Efficiencies and Pareto efficiencies of set-valued mappings on ordered spaces

Reference

• Bednarczuk E. Some stability results for vector optimization problems in partially topological vector spaces. In Proceedings of the First World Congress of Nonlinear Analysts (Tampa, Florida, 1992), pp. 2371–2382 (1992).
   Google Scholar
• Bednarczuk E. Berge-type theorems for vector optimization problems. Optimization. 1995;32:373–384. doi: 10.1080/02331939508844057
   Google Scholar
• Benker H. Upper and lower bounds for minimal norm problems under linear constraints. Banach Center Publ. 1985;14:35–45. doi: 10.4064/-14-1-35-45
   Google Scholar
• Benker H, Hamel A, Spitzner J, et al. A proximal point algorithm for location problems. In: Göpfert A., et al. editor. Methods of Multicriteria Decision theory. xxx: Deutsche Hochschulschriften 2398, Hänsel-Hohenhausen-Verlag; 1997. p. 203–211.
   Google Scholar
• Benker H, Hamel A, Tammer C. A proximal point algorithm for control approximation problems. Z Oper Res. 1996;43:261–280.
   Google Scholar
• Benker H, Hamel A, Tammer C. An algorithm for vectorial control approximation problems. In: Fandel G., Gal T., editor. Multiple Criteria Decision Making. Berlin: Springer; 1997. p. 3–12.
   Google Scholar
• Eichfelder G, Jahn J. Vector optimization problems and their solution concepts. In: Recent developments in vector optimization. Berlin: Springer; 2012. p. 1–27.
   Google Scholar
• Fan K. A generalization of Tychonoff’s fixed point theorem. Math Ann. 1961;142:305–310. doi: 10.1007/BF01353421
   Web of Science ®Google Scholar
• Gajek L, Zagrodny D. Countably orderable sets and their applications in optimization. Optimization. 1992;26:287–301. doi: 10.1080/02331939208843858
   Google Scholar
• Giannessi F. ed. Vector variational inequalities and vector equilibria. mathematical theories. Dordrecht Boston London: Kluwer Academic Publishers; 1999.
   Google Scholar
• Giannessi F, Mastroeni G, Pellegrini L. On the theory of vector optimization and variational inequalities. Image space analysis and separation. In: Giannessi F, editor. Vector variational inequalities and vector equilibria. mathematical theories. Dordrecht: Nonconvex Optimization and its Applications, Kluwer; 2000. p. 153–215.
   Google Scholar
• Goh CJ, Yang XQ. Scalarization methods for vector variational inequality. In: Giannessi F, editor. Vector variational inequalities and vector Equilibria. Mathematical Theories. Dordrecht: Nonconvex Optimization and its Applications, Kluwer; 2000. p. 217–232.
   Google Scholar
• Gong XH. Connectedness of efficient solution sets for set-valued maps in normed spaces. J Optim Theory Appl. 1994;83:83–96. doi: 10.1007/BF02191763
   Web of Science ®Google Scholar
• Gong XH, Fu WT, Liu W. Super efficiency for a vector equilibrium in locally convex topological vector spaces. In: Giannessi F, editor. Vector variational inequalities and vector equilibria. mathematical theories. Dordrecht: Nonconvex Optimization and its Applications, Kluwer; 2000. p. 233–252.
   Google Scholar
• Göpfert, A., Gerth (Tammer), C., Über die Skalarisierung und Dualisierung von Vektoroptimierungsproblemen. Z Anal Anwendungen, 5, 377–384 (1986). doi: 10.4171/ZAA/205
   Google Scholar
• Göpfert A, Tammer C. ε-approximate solutions and conical support points. A new Maximal Point Theorem ZAMM. 1995;75:595–596.
   Google Scholar
• Göpfert A, Tammer C. A new maximal point theorem. Z Anal Anwendungen. 1995;14:379–390. doi: 10.4171/ZAA/680
   Google Scholar
• Göpfert A, Tammer C. Maximal point theorems in product spaces and applications for multicriteria approximation problems. In: Haimes YY, Steuer RE, editor. Research and practice in multiple criteria decision making. Lecture Notes in Econom. and Math. Systems, 487. Berlin Heidelberg: Springer; 1999. p. 93–104.
