  Mixed integer sets have a strong modeling capacity to describe practical
systems. Nevertheless, incorporating a mixed integer set often renders an
optimization formulation drastically more challenging to compute. In this
paper, we study how to effectively solve two-stage robust mixed integer
programs built upon decision-dependent uncertainty (DDU). We particularly
consider those with a mixed integer DDU set that has a novel modeling power on
describing complex dependence between decisions and uncertainty, and those with
a mixed integer recourse problem that captures discrete recourse decisions.
  Note that there has been very limited research on computational algorithms
for these challenging problems. To this end, several exact and approximation
solution algorithms are presented, which address some critical limitations of
existing counterparts (e.g., the relatively complete recourse assumption) and
greatly expand our solution capacity. In addition to theoretical analyses on
those algorithms' convergence and computational complexities, a set of
numerical studies are performed to showcase mixed integer structures' modeling
capabilities and to demonstrate those algorithms' computational strength.
