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doi: 10.3934/jimo.2018087

Unified optimality conditions for set-valued optimizations

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.

Citation: Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 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.

[2]

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

[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.

[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.

[5]

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

[6]

G. P. CrespiI. 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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[15]

J. LiS. 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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

S. K. ZhuS. 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.

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.

[2]

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

[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.

[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.

[5]

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

[6]

G. P. CrespiI. 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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[15]

J. LiS. 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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

S. K. ZhuS. 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.

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