Continuous approximation schemes for stochastic programs

One of the main methods for solving stochastic programs is approximation by discretizing the probability distribution. However, discretization may lose differentiability of expectational functionals. The complexity of discrete approximation schemes also increases exponentially as the dimension of the random vector increases. On the other hand, stochastic methods can solve stochastic programs with larger dimensions but their convergence is in the sense of probabilty one. In this paper, the authors study the differentiability property of stochastic two-stage programs and discuss continuous approximation methods for stochastic programs. They present several ways to calculate and estimate this derivative. The authors then design several continuous approximation schemes and study their convergence behavior and implementation. The methods include several types of truncation approximation, lower dimensional approximation and limited basis approximation.