Multistage Robust Mixed-Integer Optimization Under Endogenous Uncertainty

Abstract

Endogenous, i.e. decision-dependent, uncertainty has received increased interest in the stochastic programming community. In the robust optimization context, however, it has rarely been considered. This work addresses multistage robust mixed-integer optimization with decision-dependent uncertainty sets. The proposed framework allows us to consider both continuous and integer recourse, including recourse decisions that affect the uncertainty set. We derive a tractable reformulation of the problem by leveraging recent advances in the construction of nonlinear decision rules, and introduce discontinuous piecewise linear decision rules for continuous recourse. Computational experiments are performed to gain insights on the impact of endogenous uncertainty, the benefit of discrete 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.

A Unified Framework for Adjustable Robust Optimization with Endogenous Uncertainty

“…In this section, we present a decision rule approach to (approximately) solve the general multistage robust optimization problem under endogenous uncertainty, which reduces to the two-stage case when T = 2. The proposed approach is an extension of the method presented in Feng et al 48 and relies on the concept of lifted uncertainty 55 . Here, the main innovation is the use of auxiliary uncertain parameters to model decision-dependent nonanticipativity.…”

“…Avraamidou and Pistikopoulos 47 apply multiparametric programming to address endogenous uncertainty in two-stage robust optimization. Finally, Feng et al 48 consider the multistage case with both continuous and binary recourse as well as uncertainty sets that can be affected by decisions at every stage, for which a decision rule approach is applied to derive tractable approximations.…”

“…• A decision rule approach based on previous work 48 is employed to develop tractable approximations of the given problems, which involve both continuous and binary recourse variables. One major novelty of this work is the use of auxiliary uncertain parameters to model decision-dependent nonanticipativity in the multistage setting.…”