  We consider a two-stage distributionally robust optimization (DRO) model with
multimodal uncertainty, where both the mode probabilities and uncertainty
distributions could be affected by the first-stage decisions. To address this
setting, we propose a generic framework by introducing a $\phi$-divergence
based ambiguity set to characterize the decision-dependent mode probabilities
and further consider both moment-based and Wasserstein distance-based ambiguity
sets to characterize the uncertainty distribution under each mode. We identify
two special $\phi$-divergence examples (variation distance and
$\chi^2$-distance) and provide specific forms of decision dependence
relationships under which we can derive tractable reformulations. Furthermore,
we investigate the benefits of considering multimodality in a DRO model
compared to a single-modal counterpart through an analytical analysis. We
provide a computational study over the facility location problem to illustrate
our results, which demonstrate that omission of multimodality and
decision-dependent uncertainties within DRO frameworks result in inadequately
performing solutions with worse in-sample and out-of-sample performances under
various settings.
