Risk Management in Stochastic Integer Programming

The author presents two concepts to handle the classic linear mixed-integer two-stage stochastic optimization problem. She describes mean-risk modeling and stochastic programming with first order dominance constraints. Both approaches are applied to optimize the operation of a dispersed generation system.
Two-stage stochastic optimization is a useful tool for making optimal decisions under uncertainty. Frederike Neise describes two concepts to handle the classic linear mixed-integer two-stage stochastic optimization problem: The well-known mean-risk modeling, which aims at finding a best solution in terms of expected costs and risk measures, and stochastic programming with first order dominance constraints that heads towards a decision dominating a given cost benchmark and optimizing an additional objective. For this new class of stochastic optimization problems results on structure and stability are proven. Moreover, the author develops equivalent deterministic formulations of the problem, which are efficiently solved by the presented dual decomposition method based on Lagrangian relaxation and branch-and-bound techniques. Finally, both approaches – mean-risk optimization and dominance constrained programming – are applied to find an optimal operation schedule for a dispersed generation system, a problem from energy industry that is substantially influenced by uncertainty.