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In this paper, firstly, we define a kind of new nonlinear functional with variable ordering structures and investigate its properties. Secondly, we establish Ekeland's variational principle with variable ordering structures. As the applications of Ekeland's variational principle, we obtain Caristi's fixed point theorem for set-valued maps and Takahashi's minimization theorem for set-valued maps with variable ordering structures. [ABSTRACT FROM AUTHOR]

Starting from the innovative ideas of Chicco et al. (Vector optimization problems via improvement sets. J. Optim. Theory Appl. 2011;150:516–529), in this paper, the concepts of improvement set and-efficiency are introduced in a real locally convex Hausdorff topological vector space. Furthermore, some properties of the improvement sets are given and a kind of proper efficiency, named as-Benson proper efficiency, which unifies some proper efficiency and approximate proper efficiency, is proposed via the improvement sets in vector optimization. Moreover, the concept of-subconvexlikeness of set-valued maps is introduced via the improvement sets and an alternative theorem is proved. In the end, some scalarization theorems and Lagrange multiplier theorems of-Benson proper efficiency are established for a vector optimization problem with set-valued maps. [ABSTRACT FROM PUBLISHER]