• Previous Article
    A unified parameter identification method for nonlinear time-delay systems
  • JIMO Home
  • This Issue
  • Next Article
    Joint pricing and ordering policies for deteriorating item with retail price-dependent demand in response to announced supply price increase
April  2013, 9(2): 455-470. doi: 10.3934/jimo.2013.9.455

Second-order weak composed epiderivatives and applications to optimality conditions

1. 

College of Sciences, Chongqing Jiaotong University, Chongqing, 400074, China, China

2. 

Research Institute of Information and System Computation Science, Beifang University of Nationalities, Yinchuan, 750021, China

Received  April 2011 Revised  January 2013 Published  February 2013

In this paper, one introduces the second-order weak composed contingent epiderivative of set-valued maps, and discusses some of its properties. Then, by virtue of the second-order weak composed contingent epiderivative, necessary optimality conditions and sufficient optimality conditions are obtained for set-valued optimization problems. As consequences, recent existing results are derived. Several examples are provided to show the main results obtained.
Citation: Qilin Wang, Xiao-Bing Li, Guolin Yu. Second-order weak composed epiderivatives and applications to optimality conditions. Journal of Industrial & Management Optimization, 2013, 9 (2) : 455-470. doi: 10.3934/jimo.2013.9.455
References:
[1]

J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions,, in, 7a (1981), 159.   Google Scholar

[2]

J.-P. Aubin and H. Frankowska, "Set-Valued Analysis,", Systems & Control: Foundations & Applications, 2 (1990).   Google Scholar

[3]

H. W. Corley, Optimality condition for maximizations of set-valued functions,, J. Optim. Theory Appl., 58 (1988), 1.  doi: 10.1007/BF00939767.  Google Scholar

[4]

J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization,, Math. Methods Oper. Res., 46 (1997), 193.  doi: 10.1007/BF01217690.  Google Scholar

[5]

G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems. Set-valued optimization,, Math. Methods Oper. Res., 48 (1998), 187.  doi: 10.1007/s001860050021.  Google Scholar

[6]

E. M. Bednarczuk and W. Song, Contingent epiderivative and its applications to set-valued optimization,, Control Cybernet., 27 (1998), 375.   Google Scholar

[7]

G. Bigi and M. Castellani, K-epiderivatives for set-valued functions and optimization,, Math. Methods Oper. Res., 55 (2002), 401.  doi: 10.1007/s001860200187.  Google Scholar

[8]

X.-H. Gong, H.-B. Dong and S.-Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization,, J. Math. Anal. Appl., 284 (2003), 332.  doi: 10.1016/S0022-247X(03)00360-3.  Google Scholar

[9]

J. Jahn and A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: Optimality conditions,, Numer. Funct. Anal. Optim., 23 (2002), 807.  doi: 10.1081/NFA-120016271.  Google Scholar

[10]

J.-P. Penot, Second-order conditions for optimization problems with constraints,, SIAM J. Control Optim., 37 (1999), 303.  doi: 10.1137/S0363012996311095.  Google Scholar

[11]

J. F. Bonnans, R. Cominetti and A. Shapiro, Second-order optimality conditions based on parabolic second-order tangent sets,, SIAM J. Optim., 9 (1999), 466.  doi: 10.1137/S1052623496306760.  Google Scholar

[12]

B. Jiménez and V. Novo, Second-order necessary conditions in set constrained differentiable vector optimization,, Math. Methods Oper. Res., 58 (2003), 299.  doi: 10.1007/s001860300283.  Google Scholar

[13]

B. Jiménez and V. Novo, Optimality conditions in differentiable vector optimization via second-order tangent sets,, Appl. Math. Optim., 49 (2004), 123.  doi: 10.1007/s00245-003-0782-6.  Google Scholar

[14]

J. Jahn, A. A. Khan and P. Zeilinger, Second-order optimality conditions in set optimalization,, J. Optim. Theory Appl., 125 (2005), 331.  doi: 10.1007/s10957-004-1841-0.  Google Scholar

[15]

M. Durea, First and second order optimality conditions for set-valued optimization problems,, Rend. Circ. Mat. Palermo (2), 53 (2004), 451.  doi: 10.1007/BF02875738.  Google Scholar

[16]

G. Bigi, On sufficient second order optimality conditions in multiobjective optimization,, Math. Methods Oper. Res., 63 (2006), 77.  doi: 10.1007/s00186-005-0013-9.  Google Scholar

[17]

L. Rodríguez-Marín and M. Sama, About contingent epiderivatives,, J. Math. Anal. Appl., 327 (2007), 745.  doi: 10.1016/j.jmaa.2006.04.060.  Google Scholar

[18]

S. J. Li, S. K. Zhu and K. L. Teo, New generalized second-order contingent epiderivatives and set-valued optimization problems,, J. Optim. Theory Appl., 152 (2012), 587.  doi: 10.1007/s10957-011-9915-2.  Google Scholar

[19]

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.  doi: 10.1016/j.jmaa.2005.11.035.  Google Scholar

[20]

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.  doi: 10.1007/s10957-007-9345-3.  Google Scholar

[21]

P. Q. Khanh and N. D. Tuan, Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimalization,, J. Optim. Theory Appl., 139 (2008), 243.  doi: 10.1007/s10957-008-9414-2.  Google Scholar

[22]

C. R. Chen, S. J. Li and K. L. Teo, Higher order weak epiderivatives and applications to duality and optimality conditions,, Comput. Math. Appl., 57 (2009), 1389.  doi: 10.1016/j.camwa.2009.01.012.  Google Scholar

[23]

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.  doi: 10.1007/s11590-009-0170-5.  Google Scholar

[24]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer Series in Operations Research, (2000).   Google Scholar

[25]

