Abstract
In this paper, firstly, a new generalized subconvexlike set-valued map based on the quasi-relative interior is introduced. Secondly, by a separation theorem involving the quasi-relative interior, some separation properties are obtained. Finally, some optimality conditions are established. Our results improve some results in the literature.
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Borwein, J.M., Lewis, A.S.: Partially finite convex programming, Part I: quasi relative interiors and duality theory. Math. Program. 57(1–3), 15–48 (1992)
Jeyakumar, V., Wolkowicz, H.: Generalizations of Slater’s constraint qualification for infinite convex programs. Math. Program. 57(1–3), 85–101 (1992)
Limber, M.A., Goodrich, R.K.: Quasi interiors, Lagrange multipliers and l p spectral estimation with lattice bounds. J. Optim. Theory Appl. 78(1), 143–161 (1993)
Borwein, J.M., Goebel, R.: Notions of relative interior in Banach spaces. J. Math. Sci. 115, 2542–2553 (2003)
Cammaroto, F., Di Bella, B.: Separation theorem based on the quasi relative interior and application to duality theory. J. Optim. Theory Appl. 125(1), 223–229 (2005)
Cammaroto, F., Di Bella, B.: On a separation theorem involving the quasi relative interior. Proc. Edinb. Math. Soc. 50, 605–610 (2007)
Daniele, P., Giuffré, S., Idone, G., Maugeri, A.: Infinite dimensional duality and applications. Math. Ann. 339(1), 221–239 (2007)
Bot, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi relative interior in convex programming. SIAM J. Optim. 19(1), 217–233 (2008)
Bot, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasi relative interior in convex optimization. J. Optim. Theory Appl. 139(1), 67–84 (2008)
Bot, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization. doi:10.1080/02331934.2010.505649
Bot, R.I., Grad, S.M., Wanka, G.: Duality in Vector Optimization. Springer, Berlin (2009)
Grad, A.: Quasi relative interior-type constraint qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints. J. Math. Anal. Appl. 361, 86–95 (2010)
Rong, W.D., Wu, Y.N.: Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps. Math. Methods Oper. Res. 48, 247–258 (1998)
Li, Z.M.: A theorem of the alternative and its Application to the optimization of set-valued maps. J. Optim. Theory Appl. 100(2), 365–375 (1999)
Yang, X.M., Yang, X.Q., Chen, G.Y.: Theorems of the alternative and optimization with set-valued maps. J. Optim. Theory Appl. 107(3), 627–640 (2000)
Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110(2), 413–427 (2001)
Sach, P.H.: Nearly subconvexlike set-valued maps and vector optimization problems. J. Optim. Theory Appl. 119(2), 335–356 (2003)
Xu, Y.H., Liu, S.Y.: Benson proper efficiency in the nearly cone-subconvexlike vector optimization with set-valued functions. Appl. Math. J. Chin. Univ. Ser. B 18(1), 95–102 (2003)
Borwein, J.M.: Properly efficient points for maximization with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)
Shi, S.Z.: Convex Analysis. Universitext. Shanghai Scientifical and Technical Press, Shanghai (1990)
Tiel, J.V.: Convex Analysis. An Introductory Text. Wiley, New York (1984)
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Communicated by J.-C. Yao.
The authors are grateful to Professor J.-C. Yao and the referees for valuable comments and suggestions.
X.M. Yang was supported by the National Nature Science Foundation of China (Grant 10831009).
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Zhou, Z.A., Yang, X.M. Optimality Conditions of Generalized Subconvexlike Set-Valued Optimization Problems Based on the Quasi-Relative Interior. J Optim Theory Appl 150, 327–340 (2011). https://doi.org/10.1007/s10957-011-9844-0
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DOI: https://doi.org/10.1007/s10957-011-9844-0