Minimax Theory and its Applications 09 (2024), No. 2, 225--252
Copyright Heldermann Verlag 2024
Duality Assertions in Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces
Christian Günther
Institut für Angewandte Mathematik, Leibniz Universität, Hannover, Germany
c.guenther@ifam.uni-hannover.de
Bahareh Khazayel
Institute of Mathematics, Martin Luther University, Halle-Wittenberg, Germany
bahareh.khazayel@mathematik.uni-halle.de
Christiane Tammer
Institute of Mathematics, Martin Luther University, Halle-Wittenberg, Germany
christiane.tammer@mathematik.uni-halle.de
We derive duality assertions for vector optimization problems in real linear spaces based on a scalarization using recent results concerning the concept of relative solidness for convex cones (i.e., convex cones with nonempty intrinsic cores). In our paper, we consider an abstract vector optimization problem with generalized inequality constraints and investigate Lagrangian type duality assertions for (weak, proper) minimality notions. Our interest is neither to impose a pointedness assumption nor a solidness assumption for the convex cones involved in the solution concept of the vector optimization problem. We are able to extend the well-known Lagrangian vector duality approach by J. Jahn [Duality in vector optimization, Math. Programming 25 (1983) 343--353] to such a setting.
Keywords: Vector optimization, relatively solid convex cones, intrinsic core, minimality, efficiency, weak duality, strong duality, regularity.
MSC: 90C48, 90C29, 06F20, 52A05, 49N15.
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