   Google Scholar
• Göpfert A, Tammer C, Zălinescu C. A new minimal point theorem in product spaces. Z Anal Anwendungen. 1999;18:767–770. doi: 10.4171/ZAA/911
   Web of Science ®Google Scholar
• Göpfert A, Tammer C, Zălinescu C. On the vectorial Ekeland’s variational principle and minimal points in product spaces. Nonlinear Analysis. 2000;39:909–922. doi: 10.1016/S0362-546X(98)00255-7
   Web of Science ®Google Scholar
• Göpfert A, Riahi H, Tammer C, et al. Variational methods in partially ordered spaces. Berlin Heidelberg: Springer; 2009.
   Google Scholar
• Gorochowik BB, Kirillowa FM. About scalarization of vector optimization problems. (Russian) Dokl Akad Nauk Soviet Union. 1975;19:588–591.
   Web of Science ®Google Scholar
• Grecksch W, Heyde F, Isac G, et al. A characterization of approximate solutions of multiobjective stochastic optimal control problems. Optimization. 2003;52:153–170. doi: 10.1080/0233193031000079810
   Web of Science ®Google Scholar
• Ha TXD. On the existence of efficient points in locally convex spaces. J Global Optim. 1994;4:265–278. doi: 10.1007/BF01098361
   Web of Science ®Google Scholar
• Ha TXD. A note on a class of cones ensuring the existence of efficient points in bounded complete sets. Optimization 1994;31:141–152. doi: 10.1080/02331939408844011
   Google Scholar
• Jahn J. Vector optimization. theory, applications and extensions. Berlin: Springer; 2004.
   Google Scholar
• Jahn J, Ha TXD. New order relations in set optimization. J Optim Theory Appl. 2011;148(2):209–236. doi: 10.1007/s10957-010-9752-8
   Web of Science ®Google Scholar
• Jahn J, Khan AA. Generalized contingent epiderivatives in set-valued optimization: optimality conditions. Numer Funct Anal Optim. 2002;23(7-8):807–831. doi: 10.1081/NFA-120016271
   Web of Science ®Google Scholar
• Jahn J, Khan AA. The existence of contingent epiderivatives for set-valued maps. Appl Math Lett. 2003;16(8):1179–1185. doi: 10.1016/S0893-9659(03)90114-5
   Web of Science ®Google Scholar
• Khan AA, Tammer C, Zalinescu C. Set-valued optimization an introduction with applications. Berlin Heidelberg: Springer-Verlag; 2015.
   Google Scholar
• Li JL. Inductive properties of fixed point sets of mappings on posets and on partially ordered topological spaces. Fixed Point Theory and Appl. 2015;211; doi 10.1186/s13663-015-0461-8.
   Google Scholar
• Li JL, Tammer C. Set-valued optimization problems on ordered sets. Applied Set-Valued Analysis and Optimization. 2019;1(1):77–94.
   Google Scholar
• Li JL, Tammer C. Set-valued Optimization Problems on Ordered topological vector spaces, in preparation.
   Google Scholar
• Konnov IV, Yao JC. On the generalized vector variational inequality problem. J of Math Anal And Appl. 1997;206:42–58. doi: 10.1006/jmaa.1997.5192
   Web of Science ®Google Scholar
• Kuroiwa D. Existence theorems of set optimization with set-valued Maps. Japan: Manuscript Shimane University; 1997.
   Google Scholar
• Kuroiwa D. Natural criteria of set-valued optimization. Japan: Manuscript Shimane University; 1998; (1068), 164–170.
   Google Scholar
• Kuroiwa D. On natural criteria in set-valued optimization. RIMS Kokyuroku. 1998;1048:86–92.
   Google Scholar
• Mas-Colell A, Whinston M, Green J. Microeconomic theory. New York (NY): Oxford University Press; 1995.
   Google Scholar
• Park S. Recent applications of Fan-KKM theorem. Lecture notes in Math Anal Institute. 2013;1841:58–68.
   Google Scholar
• Xie L, Li JL, Petrusel A, et al. Order-clustered fixed point theorems and their applications to Pareto equilibrium problems. Fixed Point Theory. 2017;18(2):755–772. doi: 10.24193/fpt-ro.2017.2.61
   Web of Science ®Google Scholar
• Yu W, Li XB, Liou YC. Set convergence of non-convex vector optimization problem with variable ordering structure. Optimization. 2019; doi:10.1080/02331934.2018.1561696.
   Google Scholar