Journal of Convex Analysis 24 (2017), No. 2, 393--415
Copyright Heldermann Verlag 2017
Ekeland's Variational Principle for Vector Optimization with Variable Ordering Structure
Truong Quang Bao
Dept. of Mathematics and Computer Science, Northern Michigan University, 1401 Presque Isle Avenue, 1135 NSF Marquette, MI 49855, U.S.A.
btruong@nmu.edu
Gabriele Eichfelder
Institut für Mathematik, Technische Universität, Postfach 10 05 65, 98684 Ilmenau, Germany
gabriele.eichfelder@tu-ilmenau.de
Behnam Soleimani
Institut für Mathematik, Martin-Luther-Universität, Theodor-Lieser-Str. 5, 06120 Halle, Germany
behnam.soleimani@mathematik.uni-halle.de
Christiane Tammer
Institut für Mathematik, Martin-Luther-Universität, Theodor-Lieser-Str. 5, 06120 Halle, Germany
christiane.tammer@mathematik.uni-halle.de
There are many generalizations of Ekeland's variational principle for vector optimization problems with fixed ordering structures, i.e., ordering cones. These variational principles are useful for deriving optimality conditions, ε-Kolmogorov conditions in approximation theory, and ε-maximum principles in optimal control. Here, we present several generalizations of Ekeland's variational principle for vector optimization problems with respect to variable ordering structures. For deriving these variational principles we use nonlinear scalarization techniques. Furthermore, we derive necessary conditions for approximate solutions of vector optimization problems with respect to variable ordering structures using these variational principles and the subdifferential calculus by Mordukhovich.
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