Hahn-Banach theorems and subgradients of set-valued maps∗1

Abstract. Some new results which generalize the Hahn-Banach theorem from scalar or vector-valued case to set-valued case are obtained. The existence of the Borwein-strong subgradient and Yang-weak subgradient for set-valued maps are also proven. we present a new Lagrange multiplier theorem and a new Sandwich theorem for set-valued maps