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July  2019, 15(3): 1101-1116. doi: 10.3934/jimo.2018087

Unified optimality conditions for set-valued optimizations

Geng-Hua Li and  Sheng-Jie Li ,

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

* Corresponding author: Sheng-Jie Li

Received  July 2016 Revised  May 2017 Published  July 2018

Fund Project: This research was supported by the National Natural Science Foundation of China (Grant numbers: 11171362, 11571055).

This paper is devoted to the study of unified optimality conditions for constrained set-valued optimization problems via image space analysis. Necessary and sufficient optimality conditions are given in terms of tangent cones of extended image set. By exploiting such results, we analyse the optimality conditions employing different generalized derivatives.

Keywords: Set-valued optimization problem, optimality condition, image space analysis, tangent cone, generalized derivative.
Mathematics Subject Classification: Primary: 90C31, 90C46; Secondary: 49J53.
Citation: 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
References:
[1]

J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser, Boston, 1990. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[2]

G. Bigi and M. Castellani, K-epiderivatives for set-valued function and optimization, Math. Methods, 55 (2002), 401-412.  doi: 10.1007/s001860200187.  Google Scholar

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G. Castellani and F. Giannessi, Decomposition of mathematical programs by means of theorems of alternative for linear and nonlinear systems, In Proc. Ninth Internat. Math. Programming Sympos., Budapest. Survey of Mathematical Programming, North-Holland, Amsterdam, 2 (1979), 423-439.  Google Scholar

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M. Chinaie and J. Zafarani, A new approach to constrained optimization via image space analysis, Positivity, 20 (2016), 99-114.  doi: 10.1007/s11117-015-0343-7.  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

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

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F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 42 (1984), 331-365.  doi: 10.1007/BF00935321.  Google Scholar

[8]

F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities image space analysis and separation, in Vector Variational Inequalities and Vector Equilibria, Mathematical Theories (eds F. Giannessi), Kluwer Academic, Dordrecht, 38 (2000), 153-215. doi: 10.1007/978-1-4613-0299-5_11.  Google Scholar

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F. Giannessi, Constrained Optimization and Image Space Analysis, vol. 1: Separation of Sets and Optimality Conditions, Springer, New York, 2005. doi: 10.1007/0-387-28020-0.  Google Scholar

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A. Götz and J. Jahn, The Lagrange multiplier rule in set-valued optimization, SIAM J. Optim., 10 (2000), 331-344.  doi: 10.1137/S1052623496311697.  Google Scholar

[11]

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

[12]

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

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J. Jahn, Vector Optimization. Theory, Applications, and Extensions, Springer, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.  Google Scholar

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A. A. Khan, C. Tammer and C. Zǎlinescu, Set-valued Optimization. An Introduction with Applications, Springer, Heidelberg, 2015. doi: 10.1007/978-3-642-54265-7.  Google Scholar

[15]

J. Li, S. Q. Feng and Z. Zhang, A unified approach for constrained extremum problems: Image space analysis, J. Optim. Theory Appl., 159 (2013), 69-92.  doi: 10.1007/s10957-013-0276-x.  Google Scholar

[16]

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

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

[18]

A. Moldovan and L. Pellegrini, On regularity for constrained extremum problems. Part1: Sufficient optimality conditions, J. Optim. Theory Appl., 142 (2009), 147-163.  doi: 10.1007/s10957-009-9518-3.  Google Scholar

[19]

A. Moldovan and L. Pellegrini, On regularity for constrained extremum problems. Part 2: Necessary optimality conditions, J. Optim. Theory Appl., 142 (2009), 165-183.  doi: 10.1007/s10957-009-9521-8.  Google Scholar

[20]

A. Taa, Optimality conditions for vector mathematical programming via a theorem of the alternative, J. Math. Anal. Appl., 233 (1999), 233-245.  doi: 10.1006/jmaa.1999.6288.  Google Scholar

[21]

S. K. Zhu and S. J. Li, Unified duality theory for constrained extremum problems. Part Ⅰ: image space analysis, J. Optim. Theory Appl., 161 (2014), 738-762.  doi: 10.1007/s10957-013-0468-4.  Google Scholar

[22]

S. K. Zhu and S. J. Li, Unified duality theory for constrained extremum problems. Part Ⅱ: Special duality schemes, J. Optim. Theory Appl., 161 (2014), 763-782.  doi: 10.1007/s10957-013-0467-5.  Google Scholar

