Over the past decade, decision diagrams (DDs) have been used to model and
solve integer programming and combinatorial optimization problems. Despite
successful performance of DDs in solving various discrete optimization
problems, their extension to model mixed integer programs (MIPs) such as those
appearing in energy applications has been lacking. More broadly, the question
which problem structures admit a DD representation is still open in the DDs
community. In this paper, we address this question by introducing a geometric
decomposition framework based on rectangular formations that provides both
necessary and sufficient conditions for a general MIP to be representable by
DDs. As a special case, we show that any bounded mixed integer linear program
admits a DD representation through a specialized Benders decomposition
technique. The resulting DD encodes both integer and continuous variables, and
therefore is amenable to the addition of feasibility and optimality cuts
through refinement procedures. As an application for this framework, we develop
a novel solution methodology for the unit commitment problem (UCP) in the
wholesale electricity market. Computational experiments conducted on a
stochastic variant of the UCP show a significant improvement of the solution
time for the proposed method when compared to the outcome of modern solvers