Value Function

In optimization and optimal control, a value function gives the value attained by an objective function at an optimal solution, expressed only in terms of the parameters of the problem. In optimal control it represents the supremum of the objective over all admissible controls, interpreted for a dynamical system as the optimal payoff achievable from a given state over a time interval, and it acts as a cost-to-go function when the objective is a cost to be minimized. The value function satisfies the Hamilton-Jacobi-Bellman equation, the partial differential equation that encodes Bellman's principle of optimality, and it underlies both dynamic programming and reinforcement learning; in economics the analogous object is the indirect utility function.