  The type of decision dependent uncertainties (DDUs) imposes a great challenge
in decision making, while existing methodologies are not sufficient to support
many real practices. In this paper, we present a systematic study to handle
this challenge in two-stage robust optimization~(RO). Our main contributions
include three sophisticated variants of column-and-constraint generation method
to exactly compute DDU-based two-stage RO. By a novel application of core
concepts of linear programming, we provide rigorous analyses on their
computational behaviors. Interestingly, in terms of the iteration complexity of
those algorithms, DDU-based two-stage RO is not more demanding than its
decision independent uncertainty (DIU) based counterpart. It is worth
highlighting a counterintuitive discovery that converting a DIU set into a DDU
set by making use of "deep knowledge" and then computing the resulting
DDU-based formulation may lead to a significant improvement. Indeed, as shown
in this paper, in addition to capturing the actual dependence existing in the
real world, DDU is a powerful and flexible tool to represent and leverage
analytical properties or simply domain expertise to achieve a strong solution
capacity. So, we believe it will open a new direction to solve large-scale DIU-
or DDU-based RO. Other important results include basic structural properties
for two-stage RO, an approximation scheme to deal with mixed integer recourse,
and a couple of enhancement techniques for the developed algorithms, as well as
an organized numerical study to help us appreciate all algorithms and
enhancement techniques' computational performances.
