Polynomial approximation method for stochastic programming.

摘要

Two stage stochastic programming is an important part in the whole area of stochastic programming, and is widely spread in multiple disciplines, such as financial management, risk management, and logistics. The two stage stochastic programming is a natural extension of linear programming by incorporating uncertainty into the model. This thesis solves the two stage stochastic programming using a novel approach. For most two stage stochastic programming model instances, both the objective function and constraints are convex but non-differentiable, e.g. piecewise-linear, and thereby solved by the first gradient-type methods. When encountering large scale problems, the performance of known methods, such as the stochastic decomposition (SD) and stochastic approximation (SA), is poor in practice. This thesis replaces the objective function and constraints with their polynomial approximations. That is because polynomial counterpart has the following benefits: first, the polynomial approximation will preserve the convexity; Second, the polynomial approximation will uniformly converge to the original objective/constraints with arbitrary accuracy; and third, the polynomial approximation will not only provide good estimation on the original objectives/functions but also their gradients/sub-gradients. All these features enable us to apply convex optimization techniques for large scale problems. Hence, the thesis applies SAA, polynomial approximation method and then steepest descent method in combination to solve the large-scale problems effectively and efficiently.