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.)?
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 $$h=x^*$$.