Geodesic
Functions with uniform sublevel sets on cones
Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 2, pp. 56-64 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Extended real-valued functions on a real vector space with uniform sublevel sets are important in optimization theory. Weidner studied these functions in [1]. In the present paper, we study the class of these functions, which coincides with the class of Gerstewitz functionals, on cones. These cone are not necessarily embeddable in vector spaces. Almost any Weidner's results are not true on cones without extra conditions. We show that the mentioned conditions are necessary, by nontrivial examples. Specially for element k from the cone $\mathcal{P}$, we define $k$-directional closed subsets of the cone and prove some properties of them. For a subset $A$ of the cone $\mathcal{P}$, we characterize domain of the $\varphi_{A,k}$ (function with uniform sublevel set) and show that this function is $k$-transitive. One of the important conditions for satisfying the results, is that $k$ has the symmetric element in the cone. Also, we prove that, under some conditions, the class of Gerstewitz functionals coincides with the class of $k$-translative functions on $\mathcal{P}$. 
@article{VMJ_2023_25_2_a4,
     author = {A. Dastouri and A. Ranjbari},
     title = {Functions with uniform sublevel sets on cones},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {56--64},
     year = {2023},
     volume = {25},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2023_25_2_a4/}
}
TY  - JOUR
AU  - A. Dastouri
AU  - A. Ranjbari
TI  - Functions with uniform sublevel sets on cones
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2023
SP  - 56
EP  - 64
VL  - 25
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VMJ_2023_25_2_a4/
LA  - en
ID  - VMJ_2023_25_2_a4
ER  - 
%0 Journal Article
%A A. Dastouri
%A A. Ranjbari
%T Functions with uniform sublevel sets on cones
%J Vladikavkazskij matematičeskij žurnal
%D 2023
%P 56-64
%V 25
%N 2
%U http://geodesic.mathdoc.fr/item/VMJ_2023_25_2_a4/
%G en
%F VMJ_2023_25_2_a4
A. Dastouri; A. Ranjbari. Functions with uniform sublevel sets on cones. Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 2, pp. 56-64. http://geodesic.mathdoc.fr/item/VMJ_2023_25_2_a4/

[1] Weidner P., “Construction Functions with Uniform Sublevel Sets”, Optimization Letters, 12:1 (2018), 35–41 | DOI | MR | Zbl

[2] Gerstewitz Ch., Iwanow E., “Dualitat fur Nichtkonvexe Vektoroptimierungsprobleme”, Wiss. Z. Tech. Hochsch. Ilmenau, 31:2 (1985), 61–81 | MR | Zbl

[3] Gerth C., Weidner P., “Nonconvex Separation Theorems and Some Applications in Vector Optimization”, Journal of Optimization Theory and Applications, 67 (1990), 297–320 | DOI | MR | Zbl

[4] Weidner P., Ein Trennungskonzept und seine Anwendung auf Vektoroptimierungsverfahren, Habilitation Thesis, Martin Luther Universitat Halle-Wittenberg, 1990

[5] Köbis E., Köbis M., “Treatment of Set Order Relations by Means of a Nonlinear Scalarization Functional: a Full Characterization”, Optimization: A Journal of Mathematical Programming and Operations Research, 65:10 (2016), 1805–1827 | DOI | MR | Zbl

[6] Keimel K., Roth W., Ordered Cones and Approximation, Lecture Notes in Mathematics, 1517, Springer-Verlag, Heidelberg–Berlin–New York, 1992 | DOI | MR | Zbl

[7] Roth W., Operator-Valued Measures and Integrals for Cone-Valued Functions, Lecture Notes in Mathematics, 1964, Springer-Verlag, Berlin, 2009 | MR | Zbl

[8] Roth W., “Hahn-Banach Type Theorems for Locally Convex Cones”, Journal of the Australian Mathematical Society, 68:1 (2000), 104–125 | DOI | MR | Zbl

[9] Ayaseh D., Ranjbari A., “Locally Convex Quotient Lattice Cones”, Mathematische Nachrichten, 287:10 (2014), 1083–1092 | DOI | MR | Zbl

[10] Ayaseh D., Ranjbari A., “Bornological Locally Convex Cones”, Le Matematiche (Catania), 69:2 (2014), 267–284 | DOI | MR | Zbl

[11] Jafarizad S., Ranjbari A., “Openness and Continuity in Locally Convex Cones”, Filomat, 31:16 (2017), 5093–5103 | DOI | MR | Zbl