Saddle point and duality in the optimization theory of convex set functions

Abstract

AbstractFor a set function G on an atomless finite measure space (X, , m), we define the subgradient, conjugate set of and conjugate functional of G. It is proved that a minimization problem of set function G has an optimal solution if and only if the Lagrangian on × L1(X, , m)has a saddle point (Ω0, f0) such thatwhere f0 is an element of the conjugate set (for the definition, see the later context).