  We study the geometry of convex optimization problems given in a
Domain-Driven form and categorize possible statuses of these problems using
duality theory. Our duality theory for the Domain-Driven form, which accepts
both conic and non-conic constraints, lets us determine and certify statuses of
a problem as rigorously as the best approaches for conic formulations (which
have been demonstrably very efficient in this context). We analyze the
performance of an infeasible-start primal-dual algorithm for the Domain-Driven
form in returning the certificates for the defined statuses. Our iteration
complexity bounds for this more practical Domain-Driven form match the best
ones available for conic formulations. At the end, we propose some stopping
criteria for practical algorithms based on insights gained from our analyses.
