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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 2019 Early access  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 and 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.

[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.

[3]

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

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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. 

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[30]

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

[31]

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

[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. 

[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.

[34]

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

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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. 

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[30]

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

[31]

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

[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. 

[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.

[34]

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

[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.

[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.

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