Search

In a real normed linear space, by applying algebraic interior of an ordering cone, a new kind of Minkowski functionals is introduced and their properties are discussed. A new kind of weakly efficient solutions, approximate weakly efficient solutions and approximate efficient solutions o f set optimization are introduced. It is proved that approximate efficient solutions and approximate weakly efficient solutions are generalizations of efficient solutions and weakly efficient solutions, respectivly, and the set of weakly efficient solutions is equal to the intersection of all sets of approximate weakly efficient solutions. By applying Minkowski nonlinear functionals, sufficient and necessary optimality conditions are established for approximate efficient solutions of set optimization. [ABSTRACT FROM AUTHOR]

In this paper, we employ the image space analysis to study constrained inverse variational inequalities by means of a nonlinear separation approach. We introduce such a nonlinear functional, which is based on the known Gerstewitz functional, and show its property as a weak separation function and a regular weak separation function under different parameter sets. In contrast to known results, we do not assume any topology on the considered spaces. Then, an alternative theorem is established, which leads directly to a sufficient and necessary optimality condition of the constrained inverse variational inequality. Finally, a gap function and an error bound are obtained for the constrained inverse variational inequality. [ABSTRACT FROM AUTHOR]

In this paper, we prove some properties of algebraic cone metric spaces and in- troduce the notion of algebraic distance in an algebraic cone metric space. As an application, we obtain some famous fixed point results in the framework of this algebraic distance. [ABSTRACT FROM AUTHOR]

In this paper, we show that the relative algebraic interior is a suitable replacement for both of the topological interior and the algebraic interior for the cases where these are empty. Also, we present some properties of (relative) algebraic interior and some fixed point theorems for increasing mapping. The obtained results can be viewed as an extension and improvement of the known corresponding results. Also, some examples are to support our conclusions considered. [ABSTRACT FROM AUTHOR]

In this paper, we focus on vector equilibrium problems whose final space is a real linear space not necessarily endowed with a topology. In this framework, some exact and approximate Ekeland variational principles are obtained. The main mathematical tool is a kind of strict fixed point theorem for set-valued mappings, from which the Ekeland variational principles are derived by means of algebraic notions and a concept of approximate solution for vector equilibrium problems based on free-disposal sets. The obtained results improve other recent ones of the literature, and clarify the role of some assumptions usually required on the vector bifunction to state Ekeland variational principles in vector equilibrium problems. [ABSTRACT FROM AUTHOR]

In this paper, the so-called nonconvex separation functional is studied in a real linear space not necessarily endowed with a topology. The essential properties of this functional, which are well known in the topological framework, are derived by using algebraic counterparts to the topological concepts of interior and closure. Finally, it is used to characterize, in the same algebraic setting and by scalarization, weak efficient solutions of vector equilibrium problems defined through algebraic solid ordering sets. [ABSTRACT FROM AUTHOR]

This paper explores new notions of approximate minimality in set optimization using a set approach. We propose characterizations of several approximate minimal elements of families of sets in real linear spaces by means of general functionals, which can be unified in an inequality approach. As particular cases, we investigate the use of the prominent Tammer–Weidner nonlinear scalarizing functionals, without assuming any topology, in our context. We also derive numerical methods to obtain approximate minimal elements of families of finitely many sets by means of our obtained results. [ABSTRACT FROM AUTHOR]