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April  2019, 15(2): 465-480. doi: 10.3934/jimo.2018051

Higher-order weak radial epiderivatives and non-convex set-valued optimization problems

Qilin Wang a,, , Liu He a, and  Shengjie Li b,

a. 

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China
b. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

1 Corresponding author.

Received  July 2017 Revised  December 2017 Published  April 2018

Fund Project: This research was partially supported by Chongqing Natural Science Foundation Project of CQ CSTC(Nos. 2015jcyjA30009, 2015jcyjBX0131,2017jcyjAX0382), the Program of Chongqing Innovation Team Project in University (No. CXTDX201601022) and the National Natural Science Foundation of China (No. 11571055).

In the paper, we propose the notion of the higher-order weak radial epiderivative of a set-valued map, and discuss some of its properties. Then, by virtue of the higher-order weak radial epiderivative, we establish the optimality necessary conditions and sufficient ones of weak efficient solutions (Pareto efficient solutions) for non-convex set-valued optimization problems. Some of the obtained results improve and extend the recent existing results. Several examples are provided to show the main results obtained.

Keywords: Set-valued optimization problems, higher-order weak radial epiderivatives, optimality conditions, weak efficient solutions, Pareto efficient solutions.
Mathematics Subject Classification: 90C26, 90C46.
Citation: 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
References:
[1]

N. L. H. Anh, Mixed type duality for set-valued optimization problems via higher-order radial epiderivatives, Numer. Func. Anal. Optim., 37 (2016), 823--838.  doi: 10.1080/01630563.2016.1179202.  Google Scholar

[2]

N. L. H. Anh, Higher-order optimality conditions for strict and weak efficient solutions in set-valued optimization, Positivity, 20 (2016), 499-514.  doi: 10.1007/s11117-015-0369-x.  Google Scholar

[3]

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

[4]

E. M. Bednarczuk and W. Song, Contingent epiderivative and its applications to set-valued optimization, Control Cybern, 27 (1998), 376-386.   Google Scholar

[5]

J. M. Borwein, On the existence of Pareto efficient points, Math. Oper. Res., 8 (1983), 64-73.  doi: 10.1287/moor.8.1.64.  Google Scholar

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G. Bouligand, Sur l'existence des demi-tangents á une courbe de Jordan, Fundamenta Math., 15 (1930), 215-215.   Google Scholar

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C. Certh and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J.Optim.Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.  Google Scholar

[8]

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), 1389-1399.  doi: 10.1016/j.camwa.2009.01.012.  Google Scholar

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G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res., 48 (1998), 187-200.  doi: 10.1007/s001860050021.  Google Scholar

[10]

S. Y. Cho, Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space, J. Appl. Anal. Comput., 8 (2018), 19-31.   Google Scholar

[11]

T. D. Chuong and J. C. Yao, Generalized Clarke epiderivatives of parametric vector optimization problems, J. Optim. Theory Appl., 146 (2010), 77-94.  doi: 10.1007/s10957-010-9646-9.  Google Scholar

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H. W. Corley, Optimality conditions for maximizations of set-valued functions, J. Optim. Theory Appl., 58 (1988), 1-10.  doi: 10.1007/BF00939767.  Google Scholar

[13]

G. P. Crepsi, I. Ginchev and M. Rocca, First-order optimality conditions in set-valued optimization, Math. Meth. Oper. Res., 63 (2006), 87-106.  doi: 10.1007/s00186-005-0023-7.  Google Scholar

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M. Durea, First and second order optimality conditions for set-valued optimization problems, Rend. Circ. Mat. Palermo 2, 53 (2004), 451-468.  doi: 10.1007/BF02875738.  Google Scholar

[15]

M. Durea, Optimality conditions for weak and firm efficiency in set-valued optimization, J. Math. Anal. Appl., 344 (2008), 1018-1028.  doi: 10.1016/j.jmaa.2008.03.053.  Google Scholar

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F. Flores-Bazán, Radial epiderivatives and asymptotic function in nonconvex vector optimization, SIAM J. Optim., 14 (2003), 284-305.  doi: 10.1137/S1052623401392111.  Google Scholar

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J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Springer, Berlin, 2004.  Google Scholar

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J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Meth. Oper. Res., 46 (1997), 193-211.  doi: 10.1007/BF01217690.  Google Scholar

[20]

J. Jahn and A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: Optimality conditions, Numer. Func. Anal. Optim., 23 (2002), 807-831.  doi: 10.1081/NFA-120016271.  Google Scholar

[21]

R. Kasimbeyli, Radial epiderivatives and set-valued optimization, Optimization, 58 (2009), 521-534.  doi: 10.1080/02331930902928310.  Google Scholar

