American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Single-machine bi-criterion scheduling with release times and exponentially time-dependent learning effects
  • JIMO Home
  • This Issue
  • Next Article
    Performance analysis of a cooperative flow game algorithm in ad hoc networks and a comparison to Dijkstra's algorithm
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

[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

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

[1]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022
[2]

Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35
[3]

Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019
[4]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327
[5]

Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1
[6]

Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 55-70. doi: 10.3934/jimo.2018140
[7]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673
[8]

Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031
[9]

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
[10]

Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial & Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57
[11]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
[12]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[13]

Tran Ngoc Thang, Nguyen Thi Bach Kim. Outcome space algorithm for generalized multiplicative problems and optimization over the efficient set. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1417-1433. doi: 10.3934/jimo.2016.12.1417
[14]

Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial & Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1
[15]

Behrouz Kheirfam, Kamal mirnia. Comments on ''Generalized support set invariancy sensitivity analysis in linear optimization''. Journal of Industrial & Management Optimization, 2008, 4 (3) : 611-616. doi: 10.3934/jimo.2008.4.611
[16]

Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246
[17]

Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548
[18]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309
[19]

Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435
[20]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (144)
  • HTML views (894)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences