dimensionality”. The appropriate Bellman equation can be found by introducing new state variables (state augmentation). In discrete time any multi-stage optimization problem can be solved by analyzing the appropriate Bellman equation. In continuous-time optimization problems, the analogous equation is a partial differential equation that is called the Hamilton–Jacobi–Bellman equation. The term 'Bellman equation' usually refers to the dynamic programming equation associated with discrete-time optimization problems. The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory though the basic concepts of dynamic programming are prefigured in John von Neumann and Oskar Morgenstern's Theory of Games and Economic Behavior and Abraham Wald's sequential analysis. The equation applies to algebraic structures with a total ordering for algebraic structures with a partial ordering, the generic Bellman's equation can be used. This breaks a dynamic optimization problem into a sequence of simpler subproblems, as Bellman's “principle of optimality" prescribes. It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision problem that results from those initial choices. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. ( April 2018) ( Learn how and when to remove this template message)Ī Bellman equation, named after Richard E. Please help to improve this article by introducing more precise citations. This article includes a list of general references, but it lacks sufficient corresponding inline citations.
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