March  2020, 16(2): 707-724. doi: 10.3934/jimo.2018174

Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

2. 

College of Computer Science, Chongqing University, Chongqing 400044, China

3. 

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

4. 

Center for General Education, China Medical University, Taichung 40402, Taiwan

* Corresponding author: Shengjie Li

Received  May 2017 Revised  May 2018 Published  March 2020 Early access  December 2018

Fund Project: This research was supported by the Natural Science Foundation of China (Nos: 11171362, 11401487), the Basic and Advanced Research Project of Chongqing (cstc2016jcyjA0239), the China Postdoctoral Science Foundation (No: 2015M582512), National Scholarship under the Grant of China Scholarship Council, the Fundamental Research Funds for the Central Universities(XDJK2019C073) and the grant MOST 106-2923-E-039-001-MY3

In this paper, we consider a class of constrained vector optimization problems by using image space analysis. A class of vector-valued separation functions and a $ \mathfrak{C} $-solution notion are proposed for the constrained vector optimization problems, respectively. Moreover, existence of a saddle point for the vector-valued separation function is characterized by the (regular) separation of two suitable subsets of the image space. By employing the separation function, we introduce a class of generalized vector-valued Lagrangian functions without involving any elements of the feasible set of constrained vector optimization problems. The relationships between the type-Ⅰ(Ⅱ) saddle points of the generalized Lagrangian functions and that of the function corresponding to the separation function are also established. Finally, optimality conditions for $ \mathfrak{C} $-solutions of constrained vector optimization problems are derived by the saddle-point conditions.

Citation: Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial and Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174
References:
[1]

J. W. Chen, Q. H. Ansari, Y. C. Liou and J. C. Yao, A proximal point algorithm based on decomposition method for cone constrained multiobjective optimization problems, Comput. Optim. Appl., 65 (2016), 289-308. doi: 10.1007/s10589-016-9840-2.

[2]

J. W. ChenY. J. ChoJ. K. Kim and J. Li, Multiobjective optimization problems with modified objective functions and cone constraints and applications, J. Glob. Optim., 49 (2011), 137-147.  doi: 10.1007/s10898-010-9539-3.

[3]

J. ChenL. Huang and S. J. Li, Separations and optimality of constrained multiobjective optimization via improvement sets, J. Optim. Theory Appl., 178 (2018), 794-823.  doi: 10.1007/s10957-018-1325-2.

[4]

J. W. ChenS. J. LiZ. Wan and J. C. Yao, Vector variational-like inequalities with constraints: Separation and alternative, J. Optim. Theory Appl., 166 (2015), 460-479.  doi: 10.1007/s10957-015-0736-6.

[5]

B. D. Craven, Control and Optimization, Chapman & Hall, 1995. doi: 10.1007/978-1-4899-7226-2.

[6]

B. D. Craven and X. Q. Yang, A nonsmooth version of alterative theorem and nonsmooth multiobjective programming, Utilitas Math., 40 (1991), 117-128. 

[7]

F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities. Image space analysis and separation,, In: Giannessi, F.(ed.) Vector Variational Inequalities and Vector Equilibria, Kluwer Academic, Dordrech/Boston/London, 38 (2000), 153-215. doi: 10.1007/978-1-4613-0299-5_11.

[8]

F. Giannessi, On the theory of Lagrangian duality, Optim. Lett., 1 (2007), 9-20.  doi: 10.1007/s11590-006-0013-6.

[9]

F. Giannessi and G. Mastroeni, Separation of sets and Wolfe duality, J. Glob. Optim., 42 (2008), 401-412.  doi: 10.1007/s10898-008-9301-2.

[10]

F. Giannessi and G. Mastroeni, Separation of sets and Wolfe duality, J. Glob. Optim., 44 (2009), 459-460.  doi: 10.1007/s10898-009-9406-2.

[11]

F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, In: R.W. Cottle, F. Giannessi and J.L. Lions (ed.), Variational Inequalities and Complementarity Problems, 151-186, Wiley, New York, (1980).

