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Dynamic optimal decision making for manufacturers with limited attention based on sparse dynamic programming
Higher-order weak radial epiderivatives and non-convex set-valued optimization problems
a. | College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China |
b. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
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.
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. 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. |
[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. 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. |
[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. 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. |
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. 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. |
[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. 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. |
[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. 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. |
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