Exact penalization, level function method, and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints

摘要

Level function methods and cutting plane methods have been recently proposed to solve stochastic programs with stochastic second order dominance (SSD) constraints. A level function method requires an exact penalization setup because it can only be applied to the objective function, not the constraints. Slater constraint qualification (SCQ) is often needed for deriving exact penalization. It is well known that SSD usually does not satisfy SCQ and various relaxation schemes have been proposed so that the relaxed problem satisfies the SCQ. In this paper, we show that under some moderate conditions the desired constraint qualification can be guaranteed through some appropriate reformulation of the constraints rather than relaxation. Exact penalization schemes based on L1-norm and L1-norm are subsequently derived through Robinson’s error bound on convex system and Clarke’s exact penalty function theorem. Moreover, we propose a modified cutting plane method which constructs a cutting plane through the maximum of the reformulated constraint functions. In comparison with the existing cutting plane methods, it is numerically more efficient because only a single cutting plane is constructed and added at each iteration. We have carried out a number of numerical experiments and the results show that our methods display better performances particularly in the case when the underlying functions are nonlinear w.r.t. decision variables.