[22]

C. S. Lalitha and R. Arora, Weak Clarke epiderivative in set-valued optimization, J. Math. Anal. Appl., 342 (2008), 704-714.  doi: 10.1016/j.jmaa.2007.11.057.  Google Scholar

[23]

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

[24]

S. J. Li, K. L. Teo and X. Q. Yang, Higher-order Mond-Weir duality for set-valued optimization, J. Comput. Appl. Math., 217 (2008), 339-349.  doi: 10.1016/j.cam.2007.02.011.  Google Scholar

[25]

S. J. Li, X. Q. Yang and G. Y. Chen, Nonconvex vector optimization of set-valued mappings, J. Math. Anal. Appl., 283 (2003), 337-350.  doi: 10.1016/S0022-247X(02)00410-9.  Google Scholar

[26]

X. J. Long, J. W. Peng and M. M. Wong, Generalized radial epiderivatives and nonconvex set-valued optimization problems, Applicable Analysis, 91 (2012), 1891-1900.  doi: 10.1080/00036811.2012.682057.  Google Scholar

[27]

D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989. doi: 10.1007/978-3-642-50280-4.  Google Scholar

[28]

D. T. Luc, Contingent derivatives of set-valued maps and applications to vector optimization, Math. Prog., 50 (1991), 99-111.  doi: 10.1007/BF01594928.  Google Scholar

[29]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. I Basic Theory, Springer, Berlin, 2006.  Google Scholar

[30]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. II Applications, Springer, Berlin, 2006.  Google Scholar

[31]

X. L. Qin and J. C. Yao, Projection splitting algorithms for nonself operators, J. Nonlinear Convex Anal., 18 (2017), 925-935.   Google Scholar

[32]

B. Soleimani and C. Tammer, A vector-valued Ekelands variational principle in vector optimization with variable ordering structures, J. Nonlinear Var. Anal., 1 (2017), 89-110.   Google Scholar

[33]

X. K. Sun and S. J. Li, Generalized second-order contingent epiderivatives in parametric vector optimization problems, J. Optim. Theory Appl., 58 (2014), 351-363.  doi: 10.1007/s10898-013-0054-1.  Google Scholar

[34]

A. Taa, Set-valued derivatives of multifunctions and optimality conditions, Numer. Funct. Anal. Optim., 19 (1998), 121-140.  doi: 10.1080/01630569808816819.  Google Scholar

[35]

L. T. Tung, Strong Karush-Kuhn-Tucker optimality conditions and duality for nonsmooth multiobjective semi-infinite programming via Michel-Penot subdifferential, J. Nonlinear Funct. Anal., 2017 (2017), Article ID 49. Google Scholar

[36]

Q. L. Wang and S. J. Li, Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency, Numer. Func. Anal. Optim., 30 (2009), 849-869.  doi: 10.1080/01630560903139540.  Google Scholar

show all references

References:
[1]

N. L. H. Anh, Mixed type duality for set-valued optimization problems via higher-order radial epiderivatives, Numer. Func. Anal. Optim., 37 (2016), 823--838.  doi: 10.1080/01630563.2016.1179202.  Google Scholar

[2]

N. L. H. Anh, Higher-order optimality conditions for strict and weak efficient solutions in set-valued optimization, Positivity, 20 (2016), 499-514.  doi: 10.1007/s11117-015-0369-x.  Google Scholar

[3]

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

[4]

E. M. Bednarczuk and W. Song, Contingent epiderivative and its applications to set-valued optimization, Control Cybern, 27 (1998), 376-386.   Google Scholar

[5]

J. M. Borwein, On the existence of Pareto efficient points, Math. Oper. Res., 8 (1983), 64-73.  doi: 10.1287/moor.8.1.64.  Google Scholar

[6]

G. Bouligand, Sur l'existence des demi-tangents á une courbe de Jordan, Fundamenta Math., 15 (1930), 215-215.   Google Scholar

[7]

C. Certh and P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J.Optim.Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.  Google Scholar

[8]

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), 1389-1399.  doi: 10.1016/j.camwa.2009.01.012.  Google Scholar

[9]

G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res., 48 (1998), 187-200.  doi: 10.1007/s001860050021.  Google Scholar

[10]

S. Y. Cho, Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space, J. Appl. Anal. Comput., 8 (2018), 19-31.   Google Scholar

[11]

T. D. Chuong and J. C. Yao, Generalized Clarke epiderivatives of parametric vector optimization problems, J. Optim. Theory Appl., 146 (2010), 77-94.  doi: 10.1007/s10957-010-9646-9.  Google Scholar

[12]

H. W. Corley, Optimality conditions for maximizations of set-valued functions, J. Optim. Theory Appl., 58 (1988), 1-10.  doi: 10.1007/BF00939767.  Google Scholar

