Suppose we have a two-stage stochastic program as follows: \begin{equation} \begin{split} \min \ & c^Tx + \mathbb{E}_\xi[Q(x,\xi)] \\ & \text{where}\\ &Q(x,\xi)=\min q(\xi)^Ty\\ &Tx+W(\xi)y=h(\xi)\\ &y \ge 0 \end{split} \end{equation} I am wondering how it is possible to find a set of values for the first-stage decisions such that in the L-Shape method, the same (optimality) cuts are generated. For example, suppose we have a problem like the follows:

\begin{equation} \begin{split} \min \ & 5x_1+10x_2 + \mathbb{E}_\xi[q_1y_1+q_2y_2] \\ & y_1 + 2y_2 \ge d_1 -x_1 \\ & y_1 \ge d_2 -2x_2\\ & 0 \le x_1 \le 10 \\ & 0 \le x_2\le 10 \\ & y_1,y_2 \ge 0 \end{split} \end{equation}

where $(q_1,q_2,d_1,d_2)$ takes two values $(20,10,5,10)$ and $(15,25,12,5)$ with equal probabilities.


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