In this paper we study strong duality aspects in convex conic programming over general convex cones. It is known that the duality in convex optimization is linked with specific theorems of alternatives. We formulate and prove strong alternatives to the existence of the relative interior point in the primal (dual) feasible set. We analyze the relation between the boundedness of the optimal solution sets and the existence of the relative interior points in the feasible set. We also provide conditions under which the duality gap is zero and the optimal solution sets are unbounded. As a consequence, we obtain several alternative conditions that guarantee the strong duality between primal and dual convex conic programs.
Faculty of mathematics, physics and informatics, Comenius University in Bratislava