Abstract
In this paper, we introduce three types of well-posedness for a set optimization problem (u-SOP). Some necessary and sufficient conditions for these well-posedness have been established. Two different scalar optimization problems involving a generalized oriented distance function have been considered. Characterization of u-minimal solutions of (u-SOP) in terms of solutions of these scalar optimization problems have been obtained. Finally, equivalence of well-posedness of (u-SOP) with well-posedness of these scalar optimization problems have been established.
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Aubin, J.P., Cellina, A.: Differential Inclusions, Set-Valued Maps and Viability Theory, Grundlehren der mathematischen Wissenschaften, vol. 264. Springer, Berlin (1984)
Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis, Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston (2009)
Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010)
Bao, T.Q., Mordukhovich, B.S.: Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling. Discrete Contin. Dyn. Syst. 31, 1069–1096 (2011)
Chen, G.Y., Huang, X., Yang, X.: Vector Optimization-Set Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)
Crespi, G.P., Guerraggio, A., Rocca, M.: Well-posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132(1), 213–226 (2007)
Crespi, G.P., Dhingra, M., Lalitha, C.S.: Pointwise and global well-posedness in set optimization: a direct approach. Ann. Oper. Res. (2017). https://doi.org/10.1007/s10479-017-2709-7
Crespi, G.P., Papalia, M., Rocca, M.: Extended well-posedness of quasiconvex vector optimization problems. J. Optim. Theory Appl. 141(2), 285–297 (2009)
Crespi, G.P., Papalia, M., Rocca, M.: Extended well-posedness of vector optimization problems: the convex case. Taiwan. J. Math. 15(4), 1545–1559 (2011)
Crespi, G.P., Kuroiwa, D., Rocca, M.: Convexity and global well-posedness in set optimization. Taiwan. J. Math. 18, 1897–1908 (2014)
Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems, Lecture Notes in Mathematics. Springer, Berlin (1993)
Fang, Y.P., Hu, R., Huang, N.J.: Extended B-well-posedness and property (H)- for set-valued vector optimization with convexity. J. Optim. Theory Appl. 135, 445–458 (2007)
Göpfert, A., Riahi, H., Tammer, C., et al.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)
Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pontwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, 1822–1833 (2012)
Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B., Schrage, C.: Set optimization—a rather short introduction. In: Hamel, A.H. (ed.) Set Optimization and Applications—The State of the Art, pp. 65–141. Springer, Berlin (2015)
Han, Y., Huang, N.: Well-posedness and stability of solutions for set optimization problems. Optimization 66(1), 17–33 (2017)
Hiriart-Urruty, J.B.: Tangent cone, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79–97 (1979)
Huang, X.X.: Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 53(1), 101–116 (2001)
Jahn, J.: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin (2004)
Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Application. Springer, Berlin (2015)
Khoshkhabar-amiranloo, S., Khorram, E.: Scalarization of Levitin–Polyak well-posed set optimization problems. Optimization 66(1), 113–127 (2017)
Klein, E., Thompson, A.C.: Theory of Correspondences. Wiley, New York (1984)
Kuratowski, K.: Topology, Volumes 1 and 2. Academic Press, New York (1968)
Kuroiwa, D.: Some duality theorems of set-valued optimization with natural criteria. In: Proceedings of the International Conference on Nonlinear Analysis and Convex analysis, pp. 221–228. World Scientific River Edge (1999)
Lalitha, C.S., Chatterjee, P.: Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems. J. Glob. Optim. 59, 191–205 (2014)
Long, X.J., Peng, J.W.: Generalized B-well-posedness for set optimization problems. J. Optim. Theory Appl. 157, 612–623 (2013)
Long, X.J., Peng, J.W., Peng, Z.Y.: Scalarization and pointwise well-posedness for set optimization problems. J. Glob. Optim. 62(4), 763–773 (2015)
Loridan, P.: Well-posedness in vector optimization. Math. Appl. 331, 171–192 (1995)
Luc, D.T.: Theory of Vector Optimization: Lecture Notes in Economics and Mathematics Systems, vol. 319. Springer, New York (1989)
Lucchetti, R., Revalski, J. (eds.): Recent Development in Well-Posed Variational Problems. Kluwer Academic Publishers, Dordrecht (1995)
Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391–409 (2005)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)
Tykhonov, A.N.: On the stability of the functional optimization problems. USSR Comput. Math. Phys. 6, 28–33 (1966)
Xu, Y.D., Li, S.J.: A new nonlinear scalarization function and applications. Optimization 65(1), 207–231 (2016)
Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlinear Anal. 71, 3769–3778 (2009)
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Gupta, M., Srivastava, M. Well-posedness and scalarization in set optimization involving ordering cones with possibly empty interior. J Glob Optim 73, 447–463 (2019). https://doi.org/10.1007/s10898-018-0695-1
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DOI: https://doi.org/10.1007/s10898-018-0695-1