A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds.