Decision dependent scenarios in Stochastic Programming

Question

I am new to the field of stochastic programming and saw that the general formulation of a two-stage stochastic programming problem is given by: $$\min_{x\in X}\{ g(x)= f(x) + E_{\xi}[Q(x,\xi)]\}$$

where $\xi$ is the realization of uncertain data. Techniques such as Benders' Decomposition can be applied to solve these problems. However, I did not come across examples where $\xi$ is dependent on our decision in the first stage $x$, i.e. we would deal with $\xi(x)$. Can the same solution methods be applied for such cases and if not, what are standard approaches to solve such decision dependent scenarios?

Answer 1

Stochastic programming with decision dependent uncertainty is far less a beaten track than stochastic programming with fully exogenous uncertainty.

A short, not well thought through, answer to your question would be: No, you cannot apply Benders (L-shaped method) to such problems directly, at least not in all cases.

A hopefully more useful answer would be to study the specific problem at hand. Particularly, how does your decision $x$ influence $\xi$? Is your $\xi$ continuous or discrete? (probably the latter since you say scenarios). There are at least two ways in which $x$ can influence $\xi$. It can influence the timing when $\xi$ is revealed (probably not your case), see for example this article. Or it can influence the distribution function or support of your $\xi$ (probably your case, I guess). In the latter case, one can still distinguish a couple of cases:

• A. Where you can obtain infinitely many different distributions of $\xi$ as a result of $x$
• B. Where you can obtain finitely many distributions

For class B. see this article and, particularly, this article. Both are presented for a multistage setting, but can equally well describe your case (after all you have a multistage problem with only two stages). You will probably see that in case B. you might be able to use Benders though within a broader scheme (e.g., you might have to solve several stochastic programs). In case A. things are a bit more involved and I am not sure there is much out there.