Strong duality and boundedness in conic optimization

Unlike linear programming, it is well-known that some conditions are required to achieve strong duality between a primal-dual pair of conic programs. The most common and well-known of these conditions is full-dimensionality of the cones and strict feasibility of one of the problems, also referred to as the Slater constraint qualification (CQ). Other sufficient conditions in literature for strong duality include a closedness condition for the full-dimensional case and a minimal facial property for general cones. We show that the closedness condition is also sufficient for strong duality when the cones are low-dimensional. The key step is to establish upper bounds on the duality gap through the separation of certain proximal points from the closure of an adjoint image of the cones. A consequence is a collection of specific sufficient conditions for strong duality, one of them being the generalized Slater CQ for low-dimensional cones, a few others being in terms of strict feasibility of the recession cones, and another being boundedness of the feasible region as a universal CQ. We also give various algebraic characterizations of the recession cone and its polar, thereby leading to many necessary and sufficient conditions for a bounded feasible region and also a theorem of the alternative in terms of approximate feasibility of the problem. Finally, we establish that under the generalized Slater CQ, finiteness of one problem and feasibility of the other problem are equivalent. This not only implies sufficiency of the Slater CQ for strong duality but it also allows us to characterize the projection of a conic set onto a linear subspace using extreme rays of a closed convex cone that generalizes the projection cone for polyhedral sets.