(300c) Robust Optimization for Problems with Endogenous Uncertainty

Mitigation of the risk that stems from uncertainty in model parameters is important to take into account during decision-making via rigorous mathematical optimization. To this end, Robust Optimization (RO) has emerged as a systematic approach to handle such risk, by seeking to identify solutions that remain feasible under any realization of the parameters from within an uncertainty set [1]. This set, which is chosen by the modeler a-priori, constitutes a comprehensive collection of all parameter realizations that the modeler wishes to insure against, and it can be linked to a confidence interval for probabilistic constraint satisfaction [2].

Traditionally, the uncertainty set in RO is a constant set, being unaffected by the optimizer’s decisions. In many cases, however, this is not realistic, as one’s decisions may affect the level and type of risk one is actually exposed to. These cases are referred to as cases of endogenous uncertainty [3], and they have been investigated thoroughly in the Stochastic Programming literature [4–7]. In contrast, how to best handle endogenous uncertainty has been a more recent endeavor in the field of RO [8].

In this talk, we review the state-of-the-art in RO for problems with endogenous uncertainty, and we present novel decision-dependent uncertainty sets to model these settings [9] as well as a computational framework to optimize in light of such sets. We also present extensive computational results from a number of benchmark problems, including multi-stage problems that involve decision-making with recourse [10-11].

References:

[1] A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust optimization. Princeton University Press, 2009.

[2] Y. A. Guzman, L. R. Matthews, and C. A. Floudas, “New A Priori and A Posteriori Probabilistic Bounds for Robust Counterpart Optimization: I. Unknown Probability Distributions,” Comp. Chem. Eng., vol. 84, pp. 568–598, 2016.

[3] T. W. Jonsbråten, R. J. B. Wets, and D. L. Woodruff, “A class of stochastic programs with decision dependent random elements,” Ann. Oper. Res., vol. 82, pp. 83–106, 1998.

[4] V. Gupta and I. E. Grossmann, “A new decomposition algorithm for multistage stochastic programs with endogenous uncertainties,” Comput. Chem. Eng., vol. 62, pp. 62–79, 2014.

[5] V. Gupta and I. E. Grossmann, “Solution strategies for multistage stochastic programming with endogenous uncertainties,” Comput. Chem. Eng., vol. 35, no. 11, pp. 2235–2247, 2011.

[6] S. Solak, J. P. B. Clarke, E. L. Johnson, and E. R. Barnes, “Optimization of R&D project portfolios under endogenous uncertainty,” Eur. J. Oper. Res., vol. 207, no. 1, pp. 420–433, 2010.

[7] R. M. Apap and I. E. Grossmann, “Models and computational strategies for multistage stochastic programming under endogenous and exogenous uncertainties,” Manuscr. Prep., 2015.

[8] M. Poss, “Robust combinatorial optimization with variable cost uncertainty,” Eur. J. Oper. Res., vol. 237, no. 3, pp. 836–845, 2014.

[9] N. H. Lappas and C. E. Gounaris, “The Use of Decision-dependent Uncertainty Sets in Robust Optimization,” Proc. FOCAPO-2017, paper F71, pp. 1–6, 2017.

[10] D. Bertsimas and A. Georghiou, “Design of Near Optimal Decision Rules in Multistage Adaptive Mixed-Integer Optimization,” Oper. Res., vol. 63, no. 3, pp. 610–627, 2015.

[11] N. H. Lappas and C. E. Gounaris, “Multi-stage Adjustable Robust Optimization for Process Scheduling Under Uncertainty,” AIChE J., vol. 62, no. 5, pp. 1646–1667, 2016.