Abstract
In this paper we extend the notion of minimal solutions for a vector optimization problem considered by Flores-Bazán et al. (J Optim Theory Appl 164:455–478, 2015) to a set-valued optimization problem, with both vector and set solution criteria. Also, we extend the Gerstewitz function proposed by Hernández and Rodríguez-Marín (J Math Anal Appl 325:1–18, 2007) and use it to scalarize minimal solutions with respect to set criterion. We also provide an existence result of minimal solutions with set criterion. Finally, we investigate links between the minimal solutions with respect to vector criterion and set criterion.
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References
Araya, Y.: Four types of nonlinear scalarizations and some applications in set optimization. Nonlinear Anal. 75, 3821–3835 (2012)
Bao, T.Q., Mordukhovich, B.S.: Set-valued optimization in welfare economics. Adv. Math. Econ. 13, 113–153 (2010)
Dhingra, M., Lalitha, C.: Set optimization using improvement sets. Yugosl. J. Oper. Res. 27, 153–167 (2017)
Flores-Bazán, F., Hernández, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization 60, 1399–1419 (2011)
Flores-Bazán, F., Flores-Bazán, F., Laengle, S.: Characterizing efficiency on infinite-dimensional commodity spaces with ordering cones having possibly empty interior. J. Optim. Theory Appl. 164, 455–478 (2015)
Flores-Bazán, F., Gutiérrez, C., Novo, V.: A Brézis-Browder principle on partially ordered spaces and related ordering theorems. J. Math. Anal. Appl. 375, 245–260 (2011)
Flores-Bazán, F., Hernández, E., Novo, V.: Characterizing efficiency without linear structure: a unified approach. J. Glob. Optim. 41, 43–60 (2008)
Geoffroy, M.H., Marcelin, Y., Nedelcheva, D.: Convergence of relaxed minimizers in set optimization. Optim. Lett. 11, 1677–1690 (2017)
Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)
Gerstewitz, C., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau 31, 61–81 (1985)
Hamel, A., Löhne, A.: Minimal element theorems and Ekeland’s principle with set relations. J. Nonlinear Convex Anal. 7, 19–37 (2006)
Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5, 1–28 (2011)
Han, Y., Huang, N.-J.: Continuity and convexity of a nonlinear scalarizing function in set optimization problems with applications. J. Optim. Theory Appl. 177, 679–695 (2018)
Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)
Hernández, E., Rodríguez-Marín, L., Sama, M.: Some equivalent problems in set optimization. Oper. Res. Lett. 37, 61–64 (2009)
Hernández, E., Rodríguez-Marín, L., Sama, M.: On solutions of set-valued optimization problems. Comput. Math. Appl. 60, 1401–1408 (2010)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, vol. 305, 1st edn. Springer, Berlin (1993)
Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Springer, Berlin (2015)
Khoshkhabar-amiranloo, S., Khorram, E.: Scalarization of Levitin–Polyak well-posed set optimization problems. Optimization 66, 113–127 (2017)
Khoshkhabar-amiranloo, S., Khorram, E.: Pointwise well-posedness and scalarization in set optimization. Math. Methods Oper. Res. 82, 195–210 (2015)
Khushboo, Lalitha, C.S.: Scalarizations for a unified vector optimization problem based on order representing and order preserving properties. J. Glob. Optim. 70, 903–916 (2018)
Kuroiwa, D.: Existence theorems of set optimization with set-valued maps. J. Inf. Optim. Sci. 24, 73–84 (2003)
Makarov, V.L., Levin, M.J., Rubinov, A.M.: Mathematical Economic Theory: Pure and Mixed Types of Economic Mechanisms. Advanced Textbooks in Economics, vol. 33. North-Holland Publishing Co., Amsterdam (1995)
Rubinov, A.M., Singer, I.: Topical and sub-topical functions, downward sets and abstract convexity. Optimization 50, 307–351 (2001)
Rubinov, A.M., Gasimov, R.N.: Scalarization and nonlinear scalar duality for vector optimization with preferences that are not necessarily a pre-order relation. J. Glob. Optim. 29, 455–477 (2004)
Sach, P.H.: New nonlinear scalarization functions and applications. Nonlinear Anal. 75, 2281–2292 (2012)
Sach, P.H., Tuan, L.A.: New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems. J. Optim. Theory Appl. 157, 347–364 (2013)
Shimizu, A., Nishizawa, S., Tanaka, T.: Optimality conditions in set-valued optimization using nonlinear scalarization methods. In: Proceedings of the 4th International Conference on Nonlinear Analysis and Convex Analysis (Okinawa, 2005), Yokohama Publishers, Yokohama, 565–574 (2007)
Acknowledgements
The authors wish to thank referee and associate editor for their suggestions which helped to improve the presentation of this article. This research, for the second author, was supported by MATRIC scheme of Department of Science and Technology, India.
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Khushboo, Lalitha, C.S. A unified minimal solution in set optimization. J Glob Optim 74, 195–211 (2019). https://doi.org/10.1007/s10898-019-00740-x
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DOI: https://doi.org/10.1007/s10898-019-00740-x