Abstract
In the paper, we develop sum and chain rules of the generalized contingent derivative for set-valued mappings. Then, their applications to sensitivity analysis and optimality conditions for some particular optimization problems are given. Our results extend some recent existing ones in the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anh, N.L.H., Khanh, P.Q.: Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives. J. Glob. Optim. 56, 519–536 (2013)
Anh, N.L.H., Khanh, P.Q.: Higher-order optimality conditions for proper efficiency in nonsmooth optimization using radial sets and radial derivatives. J. Glob. Optim. 58, 693–709 (2014)
Anh, N.L.H., Khanh, P.Q., Tung, L.T.: Variational sets: calculus and applications to nonsmooth vector optimization. Nonlinear Anal. Theory Methods Appl. 74, 2358–2379 (2011)
Anh, N.L.H., Khanh, P.Q., Tung, L.T.: Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization. Nonlinear Anal. Theory Methods Appl. 74, 7365–7379 (2011)
Borwein, J.M.: Epi-Lipschitz-like sets in Banach space: theorems and examples. Nonlinear Anal. Theory Methods Appl. 11, 1207–1217 (1987)
Diem, H.T.H., Khanh, P.Q., Tung, L.T.: On higher-order sensitivity analysis in nonsmooth vector optimization. J. Optim. Theory Appl. 162, 463–488 (2014)
Jahn, J., Khan, A.A.: Some calculus rules for contingent epiderivatives. Optimization 52, 113–125 (2003)
Li, S.J., Meng, K.W., Penot, J.P.: Calculus rules for derivatives of multimaps. Set-Valued Anal. 17, 21–39 (2009)
Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mapping. J. Math. Anal. Appl. 183, 250–288 (1994)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Volume I: Basic Theory. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Volume II: Applications. Springer, Berlin (2006)
Penot, J.P.: Differentiability of relations and differential stability of perturbed optimization problems. SIAM J. Control Optim. 22, 529–551 (1984)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, 3rd edn. Springer, Berlin (2009)
Taa, A.: Set-valued derivatives of multifunctions and optimality conditions. Numer. Funct. Anal. Optim. 19, 121–140 (1998)
Wang, Q.L., Li, S.J., Teo, K.L.: Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization. Optim. Lett. 4, 425–437 (2010)
Acknowledgements
This research was funded by Vietnam National University Hochiminh City (VNU-HCM) under Grant Number B2018-28-02. We are thankful to the anonymous referee for his useful comments to improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Anh, N.L.H., Thoa, N.T. Calculus rules of the generalized contingent derivative and applications to set-valued optimization. Positivity 24, 81–94 (2020). https://doi.org/10.1007/s11117-019-00667-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-019-00667-3
Keywords
- Generalized contingent derivative
- Sum rule
- Chain rule
- Set-valued optimization
- Optimality conditions
- Sensitivity analysis