Abstract
In this paper, we introduce several kinds of generalized invexity for set-valued mappings and some relationships between a set-valued optimization problem and vector variational-like inequalities are established.
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Al-Homidan, S., Ansari, Q.H.: Generalized Minty vector variational-like inequalities and vector optimization problems. J. Optim. Theory Appl. 144, 1–11 (2010)
Ansari, Q.H., Lee, G.M.: Nonsmooth vector optimization problems and Minty vector variational inequalities. J. Optim. Theory Appl. 145(1), 1–16 (2010)
Ansari, Q.H., Rezaie, M.: Existence results for Stampacchia and Minty type vector variational inequalities. Optimization 59(7), 1053–1065 (2010)
Ansari, Q.H., Rezaie, M., Zafarani, J.: Generalized variational-like inequalities and vector optimization. J. Global Optim. (2011). doi:10.1007/s10898-011-9686-1
Ansari, Q.H., Yao, J.-C.: Recent developments in vector optimization. Springer, Berlin (2012)
Aubin, J.P., Frankowska, H.: Set-valued analysis. Birkhauser, Boston (1990)
Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers to multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010)
Dos Santos Batista, L.B., Ruiz-Garzón, G., Rojas-Medar, M.A.: Some relations between variational-like inequality problems and vector optimization problems in Banach spaces. Comput. Math. Appl. 55, 1808–1814 (2008)
Cambini, A., Martein, L.: Handbook of generalized convexity and generalized monotonicity. In: Nonconvex optimization and its applications, vol. 76. Springer, The Netherlands (2005)
Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154(1), 29–50 (2007)
Fang, Y.P., Huang, N.J.: Strong vector variational inequalities in Banach spaces. Appl. Math. Lett. 19, 362–368 (2006)
Fang, Y.P., Huang, N.J.: Existence results for generalized implicit vector variational inequalities with multivalued mappings. Indian J. Pure Appl. Math. 36, 629–640 (2005)
Farajzadeh, A.P., Harandi, A.A., Kazmi, K.R.: Existence of solutions to generalized vector variational-like inequalities. J. Optim. Theory Appl. 146, 95–104 (2010)
Garzon, G.R., Gomez, R.O., Lizana, A.R.: Generalized invex monotonicity. Eur. J. Oper. Res. 144, 501–512 (2003)
Garzon, G.R., Gomez, R.O., Lizana, A.R.: Relationship between vector variational-like inequality and optimization problems. Eur. J. Oper. Res. 157, 113–119 (2004)
Giannessi, F.: On Minty variational principle. In: Giannessi, F., Komlósi, S., Tapcsáck, T. (eds.) New trends in mathematical programming, pp. 93–99. Kluwer Academic, Dordrecht (1998)
Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrum problems: nonsmooth optimization and variational inequalities models. Springer, Berlin (2002)
Hanson, M.A.: On sufficienty of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)
Jabarootian, T., Zafarani, J.: Generalized invariant monotonicity and invexity of non-differentiable functions. J. Global Optim. 36, 537–564 (2006)
Mishra, S.K., Wang, S.Y.: Vector variational-like inequalities and non-smooth vector optimization problems. Nonlinear Anal. 64, 1939–1945 (2006)
Pini, R.: Invexity and generalized convexity. Optimization 22, 513–525 (1991)
Rezaie, M., Zafarani, J.: Vector optimization and variational-like inequalities. J. Global Optim. 43, 47–66 (2009)
Soleimani-damaneh, M.: Characterizations and applications of generalized invexity and monotonicity in Asplund spaces, topology. Springer (2010). doi:10.1007/s11750-010-0150-z
Soleimani-damaneh, M.: Characterization of nonsmooth quasiconvex and pseudoconvex functions. J. Math. Anal. Appl. 330, 1387–1392 (2007)
Yang, X.M., Yang, X.Q.: Vector variational-like inequalities with pseudoinvexity. Optimization 55, 157–170 (2006)
Yang, X.M., Yang, X.Q., Teo, K.L.: Generalized invexity and generalized invariant monotonicity. J. Optim. Theory Appl. 117, 607–625 (2003)
Yang, X.M., Yang, X.Q., Teo, K.L.: Criteria for generalized invex monotonicity. Eur. J. Oper. Res. 164, 115–119 (2005)
Yang, X.M., Yang, X.Q., Teo, K.L.: Some remarks on the Minty vector variational inequality. J. Optim. Theory Appl. 121, 193–201 (2004)
Yang, X.M., Yang, X.Q., Teo, K.L.: Generalizations and applications of prequasi-invex functions. J. Optim. Theory Appl. 110, 645–668 (2001)
Zeng, J., Li, S.J.: On vector variational-like inequalities and set-valued optimization problems. Optim. Lett. 5(1), 55–69 (2011)
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This research was supported by a grant of the Romanian National Authority for Scientific Research CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0024.
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Miholca, M. On set-valued optimization problems and vector variational-like inequalities. Optim Lett 8, 463–476 (2014). https://doi.org/10.1007/s11590-012-0591-4
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DOI: https://doi.org/10.1007/s11590-012-0591-4