Abstract
In this paper, a dual problem in a wide sense is defined for a convex programming problem with multiple objective functions and the relationship between the primal problem and the dual problem is made clear.
All nondominated solutions with respect to the preference order specified by a convex cone in the objective function space are considered to be solutions of the original multiobjective programming problem.
First, it is shown that saddle points in an extended sense of the vector-valued Lagrangian provide solutions to the primal problem. Next, the primal problem is imbedded into a family of perturbed problems and a point-to-set map called perturbation map is defined as a map which gives optimal value sets of those problems. The meaning of the multiplier vector is made clear by the geometric study of the graph of this perturbation map. Furthermore, a class of problems called stable are defined by the perturbation map and their properties are investigated.
Finally, a point-to-set map called dual map is defined over the dual variable space by a minimal point set of the vector-valued Lagrangian of the primal problem. The dual problem in a wide sense is presented through this map. The relationship between the solutions of these two problems is given as a weak duality theorem and, moreover, it is shown for stable problems that there exists a solution of the dual problem corresponding to each solution of the primal problem (strong duality theorem).
