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The relationship between these two value functions is called the “Bellman equation”. In this approach, the optimal policy in the last time period is specified in advance as a function of the state variable’s value at that time, and the resulting optimal value of the objective function is thus expressed in terms of that value of the state variable.
Can a Bellman equation be found without state augmentation?
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. To understand the Bellman equation, several underlying concepts must be understood.
How to calculate the first order conditions of the Bellman equation?
By calculating the first-order conditions associated with the Bellman equation, and then using the envelope theorem to eliminate the derivatives of the value function, it is possible to obtain a system of difference equations or differential equations called the ‘Euler equations’.
How is the optimal policy in the last time period expressed?
In this approach, the optimal policy in the last time period is specified in advance as a function of the state variable’s value at that time, and the resulting optimal value of the objective function is thus expressed in terms of that value of the state variable.
How does Bellman write the value of a decision problem?
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. This breaks a dynamic optimization problem into a sequence of simpler subproblems, as Bellman’s “principle of optimality” prescribes.
Is the Bellman equation a partial differential equation?
In continuous-time optimization problems, the analogous equation is a partial differential equation that is called the Hamilton–Jacobi–Bellman equation. In discrete time any multi-stage optimization problem can be solved by analyzing the appropriate Bellman equation.