A pair of Wolfe type non-differentiable second order symmetric primal and dual problems in mathematical programming is formulated. The weak and strong duality theorems are then established under second order F-convexity assumptions. Symmetric minimax mixed integer primal and dual problems are also investigated.
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.