[12]

F. Giannessi, Semidifferentiable functions and necessary optimality conditions, J. Optim. Theory Appl., 60 (1989), 191-241.  doi: 10.1007/BF00940005.

[13]

F. Giannessi, Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions, Vol. 1., Springer, Berlin, 2005.

[14]

F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 60 (1984), 331-365. doi: 10.1007/BF00935321.

[15]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, Holland, 2000. doi: 10.1007/978-1-4613-0299-5.

[16]

F. Giannessi, G. Mastroeni and J. C. Yao, On maximum and variational principles via image space analysis, Positivity, 16 (2012), 405-427. doi: 10.1007/s11117-012-0160-1.

[17]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, New York/Berlin, 2003.

[18]

S. M. Guu and J. Li, Vector quasi-equilibrium problems: Separation, saddle points and error bounds for the solution set, J. Glob. Optim., 58 (2014), 751-767.  doi: 10.1007/s10898-013-0073-y.

[19]

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

[20]

J. Li and N. J. Huang, Image space analysis for variational inequalites with cone constraints and applications to traffic equilibria, Sci. China Math., 55 (2012), 851-868.  doi: 10.1007/s11425-011-4287-5.

[21]

S. J. LiY. D. Xu and S. K. Zhu, Nonlinear separation approach to constrained extremum problems, J. Optim. Theory Appl., 154 (2012), 842-856.  doi: 10.1007/s10957-012-0027-4.

[22]

J. Li and N. J. Huang, Image space analysis for vector variational inequalites with matrix inequality constraints and applications, J. Optim. Theory Appl., 145 (2010), 459-477.  doi: 10.1007/s10957-010-9691-4.

[23]

D. T. Luc, Theory of Vector Vptimization, Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, 1989.

[24]

G. Mastroeni and L. Pellegrini, On the image space analysis for vector variational inequalities, J. Ind. Manag. Optim., 1 (2005), 123-132.  doi: 10.3934/jimo.2005.1.123.

[25]

G. MastroeniB. PanicucciM. Passacantando and J. C. Yao, A separation approach to vector quasiequilibrium problems: Saddle point and gap function, Taiwanese J. Math., 13 (2009), 657-673.  doi: 10.11650/twjm/1500405393.

[26]

G. Mastroeni, Nonlinear separation in the image space with applications to penalty methods, Appl. Anal., 91 (2012), 1901-1914.  doi: 10.1080/00036811.2011.614603.

[27]

Q. Wang and S. J. Li, Semicontinuity of approximate solution mappings to generalized vector equilibrium problems, J. Indust. Manag. Optim., 12 (2016), 1303-1309.  doi: 10.3934/jimo.2016.12.1303.

[28]

H. X. WuH. Z. Luo and J. F. Yang, Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming, J. Glob. Optim., 59 (2014), 695-727.  doi: 10.1007/s10898-013-0093-7.

[29]

Y. D. Xu and S. J. Li, Nonlinear separation functions and constrained extremum problems, Optim. Lett., 8 (2014), 1149-1160.  doi: 10.1007/s11590-013-0644-3.

[30]

Y. D. Xu, Nonlinear separation functions, optimality conditions and error bounds for Ky Fan quasi-inequalities, Optim. Lett., 10 (2016), 527-542.  doi: 10.1007/s11590-015-0879-2.

[31]

Y. D. Xu and S. J. Li, Gap functions and error bounds for weak vector variational inequalities, Optim., 63 (2014), 1339-1352.  doi: 10.1080/02331934.2012.721115.

[32]

K. Q. Zhao and X. M. Yang, Characterizations of efficient and weakly efficient points in nonconvex vector optimization, J. Glob. Optim., 61 (2015), 575-590.  doi: 10.1007/s10898-014-0191-1.

[33]

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.

