(411e) Multistage Robust Mixed-Integer Optimization Under Endogenous Uncertainty

Endogenous, i.e. decision-dependent, uncertainty has received increased interest in the stochastic programming community in recent years. In contrast to exogenous uncertainty, endogenous uncertainty is affected by implemented decisions in terms of the underlying distribution or the time of realization. In a robust optimization framework, endogenous uncertainty translates into decision-dependent uncertainty sets whose size and shape are affected by decisions. There has been little work on robust optimization with endogenous uncertainty, and most of the few existing contributions (Lappas & Gounaris, 2018; Nohadani & Sharma, 2018) only address the static case, which does not account for recourse.

In this work, we consider multistage robust optimization with mixed-integer recourse and decision-dependent uncertainty sets that can be altered in every stage. Based on a lifting technique proposed by Georghiou et al. (2015), an optimization framework is developed that allows us to consider both continuous and integer recourse, including recourse decisions that affect the uncertainty set. This significantly expands our capability to appropriately model endogenous uncertainty in robust optimization settings. With the introduction of binary decision rules and discontinuous piecewise linear decision rules for continuous recourse variables, we derive a tractable and effective reformulation of the problem. Finally, extensive computational experiments are performed to gain insights on the impact of endogenous uncertainty, the benefit of considering both continuous and integer recourse, and computational performance. Our results indicate that the level of conservatism in the solution can be significantly reduced if endogenous uncertainty and mixed-integer recourse are properly modeled.

References

Georghiou, A., Wiesemann, W., & Kuhn, D. (2015). Generalized decision rule approximations for stochastic programming via liftings. Mathematical Programming, 152(1-2), 301–338.

Lappas, N. H. & Gounaris, C. E. (2018). Robust optimization for decision-making under endogenousuncertainty. Computers and Chemical Engineering, 111, 252–266.

Nohadani, O. & Sharma, K. (2018). Optimization under decision-dependent uncertainty. SIAM Journal on Optimization, 28(2), 1773–1795.