Duality and Self-Duality for Conic Convex Programming

This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we introduce the notions of weak/strong feasibility or infeasibility for a general primal-dual pair of conic convex programs, and then establish various relations between these notions and the duality properties of the problem. In the second half of the paper, we propose a self-dual embedding with the following properties: Any weakly centered sequence converging to a complementary pair either induces a sequence converging to a certificate of strong infeasibility, or induces a sequence of primal-dual pairs for which the amount of constraint violation converges to zero, and the corresponding objective values are in the limit not worse than the optimal objective value(s). In case of strong duality, these objective values in fact converge to the optimal value of the original problem. When the problem is neither strongly infeasible nor endowed with a complementary pair, we completely specify the asymptotic behavior of an indicator in relation to the status of the original problem, namely whether the problem (1) is weakly infeasible, (2) is feasible but with a positive duality gap, (3) has no duality gap nor complementary solution pair.