On subadditive duality for conic mixed-integer programs^∗

In this paper, we show that the subadditive dual of a feasible conic mixed-integer program (MIP) is a strong dual whenever it is feasible. Moreover, we show that this dual feasibility condition is equivalent to feasibility of the conic dual of the continuous relaxation of the conic MIP. In addition, we prove that all known conditions and other “natural” conditions for strong duality, such as strict mixed-integer feasibility, boundedness of the feasible set, or essentially strict feasibility, imply that the subadditive dual is feasible. As an intermediate result, we extend the so-called “finiteness property” from full-dimensional convex sets to intersections of full-dimensional convex sets and Dirichlet convex sets.