A new concept of semistrict quasiconvexity for vector functions

authored by: Christian Günther, Alexandru Orzan, Nicolae Popovici
Abstract: We establish a new concept of semistrict quasiconvexity for vector functions defined on a nonempty convex set in a real linear space X that take values in some real topological linear space Y, partially ordered by a proper solid convex cone C. The so-called semistrict C-quasiconvexity notion recovers the classical concept of semistrict quasiconvexity of scalar functions when (Formula presented.) and (Formula presented.). Additionally, analogous to the scalar scenario, if the cone C is closed, a vector function is both semistrictly C-quasiconvex and C-quasiconvex (in the sense of Luc, 1989) if and only if it is explicitly C-quasiconvex (in the sense of Popovici, 2007). Finally, we convey a characterization of semistrictly C-quasiconvex functions by means of scalar semistrictly quasiconvex functions that are compositions of the nonlinear scalarization functions introduced by Gerstewitz (Tammer) in 1983 with the initial vector function. In light of this characterization, the new concept of semistrict C-quasiconvexity seems to be a natural vector counterpart for the scalar concept of semistrict quasiconvexity.
Organisation(s): Institute of Applied Mathematics
Type: Article
Journal: OPTIMIZATION
ISSN: 0323-3898
Publication date: 08.2024
Publication status: E-pub ahead of print
Peer reviewed: Yes
ASJC Scopus subject areas: Control and Optimization, Applied Mathematics, Management Science and Operations Research
Electronic version(s): https://doi.org/10.1080/02331934.2024.2384919 (Access: Open)

BibTeX

@article{89a09f4ff4f84a50b26f423a3d308dfa,
title = "A new concept of semistrict quasiconvexity for vector functions",
abstract = "We establish a new concept of semistrict quasiconvexity for vector functions defined on a nonempty convex set in a real linear space X that take values in some real topological linear space Y, partially ordered by a proper solid convex cone C. The so-called semistrict C-quasiconvexity notion recovers the classical concept of semistrict quasiconvexity of scalar functions when (Formula presented.) and (Formula presented.). Additionally, analogous to the scalar scenario, if the cone C is closed, a vector function is both semistrictly C-quasiconvex and C-quasiconvex (in the sense of Luc, 1989) if and only if it is explicitly C-quasiconvex (in the sense of Popovici, 2007). Finally, we convey a characterization of semistrictly C-quasiconvex functions by means of scalar semistrictly quasiconvex functions that are compositions of the nonlinear scalarization functions introduced by Gerstewitz (Tammer) in 1983 with the initial vector function. In light of this characterization, the new concept of semistrict C-quasiconvexity seems to be a natural vector counterpart for the scalar concept of semistrict quasiconvexity.",
keywords = "Generalized convexity, nonlinear scalarization function, vector function",
author = "Christian G{\"u}nther and Alexandru Orzan and Nicolae Popovici",
note = "Publisher Copyright: {\textcopyright} 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.",
year = "2024",
month = aug,
doi = "10.1080/02331934.2024.2384919",
language = "English",
journal = "OPTIMIZATION",
issn = "0323-3898",
publisher = "Taylor and Francis Ltd.",
}