Advanced search
Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 67, 2018 - Issue 7: Set-Valued Optimization and Variational Analysis: Special Issue dedicated to Professor Johannes Jahn on the occasion of his 65th birthday
Submit an article Journal homepage
188
Views
6
CrossRef citations to date
0
Altmetric
Original Articles

An alternative theorem for set-valued maps via set relations and its application to robustness of feasible sets

Yuto OgataGraduate School of Science and Technology, Niigata University, Niigata, Japan.Correspondence[email protected]
ORCID Iconhttp://orcid.org/0000-0001-8985-4296View further author information
ORCID Icon,
Tamaki TanakaGraduate School of Science and Technology, Niigata University, Niigata, Japan.View further author information
,
Yutaka SaitoDepartment of Management Science and Engineering, Akita Prefectural University, Yurihonjo, Japan.View further author information
,
Gue Myung LeeDepartment of Applied Mathematics, Pukyong National University, Busan, Korea.View further author information
&
Jae Hyoung LeeDepartment of Applied Mathematics, Pukyong National University, Busan, Korea.View further author information
Pages 1067-1075 | Received 31 May 2017, Accepted 07 Jan 2018, Published online: 22 Jan 2018

References

  • Gordan P . Über die Auflösung linearer Gleichungen mit reellen coefficienten. Math Ann. 1873;6:23–28.
  • Jeyakumar V . A generalization of a minimax theorem of Fan via theorem of the alternative. J Optim Theory Appl. 1986;48:525–533.
  • Li Z . A theorem of the alternative and its application to the optimization of set-valued maps. J Optim Theory Appl. 1999;100:365–375.
  • Yang XM , Yang XQ , Chen GY . Theorems of the alternative and optimization with set-valued maps. J Optim Theory Appl. 2000;107:627–640.
  • Nishizawa S , Onodsuka M , Tanaka T . Alternative theorems for set-valued maps based on a nonlinear scalarization. Pacific J Optim. 2005;1:147–159.
  • Göpfert A , Riahi H , Tammer C , Zălinescu C . Variational methods in partially ordered spaces. New York (NY): Springer; 2003.
  • Gerstewitz C (Tammer C). Nichtkonvexe dualität in der vektoroptimierung. Wiss Tech Hochsch Leuna-Merseburg. 1983;25:357–364.
  • Hamel A , Löhne A . Minimal element theorems and Ekeland’s principle with set relations. J Nonlinear Convex Anal. 2006;7:19–37.
  • Kuwano I , Tanaka T , Yamada S . Characterization of nonlinear scalarizing functions for set-valued maps. In: Proceedings of the Asian Conference on Nonlinear Analysis and Optimization. Yokohama: Yokohama Publishers; 2008. p. 193–204.
  • Jahn J , Ha TXD . New order relations in set optimization. J Optim Theory appl. 2011;148:209–236.
  • Kuroiwa D , Tanaka T , Ha TXD . On cone convexity of set-valued maps. Nonlinear Anal. 1997;30:1487–1496.
  • Gutiérrez C , Jiménez B , Miglierina E , Molho E . Scalarization in set optimization with solid and nonsolid ordering cones. J Glob Optim. 2015;61:525–552.
  • Chen G-y , Huang X , Yang X . Vector optimization. Set-valued and variational analysis. Vol. 541, Lecture notes in economics and mathematical systems. Berlin: Springer; 2005.
  • Yu H , Ike K , Ogata Y , Saito Y , Tanaka T . Computational methods for set-relation-based scalarizing functions. Nihonkai Math J. 2017;28:139–149.
  • Borwein JM , Lewis AS . Convex analysis and nonlinear optimization. New York (NY): Springer; 2000.
  • de Klerk E . Aspects of semidefinite programming: interior point algorithms and selected applications. Dordrecht: Kluwer; 2002.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.