This paper is devoted to the Hamiltonian approach for extremal problems concerning convex (multi-valued) mapping. The approach exploits the concept of a Hamiltonian function permitting simplified proofs and useful mathematical insights. Moreover it provides in a duality framework a common point ox view upon the methods used. by Rockafellar, (the theory of convex processes), Pshenichnyi (the conjugate transformation method) and CASS (the symmetric duality scheme) to construct optimality conditions. The theory is used to develop a complete characterization of optimal solutions for multi-period convex programming problems.