A Perturbation Approach to Vector Optimization Problems: Lagrange and Fenchel–Lagrange Duality

In this paper, we study a general minimization vector problem which is expressed in terms of a perturbation mapping defined on a product of locally convex Hausdorff topological vector spaces with values in another locally convex topological vector space. Several representations of the epigraph of the conjugate of the perturbation mapping are given, and then, variants vector Farkas lemmas associated with the system defined by this mapping are established. A dual problem and another so-called loose dual problem of the mentioned problem are defined and stable strong duality results between these pairs of primal–dual problems are established. The results just obtained are then applied to a general class of composed constrained vector optimization problems. For this class of problems, two concrete perturbation mappings are proposed. These perturbation mappings give rise to variants of dual problems including the Lagrange dual problem and several kinds of Fenchel–Lagrange dual problems of the problem under consideration. Stable strong duality results for these pairs of primal–dual problems are derived. Several classes of concrete vector (and scalar) optimization problems are also considered at the end of the paper to illustrate the significance of our approach.