Lagrangian duality in convex conic programming with simple proofs

Abstract In this paper we study Lagrangian 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 strict feasibility and 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 sufficient conditions under which the duality gap is zero and the optimal solution sets are unbounded. As a consequence, we obtain several new sufficient conditions that guarantee the strong duality between primal and dual convex conic programs. Our proofs are based only on fundamental convex analysis and linear algebra results.