[13]

G. P. Crepsi, I. Ginchev and M. Rocca, First-order optimality conditions in set-valued optimization, Math. Meth. Oper. Res., 63 (2006), 87-106.  doi: 10.1007/s00186-005-0023-7.  Google Scholar

[14]

M. Durea, First and second order optimality conditions for set-valued optimization problems, Rend. Circ. Mat. Palermo 2, 53 (2004), 451-468.  doi: 10.1007/BF02875738.  Google Scholar

[15]

M. Durea, Optimality conditions for weak and firm efficiency in set-valued optimization, J. Math. Anal. Appl., 344 (2008), 1018-1028.  doi: 10.1016/j.jmaa.2008.03.053.  Google Scholar

[16]

F. Flores-Bazán, Optimality conditions in nonconvex set-valued optimization, Math. Methods Oper. Res., 53 (2001), 403-417.  doi: 10.1007/s001860100130.  Google Scholar

[17]

F. Flores-Bazán, Radial epiderivatives and asymptotic function in nonconvex vector optimization, SIAM J. Optim., 14 (2003), 284-305.  doi: 10.1137/S1052623401392111.  Google Scholar

[18]

J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Springer, Berlin, 2004.  Google Scholar

[19]

J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Meth. Oper. Res., 46 (1997), 193-211.  doi: 10.1007/BF01217690.  Google Scholar

[20]

J. Jahn and A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: Optimality conditions, Numer. Func. Anal. Optim., 23 (2002), 807-831.  doi: 10.1081/NFA-120016271.  Google Scholar

[21]

R. Kasimbeyli, Radial epiderivatives and set-valued optimization, Optimization, 58 (2009), 521-534.  doi: 10.1080/02331930902928310.  Google Scholar

[22]

C. S. Lalitha and R. Arora, Weak Clarke epiderivative in set-valued optimization, J. Math. Anal. Appl., 342 (2008), 704-714.  doi: 10.1016/j.jmaa.2007.11.057.  Google Scholar

[23]

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

[24]

S. J. Li, K. L. Teo and X. Q. Yang, Higher-order Mond-Weir duality for set-valued optimization, J. Comput. Appl. Math., 217 (2008), 339-349.  doi: 10.1016/j.cam.2007.02.011.  Google Scholar

[25]

S. J. Li, X. Q. Yang and G. Y. Chen, Nonconvex vector optimization of set-valued mappings, J. Math. Anal. Appl., 283 (2003), 337-350.  doi: 10.1016/S0022-247X(02)00410-9.  Google Scholar

[26]

X. J. Long, J. W. Peng and M. M. Wong, Generalized radial epiderivatives and nonconvex set-valued optimization problems, Applicable Analysis, 91 (2012), 1891-1900.  doi: 10.1080/00036811.2012.682057.  Google Scholar

[27]

D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989. doi: 10.1007/978-3-642-50280-4.  Google Scholar

[28]

D. T. Luc, Contingent derivatives of set-valued maps and applications to vector optimization, Math. Prog., 50 (1991), 99-111.  doi: 10.1007/BF01594928.  Google Scholar

[29]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. I Basic Theory, Springer, Berlin, 2006.  Google Scholar

[30]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. II Applications, Springer, Berlin, 2006.  Google Scholar

[31]

X. L. Qin and J. C. Yao, Projection splitting algorithms for nonself operators, J. Nonlinear Convex Anal., 18 (2017), 925-935.   Google Scholar

[32]

B. Soleimani and C. Tammer, A vector-valued Ekelands variational principle in vector optimization with variable ordering structures, J. Nonlinear Var. Anal., 1 (2017), 89-110.   Google Scholar

[33]

X. K. Sun and S. J. Li, Generalized second-order contingent epiderivatives in parametric vector optimization problems, J. Optim. Theory Appl., 58 (2014), 351-363.  doi: 10.1007/s10898-013-0054-1.  Google Scholar

[34]

A. Taa, Set-valued derivatives of multifunctions and optimality conditions, Numer. Funct. Anal. Optim., 19 (1998), 121-140.  doi: 10.1080/01630569808816819.  Google Scholar

[35]

L. T. Tung, Strong Karush-Kuhn-Tucker optimality conditions and duality for nonsmooth multiobjective semi-infinite programming via Michel-Penot subdifferential, J. Nonlinear Funct. Anal., 2017 (2017), Article ID 49. Google Scholar

[36]

Q. L. Wang and S. J. Li, Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency, Numer. Func. Anal. Optim., 30 (2009), 849-869.  doi: 10.1080/01630560903139540.  Google Scholar

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