Suboptimal Policies for Stochastic $$N$$ N -Stage Optimization: Accuracy Analysis and a Case Study from Optimal Consumption

Dynamic Programming formally solves stochastic optimization problems with an objective that is additive over a finite number of stages. However, it provides closed-form solutions only in particular cases. In general, one has to resort to approximate methodologies. In this chapter, suboptimal solutions are searched for by approximating the decision policies via linear combinations of Gaussian and sigmoidal functions containing adjustable parameters, to be optimized together with the coefficients of the combinations. These approximation schemes correspond to Gaussian radial-basis-function networks and sigmoidal feedforward neural networks, respectively. The accuracies of the suboptimal solutions are investigated by estimating the error propagation through the stages. As a case study, we address a multidimensional problem of optimal consumption under uncertainty, modeled as a stochastic optimization task with an objective that is additive over a finite number of stages. In the classical one-dimensional context, a consumer aims at maximizing over a given time horizon the discounted expected value of consumption of a good, where the expectation is taken with respect to a stochastic interest rate. The consumer has an initial wealth and at each time period earns an income, modeled as an exogenous input. We consider a multidimensional framework, in which there are $$d > 1$$ d > 1 consumers that aim at maximizing a social utility function. First we provide conditions that allow one to apply our estimates to such a problem; then we present a numerical analysis.