Pragmatic convex approaches for risk-averse and distributionally robust mixed-integer recourse models

In this thesis I consider two classes of stochastic optimization models: risk-averse mixed-integer recourse (MIR) models and distributionally robust MIR models. These classes of models can be used to support decision making in situations where uncertainty about the future plays an important role. For example, one might need to invest in a new production facility while future demand for the produced goods is uncertain. Typically, these two classes of MIR models are non-convex as a result of the integer restrictions in the model. This makes these models extremely hard to solve from a computational point of view. My aim is to overcome this issue and find efficient solution approaches.

In this thesis, I propose pragmatic convex approaches for risk-averse and distributionally robust MIR models. These approaches are based on the pragmatic idea of solving a convex model in order to find a reasonably good solution to the original, non-convex, model. For risk-averse MIR models this is achieved by constructing a convex approximation model, which can be solved efficiently. I prove that the approximation model is close to the original model in settings where probability distribution of the uncertain parameters is highly dispersed and I explicitly show the effect of the selected risk measure on the quality of the approximation. For distributionally robust MIR models I propose to restrict the uncertainty set to a class of special "convexifying" distributions. The resulting model is convex and, in particular settings, also resolves overfitting issues.