Duality for Multiple Objective Convex Programs

A duality theory for the problem of finding the properly efficient values of a multiple objective convex program is obtained using arguments and concepts analogous to those used by Rockafellar for scalar valued problems. A key result is that x̄ minimizes a convex, vector valued function f if and only if there is a “zero-like” subgradient of f at x̄. Consequently, if a vector valued convex program with a given perturbation is represented by a convex bifunction, then the properly efficient values of the program are equal to the perturbation function evaluated at zero-like perturbations. A dual program is defined in terms of the adjoint bifunction. Certain closure conditions insure the absence of a duality gap. The dual variables are matrices with the interpretation that the (i, j)-th element is the value of the ith resource in terms of the jth objective.

A concept of optimality called Isermann efficiency is introduced and compared with efficiency and proper efficiency. Isermann efficiency generalizes a concept from Isermann's work on multiple objective linear programs.