[23]

S. K. Zhu, S. J. Li and K. L. Teo, Second-order Karush-Kuhn-Tucker optimality conditions for set-valued optimization, J. Glob. Optim., 58 (2014), 673-692.  doi: 10.1007/s10898-013-0067-9.  Google Scholar

show all references

References:
[1]

J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser, Boston, 1990. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[2]

G. Bigi and M. Castellani, K-epiderivatives for set-valued function and optimization, Math. Methods, 55 (2002), 401-412.  doi: 10.1007/s001860200187.  Google Scholar

[3]

G. Castellani and F. Giannessi, Decomposition of mathematical programs by means of theorems of alternative for linear and nonlinear systems, In Proc. Ninth Internat. Math. Programming Sympos., Budapest. Survey of Mathematical Programming, North-Holland, Amsterdam, 2 (1979), 423-439.  Google Scholar

[4]

M. Chinaie and J. Zafarani, A new approach to constrained optimization via image space analysis, Positivity, 20 (2016), 99-114.  doi: 10.1007/s11117-015-0343-7.  Google Scholar

[5]

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

[6]

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

[7]

F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 42 (1984), 331-365.  doi: 10.1007/BF00935321.  Google Scholar

[8]

F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities image space analysis and separation, in Vector Variational Inequalities and Vector Equilibria, Mathematical Theories (eds F. Giannessi), Kluwer Academic, Dordrecht, 38 (2000), 153-215. doi: 10.1007/978-1-4613-0299-5_11.  Google Scholar

[9]

F. Giannessi, Constrained Optimization and Image Space Analysis, vol. 1: Separation of Sets and Optimality Conditions, Springer, New York, 2005. doi: 10.1007/0-387-28020-0.  Google Scholar

[10]

A. Götz and J. Jahn, The Lagrange multiplier rule in set-valued optimization, SIAM J. Optim., 10 (2000), 331-344.  doi: 10.1137/S1052623496311697.  Google Scholar

[11]

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

[12]

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

[13]

J. Jahn, Vector Optimization. Theory, Applications, and Extensions, Springer, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.  Google Scholar

[14]

A. A. Khan, C. Tammer and C. Zǎlinescu, Set-valued Optimization. An Introduction with Applications, Springer, Heidelberg, 2015. doi: 10.1007/978-3-642-54265-7.  Google Scholar

[15]

J. Li, S. Q. Feng and Z. Zhang, A unified approach for constrained extremum problems: Image space analysis, J. Optim. Theory Appl., 159 (2013), 69-92.  doi: 10.1007/s10957-013-0276-x.  Google Scholar

[16]

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

[17]

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

[18]

A. Moldovan and L. Pellegrini, On regularity for constrained extremum problems. Part1: Sufficient optimality conditions, J. Optim. Theory Appl., 142 (2009), 147-163.  doi: 10.1007/s10957-009-9518-3.  Google Scholar

[19]

A. Moldovan and L. Pellegrini, On regularity for constrained extremum problems. Part 2: Necessary optimality conditions, J. Optim. Theory Appl., 142 (2009), 165-183.  doi: 10.1007/s10957-009-9521-8.  Google Scholar

[20]

A. Taa, Optimality conditions for vector mathematical programming via a theorem of the alternative, J. Math. Anal. Appl., 233 (1999), 233-245.  doi: 10.1006/jmaa.1999.6288.  Google Scholar

[21]

S. K. Zhu and S. J. Li, Unified duality theory for constrained extremum problems. Part Ⅰ: image space analysis, J. Optim. Theory Appl., 161 (2014), 738-762.  doi: 10.1007/s10957-013-0468-4.  Google Scholar

[22]

S. K. Zhu and S. J. Li, Unified duality theory for constrained extremum problems. Part Ⅱ: Special duality schemes, J. Optim. Theory Appl., 161 (2014), 763-782.  doi: 10.1007/s10957-013-0467-5.  Google Scholar

[23]

S. K. Zhu, S. J. Li and K. L. Teo, Second-order Karush-Kuhn-Tucker optimality conditions for set-valued optimization, J. Glob. Optim., 58 (2014), 673-692.  doi: 10.1007/s10898-013-0067-9.  Google Scholar

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