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

[26]

D. T. Luc, "Theory of Vector Optimization,", Lecture Notes in Economics and Mathematical Systems, 319 (1989).   Google Scholar

[27]

Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps,, J. Optim. Theory Appl., 100 (1999), 365.  doi: 10.1023/A:1021786303883.  Google Scholar

show all references

References:
[1]

J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions,, in, 7a (1981), 159.   Google Scholar

[2]

J.-P. Aubin and H. Frankowska, "Set-Valued Analysis,", Systems & Control: Foundations & Applications, 2 (1990).   Google Scholar

[3]

H. W. Corley, Optimality condition for maximizations of set-valued functions,, J. Optim. Theory Appl., 58 (1988), 1.  doi: 10.1007/BF00939767.  Google Scholar

[4]

J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization,, Math. Methods Oper. Res., 46 (1997), 193.  doi: 10.1007/BF01217690.  Google Scholar

[5]

G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems. Set-valued optimization,, Math. Methods Oper. Res., 48 (1998), 187.  doi: 10.1007/s001860050021.  Google Scholar

[6]

E. M. Bednarczuk and W. Song, Contingent epiderivative and its applications to set-valued optimization,, Control Cybernet., 27 (1998), 375.   Google Scholar

[7]

G. Bigi and M. Castellani, K-epiderivatives for set-valued functions and optimization,, Math. Methods Oper. Res., 55 (2002), 401.  doi: 10.1007/s001860200187.  Google Scholar

[8]

X.-H. Gong, H.-B. Dong and S.-Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization,, J. Math. Anal. Appl., 284 (2003), 332.  doi: 10.1016/S0022-247X(03)00360-3.  Google Scholar

[9]

J. Jahn and A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: Optimality conditions,, Numer. Funct. Anal. Optim., 23 (2002), 807.  doi: 10.1081/NFA-120016271.  Google Scholar

[10]

J.-P. Penot, Second-order conditions for optimization problems with constraints,, SIAM J. Control Optim., 37 (1999), 303.  doi: 10.1137/S0363012996311095.  Google Scholar

[11]

J. F. Bonnans, R. Cominetti and A. Shapiro, Second-order optimality conditions based on parabolic second-order tangent sets,, SIAM J. Optim., 9 (1999), 466.  doi: 10.1137/S1052623496306760.  Google Scholar

[12]

B. Jiménez and V. Novo, Second-order necessary conditions in set constrained differentiable vector optimization,, Math. Methods Oper. Res., 58 (2003), 299.  doi: 10.1007/s001860300283.  Google Scholar

[13]

B. Jiménez and V. Novo, Optimality conditions in differentiable vector optimization via second-order tangent sets,, Appl. Math. Optim., 49 (2004), 123.  doi: 10.1007/s00245-003-0782-6.  Google Scholar

[14]

J. Jahn, A. A. Khan and P. Zeilinger, Second-order optimality conditions in set optimalization,, J. Optim. Theory Appl., 125 (2005), 331.  doi: 10.1007/s10957-004-1841-0.  Google Scholar

[15]

M. Durea, First and second order optimality conditions for set-valued optimization problems,, Rend. Circ. Mat. Palermo (2), 53 (2004), 451.  doi: 10.1007/BF02875738.  Google Scholar

[16]

G. Bigi, On sufficient second order optimality conditions in multiobjective optimization,, Math. Methods Oper. Res., 63 (2006), 77.  doi: 10.1007/s00186-005-0013-9.  Google Scholar

[17]

L. Rodríguez-Marín and M. Sama, About contingent epiderivatives,, J. Math. Anal. Appl., 327 (2007), 745.  doi: 10.1016/j.jmaa.2006.04.060.  Google Scholar

[18]

S. J. Li, S. K. Zhu and K. L. Teo, New generalized second-order contingent epiderivatives and set-valued optimization problems,, J. Optim. Theory Appl., 152 (2012), 587.  doi: 10.1007/s10957-011-9915-2.  Google Scholar

[19]

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.  doi: 10.1016/j.jmaa.2005.11.035.  Google Scholar

[20]

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.  doi: 10.1007/s10957-007-9345-3.  Google Scholar

[21]

P. Q. Khanh and N. D. Tuan, Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimalization,, J. Optim. Theory Appl., 139 (2008), 243.  doi: 10.1007/s10957-008-9414-2.  Google Scholar

[22]

C. R. Chen, S. J. Li and K. L. Teo, Higher order weak epiderivatives and applications to duality and optimality conditions,, Comput. Math. Appl., 57 (2009), 1389.  doi: 10.1016/j.camwa.2009.01.012.  Google Scholar

[23]

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.  doi: 10.1007/s11590-009-0170-5.  Google Scholar

[24]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer Series in Operations Research, (2000).   Google Scholar

[25]

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

[26]

D. T. Luc, "Theory of Vector Optimization,", Lecture Notes in Economics and Mathematical Systems, 319 (1989).   Google Scholar

[27]

Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps,, J. Optim. Theory Appl., 100 (1999), 365.  doi: 10.1023/A:1021786303883.  Google Scholar

[1]

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

[2]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051

[3]

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

[4]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053

[5]

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 & Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111

[6]

B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, 2007, 2007 (Special) : 145-154. doi: 10.3934/proc.2007.2007.145

[7]

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

[8]

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

[9]

Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745

[10]

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

[11]

Maria Do Rosario Grossinho, Rogério Martins. Subharmonic oscillations for some second-order differential equations without Landesman-Lazer conditions. Conference Publications, 2001, 2001 (Special) : 174-181. doi: 10.3934/proc.2001.2001.174

[12]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003

[13]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445

[14]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087

[15]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064

[16]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209

[17]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[18]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

[19]

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

[20]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]