Smooth Sample Average Approximation Method For Nonsmooth Stochastic Programming With Constraints

Inspired by a recent work by Xu and Zhang which proposes a smoothing sample average approximation (SAA) method for solving a general class of one stage nonsmooth stochastic problems with abstract constraint, we firstly consider the case where the stochastic programming program may have locally Lipschitz inequality and equality constraints in addition. In order to solve such a problem, the SAA method is employed to approximate the expectations and the smoothing technique is used to deal with non-smoothness. Limiting behaviors of the proposed approach are discussed, for what an Jourani constraint qualification and stability of the smooth approximation are introduced for such discussions.Besides, inspired by the picnic-vender model discussed by Lin et al., we propose a new class of stochastic mathematical programs with equilibrium constraints which is named as Multi-Choice model.The main contribution of this paper as far as we are concerned can be summarized as follows: firstly we bring the smoothing SAA method in solving a general class of one stage nonsmooth stochastic problems with function constraints. We will see first that the true problem is stable under the smoothing perturbation if a given regular condition, which can be seen as an extension of the well known Robinson constraint qualification, holds with respect to the constraint system. And we will show that under moderate qualification, even though weakening the conditions on target function assumed by Xu and Zhang, we shall establish theorems that w.p.1 the K-K-T stationary points of smoothed SAA problem convergent to the weak K-K-T stationary points of the true problem and, in probability the optimal value of smoothed SAA problem convergent to that of the true problem. It is interesting that parts of the approaches how to weaken the assumptions had been checked in [45] but not applied in convergence analysis. Indeed, by providing sufficient conditions, we discuss almost all the conditions and assumptions used by Xu and Zhang and those introduced additionally for ourselves consideration.Generally, the paper is organized as follows. Nonsmooth stochastic programming problem is discussed in Chapter 1. In Section 1, we give some concepts and definitions used in our discussion. Then, stability of smooth approximation and w.p.1 the convergence analysis of smooth approximation Karush-Kuhn-Tucker(K-K-T) stationary points will be presented in Section 2, the convergence results of SAA K-K-T stationary points and in probability the convergence results of optimal values will be established in Section 3. In Section 4 we apply the smoothing SAA method and convergence result discussed in the previous sections to a stochastic mathematical program with equilibrium constraints. In Chapter 2, we introduce the origin Picnic-Vendor model, based on which the improved Multi-Choice model is established.