• 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]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[2]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[3]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[4]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[5]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[6]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[7]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[8]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[9]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[10]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[11]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[12]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[13]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[14]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[15]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[16]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[17]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[18]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[19]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[20]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]