Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality
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In this paper, we introduce upper and lower Studniarski derivatives of set-valued maps. By virtue of these derivatives, higher-order necessary and sufficient optimality conditions are obtained for several kinds of minimizers of a set-valued optimization problem. Then, applications to duality are given. Some remarks on several existent results and examples are provided to illustrate our results.
KeywordsStudniarski derivatives Optimality conditions Set-valued optimization problem Efficiency Generalized subconvexlike Duality
Mathematics Subject Classification (2010)32F17 46G05 90C29 90C46
We acknowledge Professor Szymon Dolecki’s very helpful discussions during our working. We are also grateful to an anonymous referee for his valuable remarks which helped to improve our previous manuscript.
- 4.Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, 3rd edn. Springer, Berlin (2009)Google Scholar
- 5.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. I. Basic Theory. Springer, Berlin (2006)Google Scholar
- 6.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. II. Applications. Springer, Berlin (2006)Google Scholar
- 14.Khanh, P.Q., Tuan, N.D.: Variational sets of multivalued mappings and a unified study of optimality conditions. J. Optim. Theory Appl. 139, 45–67 (2008)Google Scholar
- 18.Li, S.J., Teo, K.L., Yang, X.Q.: Higher-order optimality conditions for set-valued optimization. J. Optim. Theory Appl. 137, 533–553 (2008)Google Scholar
- 19.Anh, N.L.H., Khanh, P.Q., Tung, L.T.: Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization. Nonlinear Anal. TMA 74, 7365–7379 (2011)Google Scholar
- 24.Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P., Rassis, Th.M., Khan, A.A. (eds.) Nonlinear Analysis and Variational Problems, Chap. 21, pp. 305–324. Springer, Berlin (2009)Google Scholar