  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.
