Multiobjective Duality for Convex-Linear Problems

Abstract

A multiobjective programming problem characterized by convex goal functions and linear inequality constraints is studied. The investigation aims to the construction of a multiobjective dual problem permitting the verification of strong duality as well as optimality conditions. For the original primal problem properly efficient (minimal) solutions are considered. This allows to deal with the linearly scalarized programming problem. Different from the usual Lagrange dual problem a dual problem for the scalarized is derived applying the Fenchel-Rockafellar duality approach and using special and appropriate perturbations. The dual problem is formulated in terms of conjugate functions. That dual problem has the advantage that its structure gives an idea for the formulation of a multiobjective dual problem to the original problem in a natural way. Considering efficient (maximal) solutions for that vector dual problem it succeeds to prove the property of so-called strong duality. Moreover, duality corresponds with necessary and sufficient optimality conditions for both the scalar and the multiobjective problems.