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Compact-Like Operators in Vector Lattices Normed by Locally Solid Lattices
Published in Hemen Dutta, Topics in Contemporary Mathematical Analysis and Applications, 2020
First of all, let us recall some notations and terminologies used in this paper. In this chapter, all vector spaces are supposed to be real. Let E be a vector space. Then, E is called ordered vector space if it has an order relation “≤” (i.e, it is reflexive, antisymmetric, and transitive) that is compatible with the algebraic structure of E,it means that y ≤ x implies y + z ≤ x + z for all z ∈ E and λ y ≤ λ x for each positive scalar λ ≥ 0.
Hahn-Banach and Duality Type Theorems for Vector Lattice-Valued Operators and Applications to Subdifferential Calculus and Optimization
Published in S. A. Mohiuddine, Bipan Hazarika, Sequence Space Theory with Applications, 2023
A partially ordered vector space R is called a vector lattice iff for each r1, r2∈R there is in R the supremum r1∨r2.
Preference relations and coradiants in ℝ n
Published in Journal of Control and Decision, 2023
Alireza Hosseini Dehmiry, Abbas Askarizadeh
Given a vector space V over the real numbers and a partial order ≼ on the set V, the pair is called a partially ordered vector space (or, ≼ is said to be compatible with the linear structure of V) if for all x, y, z in V and , the following axioms are satisfied. implies . implies .
Robust efficiency and well-posedness in uncertain vector optimization problems
Published in Optimization, 2023
In this formulation, the objective function depends on both decision variables and scenarios, which are not exactly known at the time a decision is made. As a result, evaluating a feasible solution according to conventional notions of efficiency in Definition 2.3 is no longer meaningful. Robust optimization is one of the popular methodologies for treating the case in which the scenario u is not exactly known. In this direction, the uncertain vector problem is now transformed into its robust counterpart, which is a (deterministic) set-valued optimization problem: where Based on the upper set less order relation (that is, for each pair of , if and only if ; see, for instance, [17]) Ehrgott et al. [18] introduced a concept of robust efficient solution to the robust counterpart of when , later Ide et al. [19] extended this concept to a partially ordered vector space with the partial order generated by a closed convex and pointed cone.
Fenchel–Rockafellar theorem in infinite dimensions via generalized relative interiors
Published in Optimization, 2023
D. V. Cuong, B. S. Mordukhovich, N. M. Nam, G. Sandine
Consider given by for all and the non-empty convex cone Suppose that is partially ordered by C. Then we can see that , , and the resulting ordering is not total however with this partial ordering is Archimedean. One has Furthermore, consider the same function f and the lexicographical partial ordering on , which is induced by the non-empty convex cone C defined in (29). Here we have , , and the resulting ordered vector space is totally ordered while not being Archimedean. In this case, we have