## A calculation approach to scalarization for polyhedral sets by means of set relations.(English)Zbl 1406.90113

Summary: In this paper, we focus on certain functions as scalarization for six types of set relations and discuss calculation algorithms for them between polyhedral sets, while those between polytopes have been already investigated. A major difference between polyhedral sets and polytopes is in boundedness. Polyhedral sets are no longer necessarily bounded. Methods for calculating types (1), (2), (4), (6) are easily available by a similar way to existing ideas. However, those for types (3) and (5), which are actually the most famous and long-standing types, require some technical ways approaching to the value of them by using the fact that finitely generatedness and polyhedrality coincide and can be algorithmically switched in finite-dimensional spaces. As a result, we show all types are reduced to a finite number of linear programming problems. Also, we demonstrate our methods through an example and give detailed calculation process.

### MSC:

 90C29 Multi-objective and goal programming 49J53 Set-valued and variational analysis
Full Text:

### References:

 [1] Y. Araya, Four types of nonlinear scalarizations and some applications in set optimization, Nonlinear Anal. 75 (2012), no. 9, 3821–3835. · Zbl 1237.49023 [2] G. B. Dantzig and B. C. Eaves, Fourier–Motzkin elimination and its dual, J. Combinatorial Theory Ser. A 14 (1973), 288–297. · Zbl 0258.15010 [3] M. Dhingra and C. S. Lalitha, Well-setness and scalarization in set optimization, Optim. Lett. 10 (2016), no. 8, 1657–1667. · Zbl 1391.90546 [4] A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 17, Springer-Verlag, New York, 2003. [5] C. Gutiérrez, B. Jiménez, E. Miglierina and E. Molho, Scalarization in set optimization with solid and nonsolid ordering cones, J. Global Optim. 61 (2015), no. 3, 525–552. · Zbl 1311.49041 [6] C. Gutiérrez, E. Miglierina, E. Molho and V. Novo, Pointwise well-posedness in set optimization with cone proper sets, Nonlinear Anal. 75 (2012), no. 4, 1822–1833. · Zbl 1237.49024 [7] E. Hernández and L. Rodríguez-Marín, Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl. 325 (2007), no. 1, 1–18. [8] K. Ike and T. Tanaka, Convex-cone-based comparisons of and difference evaluations for fuzzy sets, Optimization 67 (2018), no. 7, 1051–1066. · Zbl 1393.03031 [9] J. Jahn and T. X. D. Ha, New order relations in set optimization, J. Optim. Theory Appl. 148 (2011), no. 2, 209–236. · Zbl 1226.90092 [10] S. Khoshkhabar-amiranloo, E. Khorram and M. Soleimani-damaneh, Nonlinear scalarization functions and polar cone in set optimization, Optim. Lett. 11 (2017), no. 3, 521–535. · Zbl 1393.90109 [11] E. Köbis and M. A. Köbis, Treatment of set order relations by means of a nonlinear scalarization functional: a full characterization, Optimization 65 (2016), no. 10, 1805–1827. · Zbl 1384.90092 [12] D. Kuroiwa, T. Tanaka and T. X. D. Ha, On cone convexity of set-valued maps, Nonlinear Anal. 30 (1997), no. 3, 1487–1496. · Zbl 0895.26010 [13] I. Kuwano, T. Tanaka and S. Yamada, Unified scalarization for sets and set-valued Ky Fan minimax inequality, J. Nonlinear Convex Anal. 11 (2010), no. 3, 513–525. · Zbl 1221.49025 [14] Y. Ogata, T. Tanaka, Y. Saito, G. M. Lee and J. H. Lee, An alternative theorem for set-valued maps via set relations and its application to robustness of feasible sets, Optimization 67 (2018), no. 7, 1067–1075. · Zbl 1407.90238 [15] Y. D. Xu and S. J. Li, A new nonlinear scalarization function and applications, Optimization 65 (2016), no. 1, 207–231. · Zbl 1334.49062 [16] H. Yu, K. Ike, Y. Ogata, Y. Saito and T. Tanaka, Computational methods for set-relation-based scalarizing functions, Nihonkai Math. J. 28 (2017), no. 2, 139–149. · Zbl 1417.90135 [17] W. Y. Zhang, S. J. Li and K. L. Teo, Well-posedness for set optimization problems, Nonlinear Anal. 71 (2009), no. 9, 3769–3778. · Zbl 1165.90680 [18] G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995. · Zbl 0823.52002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.