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

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.
Citation: Qilin Wang, Shengji Li, Kok Lay Teo. Continuity of second-order adjacent derivatives for weak perturbation maps in vector optimization. Numerical Algebra, Control and 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.

[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.

show all references

References:
[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.

[1]

Rubén Figueroa, Rodrigo López Pouso, Jorge Rodríguez–López. Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 617-633. doi: 10.3934/dcdsb.2019257

[2]

Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252

[3]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

[4]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136

[5]

Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89

[6]

Yi Zhang, Yong Jiang, Liwei Zhang, Jiangzhong Zhang. A perturbation approach for an inverse linear second-order cone programming. Journal of Industrial and Management Optimization, 2013, 9 (1) : 171-189. doi: 10.3934/jimo.2013.9.171

[7]

Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111

[8]

Qilin Wang, Xiao-Bing Li, Guolin Yu. Second-order weak composed epiderivatives and applications to optimality conditions. Journal of Industrial and Management Optimization, 2013, 9 (2) : 455-470. doi: 10.3934/jimo.2013.9.455

[9]

Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225

[10]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[11]

Doyoon Kim, Seungjin Ryu. The weak maximum principle for second-order elliptic and parabolic conormal derivative problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 493-510. doi: 10.3934/cpaa.2020024

[12]

Guanrong Li, Yanping Chen, Yunqing Huang. A hybridized weak Galerkin finite element scheme for general second-order elliptic problems. Electronic Research Archive, 2020, 28 (2) : 821-836. doi: 10.3934/era.2020042

[13]

Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1041-1054. doi: 10.3934/jimo.2017089

[14]

Lin Zhu, Xinzhen Zhang. Semidefinite relaxation method for polynomial optimization with second-order cone complementarity constraints. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1505-1517. doi: 10.3934/jimo.2021030

[15]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control and Related Fields, 2021, 11 (3) : 653-679. doi: 10.3934/mcrf.2021017

[16]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial and Management Optimization, 2022, 18 (2) : 731-745. doi: 10.3934/jimo.2020176

[17]

Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010

[18]

José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1

[19]

Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 339-358. doi: 10.3934/dcds.2003.9.339

[20]

Ryo Ikehata, Marina Soga. Asymptotic profiles for a strongly damped plate equation with lower order perturbation. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1759-1780. doi: 10.3934/cpaa.2015.14.1759

 Impact Factor: 

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]