Computational methods for set-relation-based scalarizing functions. (English) Zbl 1417.90135

Yu, Hui; Ike, Koichiro; Ogata, Yuto; Saito, Yutaka; Tanaka, Tamaki

Nihonkai Math. J. 28, No. 2, 139-149 (2017).

Summary: In this research, we propose computational methods to evaluate scalarizing functions, which are defined via set-relations. In recent years, many theoretical results of the scalarizing functions for sets have been published. The aim of this paper is to show that each value of the scalarizing functions can be computed and to introduce computational algorithms of them for convex polytopes in a finite dimensional space.

MSC:

90C29	Multi-objective and goal programming
46N10	Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
49J53	Set-valued and variational analysis
54C60	Set-valued maps in general topology

Keywords:

sublinear scalarization; set-relation; scalarizing function; computational method; set optimization

References:

[1]	G.-y. Chen, X. Huang and X. Yang, Vector optimization: Set-valued and variational analysis, Lecture Notes in Economics and Mathematical Systems, 541, Springer, Berlin, 2005. · Zbl 1104.90044
[2]	P. Gr. Georgiev and T. Tanaka, Vector-valued set-valued variants of Ky Fan’s inequality, J. Nonlinear Convex Anal. 1 (2000), 245-254. · Zbl 0987.49010
[3]	P. Gr. Georgiev and T. Tanaka, Fan’s inequality for set-valued maps, Nonlinear Anal. 47 (2001), 607-618. · Zbl 1042.49512 · doi:10.1016/S0362-546X(01)00205-X
[4]	C. Gerstewitz (C. Tammer), Nichtkonvexe dualität in der vektoroptimierung, Wiss. Z. Tech. Hochsch. Leuna-Merseburg 25 (1983), 357-364. · Zbl 0548.90081
[5]	C. Gerstewitz (C. Tammer) and E. Iwanow, Duality for nonconvex vector optimization problems, Wiss. Z. Tech. Hochsch. Ilmenau 31 (1985), 61-81. · Zbl 0562.90091
[6]	A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational methods in partially ordered spaces, Springer-Verlag, New York, 2003. · Zbl 1140.90007
[7]	A. H. Hamel, F. Heyde, A. Löhne, B. Rudloff and C. Schrage, Set Optimization and Applications - The State of the Art, Springer Proc. Math. Stat., 151, Springer, Heidelberg, 2015. · Zbl 1335.49003
[8]	A. H. Hamel and A. Löhne, Minimal element theorems and Ekeland’s principle with set relations, J. Nonlinear Convex Anal. 7 (2006), 19-37. · Zbl 1100.58007
[9]	D. Kuroiwa, T. Tanaka and T. X. D. Ha, On cone convexity of set-valued maps, Nonlinear Anal. 30 (1997), 1487-1496. · Zbl 0895.26010
[10]	I. Kuwano, T. Tanaka and S. Yamada, Characterizaton of nonlinear scalarizing functions for set-valued maps, in Proceedings of the Asian Conference on Nonlinear Analysis and Optimization, S. Akashi, W. Takahashi and T. Tanaka (eds.), Yokohama Publishers, Yokohama, 193-204, 2009. · Zbl 1222.49024
[11]	I. Kuwano, T. Tanaka and S. Yamada, Unified scalarization for sets and set-valued Ky Fan minimax inequality, J. Nonlinear Convex Anal. 11 (2010), 513-525. · Zbl 1221.49025
[12]	D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989.
[13]	S. Nishizawa and T. Tanaka, On inherited properties for set-valued maps and existence theorems for generalized vector equilibrium problems, J. Nonlinear Convex Anal. 5 (2004), 187-197. · Zbl 1052.49021
[14]	Y. Ogata, Y. Saito, T. Tanaka and S. Yamada, Sublinear scalarization methods for sets with respect to set-relations, Linear and Nonlinear Analysis 3 (2017), 121-132. · Zbl 1400.90272
[15]	Y. Sonda, T. Tanaka and S. Yamada, Computational methods on scalarizing functions for sets in a vector space, Session ID: B3-5 in “Proceedings of the Japan Joint Automatic Control Conference”, The Japan Joint Automatic Control Conference, Kyoto (Japan), 2010. DOI: 10.11511/jacc.52.0.137.0.