Optimization under Decision-Dependent Uncertainty

TL;DR

This paper extends robust optimization to include decision-dependent uncertainties, introduces new reformulations for better computational efficiency, and demonstrates advantages through a shortest path problem example.

Contribution

It generalizes robust linear optimization to decision-dependent uncertainties and proposes reformulations that reduce conservatism and improve computational performance.

Findings

Decision-dependent robust optimization problems are NP-complete.

New reformulations outperform standard linearization techniques.

Proactive uncertainty control reduces over conservatism.

Abstract

The efficacy of robust optimization spans a variety of settings with uncertainties bounded in predetermined sets. In many applications, uncertainties are affected by decisions and cannot be modeled with current frameworks. This paper takes a step towards generalizing robust linear optimization to problems with decision-dependent uncertainties. In general settings, we show these problems to be NP-complete. To alleviate the computational inefficiencies, we introduce a class of uncertainty sets whose size depends on binary decisions. We propose reformulations that improve upon alternative standard linearization techniques. To illustrate the advantages of this framework, a shortest path problem is discussed, where the uncertain arc lengths are affected by decisions. Beyond the modeling and performance advantages, the proposed notion of proactive uncertainty control also mitigates over conservatism of current robust optimization approaches.