Article Contents
Article Contents

# Continuity of second-order adjacent derivatives for weak perturbation maps in vector optimization

• In this paper, some properties are established for second-order adjacent derivatives of set-valued maps. Upper and lower semicontinuity and closedness are obtained for second-order adjacent derivatives of weak perturbation maps in vector optimization problems. Several examples are given for illustrating our results.
Mathematics Subject Classification: 49K40, 90C31, 54C60.

 Citation:

•  [1] A. V. Fiacco, "Introduction to Sensitivity and Stability Analysis in Nonlinear Programming," Academic Press, New York, 1983. [2] J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Biekhäuser, Boston, 1990. [3] J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," John Wiley, New York, 1984. [4] F. Ferro, An optimization result for set-valued mappings and a stability property in vector problems with constraints, J. Optim. Theory Appl., 90 (1996), 63-77.doi: 10.1007/BF02192246. [5] R. B. Holmes, "Geometric Functional Analysis and Its Applications," Springer-Verlag, New York, 1975. [6] T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499.doi: 10.1007/BF00939554. [7] T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control Optim., 26 (1988), 521-536.doi: 10.1137/0326031. [8] H. Kuk, T. Tanino and M. Tanaka, Sensitivity analysis in parametrized convex vector optimization, J. Math. Anal. Appl., 202 (1996), 511-522.doi: 10.1006/jmaa.1996.0331. [9] H. Kuk, T. Tanino and M. Tanaka, Sensitivity analysis in vector optimization, J. Optim. Theory Appl., 89 (1996), 713-730.doi: 10.1007/BF02275356. [10] S. J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53. [11] D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, J. Optim. Theory Appl., 70 (1991), 385-396.doi: 10.1007/BF00940634. [12] D. S. Shi, Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77 (1993), 145-159.doi: 10.1007/BF00940783. [13] J. Jahn, A. A. Khan and P. Zeilinger, Second-order optimality conditions in set optimalization, J. Optim. Theory Appl., 125 (2005), 331-347.doi: 10.1007/s10957-004-1841-0. [14] J. Jahn, "Vector Optimization-Theory, Applications and Extensions," Springer, 2004. [15] V. Kalashnikov, B.Jadamba and A.A. Khan, First and second-order optimality conditions in set optimization, in "Optimization with Multivalued Mappings"(eds. S. Dempe and V. Kalashnikov), Spring Science+Business Media, LLC, (2006), 265-276. [16] P. Q. Khanh and N. D. Tuan, Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimization, J. Optim.Theory Appl., 139 (2008), 243-261.doi: 10.1007/s10957-008-9414-2. [17] S. J. Li, K. L. Teo and X. Q. Yang, Higher-order optimality conditions for set-valued optimization, J. Optim. Theory Appl., 137 (2008), 533-553.doi: 10.1007/s10957-007-9345-3. [18] S. J. Li and C. R. Chen, Higher-order optimality conditions for Henig efficient solutions in set-valued optimization, J. Math. Anal. Appl., 323 (2006), 1184-1200.doi: 10.1016/j.jmaa.2005.11.035. [19] Q. L. Wang and S. J. Li, Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency, Numer. Funct. Anal. Optim., 30 (2009), 849-869.doi: 10.1080/01630560903139540. [20] Q. L. Wang and S. J. Li, Higher-Order Weakly Generalized Adjacent Epiderivatives and Applications to Duality of Set-Valued Optimization, J. Inequal. Appl., 2009, 18 pages. [21] Q. L. Wang, S. J. Li and K. L. Teo, Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization, Optim. Lett. , 4 (2010), 425-437.doi: 10.1007/s11590-009-0170-5. [22] D. T. Luc, "Theory of Vector Optimization," Springer, Berlin, 1989. [23] S. W. Xiang and W. S. Yin, Stability results for efficient solutions of vector optimization problems, J. Optim. Theory Appl., 134 (2007), 385-398.doi: 10.1007/s10957-007-9214-0. [24] J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer-Verlag, New York, 2000.