Abstract
In this paper, we characterize the nonemptiness of the set of weak minimal elements for a nonempty subset of a linear space. Moreover, we obtain some existence results for a nonconvex set-valued optimization problem under weaker topological conditions.
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Adán, M., Novo, V.: Weak efficiency in vector optimization using a closure of algebraic type under some cone-convexlikeness. Eur. J. Oper. Res. 149, 641–653 (2003)
Adán, M., Novo, V.: Proper efficiency in vector optimization on real linear spaces. J. Optim. Theory Appl. 121, 515–540 (2004)
Chen, G., Huang, X., Yang, X.: Vector Optimization. Springer, Berlin (2005)
Conway, J.: A Course in Functional Analysis. Springer, New York (1985)
Dhingra, M., Lalitha, C.S.: Approximate solutions and scalarization in set-valued optimization. Optimization 66, 1793–1805 (2017)
Gerstewitz, Ch., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau. 31, 61–81 (1985)
Gutiérrez, C., Jiménez, B., Miglierina, E., Molho, E.: Scalarization in set optimization with solid and nonsolid ordering cones. J. Glob. Optim. 61, 525–552 (2015)
Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, 1822–1833 (2012)
Gutiérrez, C., Novo, V., Ródenas-Pedregosa, J.L., Tanaka, T.: Nonconvex separation functional in linear spaces with applications to vector equilibria. SIAM J. Optim. 26, 2677–2695 (2016)
Gutiérrez, C., Kassay, G., Novo, V., Ródenas-Pedregosa, J.L.: Ekeland variational principles in vector equilibrium problems. SIAM J. Optim. 27, 2405–2425 (2017)
Göpfert, A., Riahi, H., Tammer, Ch., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)
Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)
Holmes, R.B.: Geometric Functional Analysis and Its Applications. Springer, New York-Heidelberg (1975)
Jahn, J.: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin (2011)
Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Vector Optimization. Springer, Heidelberg (2015)
Kuroiwa, D.: The natural criteria in set-valued optimization. Surikaisekikenkyusho Kokyuroku 1031, 85–90 (1998)
Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)
Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499–524 (1984)
Qiu, J.H., He, F.: A general vectorial Ekeland’s variational principle with a \(p\)-distance. Acta Math. Sin. 29, 1655–1678 (2013)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Weidner, P.: Gerstewitz functionals on linear spaces and functionals with uniform sublevel sets. J. Optim. Theory Appl. 173, 812–827 (2017)
Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlinear Anal. 71, 3769–3778 (2009)
Acknowledgements
The authors would like to thank the associate editor and reviewers for their constructive comments, which helped us to improve the paper. We also thank Professor Nicolas Hadjisavvas and Professor Constantin Zalinescu who read our manuscript and provided us with valuable comments. The third and fourth authors were partially supported by a Grant from IPM (Nos. 96550414, 98460038).
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Chinaie, M., Fakhar, F., Fakhar, M. et al. Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem. J Glob Optim 75, 131–141 (2019). https://doi.org/10.1007/s10898-019-00810-0
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DOI: https://doi.org/10.1007/s10898-019-00810-0