Dynamic Programming And Value-Function Approximation With Application To Optimal Consumption

Sequential decision problems are considered, where a reward additive over a number of stages has to be maximized. Instances arise in scheduling eets of vehicles, allocating resources, selling assets, optimizing transportation or telecommunication networks, inventory forecasting, financial planning, etc. At each stage, Dynamic Programming (DP) introduces the value function, which gives the value of the reward to be incurred at the next stage, as a function of the state at the current stage. The solution is formally obtained via recursive equations. However, closed-form solutions can be derived only in particular cases. We investigate how DP and suitable approximations of the value functions can be combined, providing a methodology to face high-dimensional sequential decision problems. Approximations of the value functions are considered, expressed as linear combinations of basis functions obtained from a "mother function" (e.g., the Gaussian), by varying some "inner parameters" (e.g., variance and center coordinates) [1-5]. The accuracies of such suboptimal solutions are estimated. It is shown that one can cope with the \curse of dimensionality" in value-function approximation (i.e., an exponential growth of the number of basis functions, required to guarantee a desired solution accuracy). The theoretical analysis is applied to a multidimensional version of the optimal consumption problem. (In the classical version, a consumer aims at maximizing the discounted value of the consumption of a good, given a time horizon, a sequence of interest rates, an initial wealth, and an income earned at each stage. Here, more consumers are considered.) The proposed approximation scheme is compared with classical linear approximators, i.e., linear combinations of a-priori fixed basis functions. It is shown via simulations that the our approach provides a better solution accuracy, the number of computational units being the same as in fixed-basis approximation