A duality theorem for a convex programming problem in order complete vector lattices

Abstract

The problem “minimize f(x) -g(x)” is studied, where f is a convex function and g a concave function of a real vector space X into an order complete vector lattice Y. We define Y-valued functions f”(T) and gc( T) on the real vector space of linear mappings of X into Y and associate with the minimization problem the maximization problem “maximize gC(T) fC(T).” For X = R” and Y = R Fenchel [2, 31 showed that these programs are dual to each other. Our main result will be that, iff g is bounded below, the maximization problem has an optimal solution and the extreme values of both objective functions are equal. As a direct consequence we get a condition, saying that a point x is an optimal solution of our minimization problem if and only if f and g have a common subgradient at x. This result extends a theorem by Valadier [9], concerning the subdifferentiability of a convex function with values in an order complete vector lattice. Furthermore, we apply our theorem to prove an extension of the KuhnTucker Theorem for order complete vector lattices. To this we generalize an idea from Rockafellar [7]. Our result is similar to one given by Ritter [6]. An example is given, showing that the hypothesis of order completeness is necessary for our theorems.