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When reading various papers about two-stage (or multi-stage) stochastic programs with recourse, a common representation of the non-anticipativity constraints is: $$ \sum_{i}H_ix_i=h, $$ where $i$ indexes the scenarios, and the matrices $H_i$ and vector $h$ are typically not specified (see, e.g., this paper by Carøe and Schultz).

Question: What are some common choices of $H_i$ and $h$ used to enforce non-anticipativity? What advantages and disadvantages do these choices have when used algorithmically (in decomposition approaches, etc.)?

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Edit: If one define $H_i$ as an operator, not necessarily a matrix, I can think of what happens in progressive hedging and go for something like: $H_i(x_i)=I(x_i−x^*)+1/2(x_i−x^*)^TI(x_i−x^*)$. If in the matrix format: $H_i=1/rIx_i$ and $h=x^*$.

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