Adaptive Robust Optimization

In Chap. 2 , we have studied static robust optimization with the following robust inequality constraint f ( x , ξ ) ≤ 0 , ∀ ξ ∈ U , $$\displaystyle \begin{aligned} f(\boldsymbol {x},\boldsymbol {\xi })\le 0, \quad \forall \boldsymbol {\xi }\in \mathcal {U},{} \end{aligned} $$ where U $$\mathcal {U}$$ is an uncertainty set, ξ is the uncertain parameter taking values in U $$\mathcal {U}$$ , and x is the decision. This robust model seeks one robust decision x that is feasible for all uncertainty realizations in U $$\mathcal {U}$$ . This embodies the canonical meaning of “robustness” in decision making. We call such a robust decision a static policy. However, in many situations, static policy may not be the best choice. More specifically, (1) we may need to deal with equality constraints involving uncertain parameters, or (2) the uncertainty may have some dynamic structures that call for the decisions to adapt accordingly. Both situations can be illustrated by examples from electric power systems. For the first situation, we know that the electricity supply must be equal to the electricity demand all the time. In other words, there is an equality constraint between supply and demand. As the electricity demand, viewed as an uncertain parameter, fluctuates between peaks and troughs everyday, the generation decision must also follow the demand, i.e., must be a function of the demand. A static policy for electricity generation in this case would amount to a fixed supply for all varying levels of demand, which clearly would lead to violation of the balance between generation and consumption.