Second and Higher Order Duality

The purpose of this chapter is to show that the duality theory, which has evolved with the traditional (first order) duals and convexity assumptions, can be developed further in two ways: one is in a more general setting of a modified dual (namely, a second order and a higher order dual), the other is in the generalized convexity. The benefit of doing this not only that results obtained by these kinds of duals under generalized convexity extend some well-known classical results of (first order) duality for convex optimization problems, but also that higher order duality can provide a lower bound to the infimum of a primal optimization problem when it is difficult to find a feasible solution for the first order dual.