Regularized Methods for a Two-Stage Robust Production Planning Problem and its Sample Average Approximation

In this paper, we consider a two-stage robust production planning model where the first stage problem determines the optimal production quantity upon considering the worst-case revenue generated by the uncertain future demand, and the second stage problem determines the possible demand of consumers by using a utility-based model given the production quantity and a realization of the random variable. We derive an equivalent single-stage reformulation of the two-stage problem. However, it fails the convergence analysis of the sample average approximation (SAA) approach for the reformulation directly. Thus we develop a regularized approximation of the second stage problem and derive its closed-form solution. We then present conditions under which the optimal value and the optimal solution set of the proposed SAA regularized approximation problem converge to those of the single-stage reformulation problem as the regularization parameter shrinks to zero and the sample size tends to infinity. Finally, some preliminary numerical examples are presented to illustrate our theoretical results.