(441g) An Algorithmic Cutting Plane Method for Solving Robust Optimization Problems with Endogenous Uncertainty
Traditionally in the RO literature, uncertainty sets are constructed under the assumption that the decision makerâs strategy cannot have an impact on the realization of uncertainty (âexogenous uncertaintyâ). However, there are well documented cases from the Stochastic Programming literature where the realization of uncertainty and/or its underlying distributional support can be affected by the decisions of the policy makers (âendogenous uncertaintyâ) , with examples including oilfield production planning [3,4], R&D portfolio optimization , capacity expansion , clinical trial planning  and network interdiction , among others.
Recently, Lappas & Gounaris  proposed the extension of the traditional polyhedral uncertainty sets to decision-dependent uncertainty sets (DDUS), and showcased the flexibility of the introduced modelling capabilities in terms of capturing the effects of endogenous uncertainty in a variety of relevant problems, while maintaining the favorable computational tractability characteristics of RO. The solution approach utilized was based on a reformulation of the original problem to a monolithic robust counterpart. While the presented methodology is sufficient for small sized problems and continuous valued uncertain parameters, it can be limiting if the reformulated model is of size prohibitive to be loaded in memory, or when the uncertainty set involves uncertain parameters of discrete nature. The limitations of the reformulation approach become even more pronounced when a decision rule scheme is desired in the context of multi-stage decision-making contexts .
To alleviate the above limitations of the reformulation approach, we propose a novel algorithmic cutting plane method for solving robust optimization problems with endogenous uncertainty. In particular, we develop a branch & bound algorithm that gradually enforces the robustness of the solution by hedging against violating scenarios that are relevant to the optimal decision strategy. A comprehensive computational study across various problems illustrates the computational advantages of the proposed algorithm over the old reformulation approach, while the impact of various algorithmic enhancements is also investigated. Finally, the flexibility of the new algorithmic approach is presented with problems involving discrete uncertain parameters and elaborate decision-rule schemes.
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