American Institute of Mathematical Sciences

Advanced
2011, 1(3): 417-433. doi: 10.3934/naco.2011.1.417

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

Qilin Wang 1, , Shengji Li 2, and  Kok Lay Teo 3,

1. 

College of Mathematics and Science, Chongqing University, Chongqing 400044
2. 

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331
3. 

Department of Mathematics and Statistics, Curtin University, G.P.O. Box U1987, Perth, WA 6845

Received  April 2011 Revised  July 2011 Published  September 2011

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.
Keywords: Vector optimization, second-order adjacent derivatives, weak perturbation maps, upper and lower semicontinuity..
Mathematics Subject Classification: 49K40, 90C31, 54C60.
Citation: Qilin Wang, Shengji Li, Kok Lay Teo. Continuity of second-order adjacent derivatives for weak perturbation maps in vector optimization. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 417-433. doi: 10.3934/naco.2011.1.417
References:
[1]

A. V. Fiacco, "Introduction to Sensitivity and Stability Analysis in Nonlinear Programming," Academic Press, New York, 1983. Google Scholar

[2]

J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Biekhäuser, Boston, 1990.  Google Scholar

[3]

J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," John Wiley, New York, 1984.  Google Scholar

[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.  Google Scholar

[5]

R. B. Holmes, "Geometric Functional Analysis and Its Applications," Springer-Verlag, New York, 1975.  Google Scholar

[6]

T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499. doi: 10.1007/BF00939554.  Google Scholar

[7]

T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control Optim., 26 (1988), 521-536. doi: 10.1137/0326031.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53.  Google Scholar

[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.  Google Scholar

[12]

D. S. Shi, Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77 (1993), 145-159. doi: 10.1007/BF00940783.  Google Scholar

[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.  Google Scholar

[14]

J. Jahn, "Vector Optimization-Theory, Applications and Extensions," Springer, 2004.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. T. Luc, "Theory of Vector Optimization," Springer, Berlin, 1989.  Google Scholar

[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.  Google Scholar

[24]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer-Verlag, New York, 2000.  Google Scholar

show all references

References:
[1]

A. V. Fiacco, "Introduction to Sensitivity and Stability Analysis in Nonlinear Programming," Academic Press, New York, 1983. Google Scholar

[2]

J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Biekhäuser, Boston, 1990.  Google Scholar

[3]

J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," John Wiley, New York, 1984.  Google Scholar

[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.  Google Scholar

[5]

R. B. Holmes, "Geometric Functional Analysis and Its Applications," Springer-Verlag, New York, 1975.  Google Scholar

[6]

T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499. doi: 10.1007/BF00939554.  Google Scholar

[7]

T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control Optim., 26 (1988), 521-536. doi: 10.1137/0326031.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53.  Google Scholar

[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.  Google Scholar

[12]

D. S. Shi, Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77 (1993), 145-159. doi: 10.1007/BF00940783.  Google Scholar

[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.  Google Scholar

[14]

J. Jahn, "Vector Optimization-Theory, Applications and Extensions," Springer, 2004.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. T. Luc, "Theory of Vector Optimization," Springer, Berlin, 1989.  Google Scholar

[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.  Google Scholar

[24]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer-Verlag, New York, 2000.  Google Scholar

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