Abstract
Based on the notation of Mordukhovich subdifferentials (Mordukhovich in Variational analysis and generalized differentiation I: basic theory, Springer, Berlin, 2006; Variational analysis and generalized differentiation II: applications, Springer, Berlin, 2006; Variational analysis and applications, Springer, Berlin, 2018), we establish strong Karush–Kuhn–Tucker type necessary optimality conditions for the weak efficiency of a nonsmooth nonconvex multiobjective programming problem with set, inequality and equality constraints. We also provide several new definitions for the Mordukhovich-pseudoconvexity and Mordukhovich-quasiconvexity with extended-real-valued functions, and then provide sufficient optimality conditions for weak efficiency to such problem in terms of Mordukhovich subdifferentials.
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Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)
Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122(2), 301–247 (2010)
Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Global. Optim. 43(4), 533–552 (2009)
Bao, T.Q., Mordukhovich, B.S.: Necessary conditions in multiobjective optimization with equilibrium constraints. J. Optim. Theory Appl. 135(2), 179–203 (2007)
Burachik, R.S., Rizvi, M.M.: On weak and strong KuhnTucker conditions for smooth multiobjective optimization. J. Optim. Theory Appl. 155, 477–491 (2012)
Constantin, E.: First-order necessary conditions in locally Lipschitz multiobjective optimization. Optimization 67, 1447–1460 (2018)
Chuong, T.D., Kim, D.S.: Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160, 748–762 (2014)
Goberna, M.A., Lopéz, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993)
Jiménez, B., Novo, V.: First order optimality conditions in vector optimization involving stable functions. Optimization 57(3), 449–471 (2008)
Jiménez, B., Novo, V.: A finite dimensional extension of Lyusternik theorem with applications to multiobjective optimization. J. Math. Anal. Appl. 270, 340–356 (2002)
Khan, A., Tammer, C., Zalinescu, C.: Set-Valued Optimization. Springer, Berlin (2015)
Liu, J.J., Zhao, K.Q., Yang, X.M.: Optimality and regularity conditions using Mordukhovich’s subdifferential. J. Nonlinear Convex Anal. 13, 43–53 (2017)
Luu, D.V.: Necessary and sufficient conditions for efficiency via convexificators. J. Optim. Theory Appl. 160, 510–526 (2014)
Luu, D.V.: Necessary efficiency conditions for vector equilibrium problems with general inequality constraints via convexificators. Bull. Braz. Math. Soc. New Ser. 50, 685–704 (2019)
Luu, D.V.: Necessary conditions for efficiency in terms of the Michel–Penot subdifferentials. Optimization 61, 1099–1117 (2012)
Luu, D.V.: Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems. J. Glob. Optim. 70, 437–453 (2018)
Luu, D.V., Su, T.V.: Contingent derivatives and necessary efficiency conditions for vector equilibrium problems with constraints. RAIRO - Oper. Res. bf 52, 543–559 (2018)
Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II: Applications. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Berlin (2018)
Rockafellar, R.T.: Convex Analysis Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970)
Su, T.V., Hien, N.D.: Studniarski’s derivatives and efficiency conditions for constrained vector equilibrium problems with applications. Optimization (2019). https://doi.org/10.1080/02331934.2019.1702985
Su, T.V., Hang, D.D.: Optimality conditions for the efficient solutions of vector equilibrium problems with constraints in terms of directional derivatives and applications. Bull. Iran. Math. Soc. 45(6), 1619–1650 (2019)
Zhao, K.Q.: Strong Kuhn-Tucker optimality in nonsmooth multiobjective optimization problems. Pac. J. Optim. 11, 483–494 (2015)
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Van Su, T., Hien, N.D. Strong Karush–Kuhn–Tucker optimality conditions for weak efficiency in constrained multiobjective programming problems in terms of mordukhovich subdifferentials. Optim Lett 15, 1175–1194 (2021). https://doi.org/10.1007/s11590-020-01620-0
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DOI: https://doi.org/10.1007/s11590-020-01620-0