[34]

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.

show all references

References:
[1]

J. W. Chen, Q. H. Ansari, Y. C. Liou and J. C. Yao, A proximal point algorithm based on decomposition method for cone constrained multiobjective optimization problems, Comput. Optim. Appl., 65 (2016), 289-308. doi: 10.1007/s10589-016-9840-2.

[2]

J. W. ChenY. J. ChoJ. K. Kim and J. Li, Multiobjective optimization problems with modified objective functions and cone constraints and applications, J. Glob. Optim., 49 (2011), 137-147.  doi: 10.1007/s10898-010-9539-3.

[3]

J. ChenL. Huang and S. J. Li, Separations and optimality of constrained multiobjective optimization via improvement sets, J. Optim. Theory Appl., 178 (2018), 794-823.  doi: 10.1007/s10957-018-1325-2.

[4]

J. W. ChenS. J. LiZ. Wan and J. C. Yao, Vector variational-like inequalities with constraints: Separation and alternative, J. Optim. Theory Appl., 166 (2015), 460-479.  doi: 10.1007/s10957-015-0736-6.

[5]

B. D. Craven, Control and Optimization, Chapman & Hall, 1995. doi: 10.1007/978-1-4899-7226-2.

[6]

B. D. Craven and X. Q. Yang, A nonsmooth version of alterative theorem and nonsmooth multiobjective programming, Utilitas Math., 40 (1991), 117-128. 

[7]

F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities. Image space analysis and separation,, In: Giannessi, F.(ed.) Vector Variational Inequalities and Vector Equilibria, Kluwer Academic, Dordrech/Boston/London, 38 (2000), 153-215. doi: 10.1007/978-1-4613-0299-5_11.

[8]

F. Giannessi, On the theory of Lagrangian duality, Optim. Lett., 1 (2007), 9-20.  doi: 10.1007/s11590-006-0013-6.

[9]

F. Giannessi and G. Mastroeni, Separation of sets and Wolfe duality, J. Glob. Optim., 42 (2008), 401-412.  doi: 10.1007/s10898-008-9301-2.

[10]

F. Giannessi and G. Mastroeni, Separation of sets and Wolfe duality, J. Glob. Optim., 44 (2009), 459-460.  doi: 10.1007/s10898-009-9406-2.

[11]

F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, In: R.W. Cottle, F. Giannessi and J.L. Lions (ed.), Variational Inequalities and Complementarity Problems, 151-186, Wiley, New York, (1980).

[12]

F. Giannessi, Semidifferentiable functions and necessary optimality conditions, J. Optim. Theory Appl., 60 (1989), 191-241.  doi: 10.1007/BF00940005.

[13]

F. Giannessi, Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions, Vol. 1., Springer, Berlin, 2005.

[14]

F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 60 (1984), 331-365. doi: 10.1007/BF00935321.

[15]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, Holland, 2000. doi: 10.1007/978-1-4613-0299-5.

[16]

F. Giannessi, G. Mastroeni and J. C. Yao, On maximum and variational principles via image space analysis, Positivity, 16 (2012), 405-427. doi: 10.1007/s11117-012-0160-1.

[17]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, New York/Berlin, 2003.

[18]

S. M. Guu and J. Li, Vector quasi-equilibrium problems: Separation, saddle points and error bounds for the solution set, J. Glob. Optim., 58 (2014), 751-767.  doi: 10.1007/s10898-013-0073-y.

[19]

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

[20]

J. Li and N. J. Huang, Image space analysis for variational inequalites with cone constraints and applications to traffic equilibria, Sci. China Math., 55 (2012), 851-868.  doi: 10.1007/s11425-011-4287-5.

[21]

S. J. LiY. D. Xu and S. K. Zhu, Nonlinear separation approach to constrained extremum problems, J. Optim. Theory Appl., 154 (2012), 842-856.  doi: 10.1007/s10957-012-0027-4.

[22]

J. Li and N. J. Huang, Image space analysis for vector variational inequalites with matrix inequality constraints and applications, J. Optim. Theory Appl., 145 (2010), 459-477.  doi: 10.1007/s10957-010-9691-4.

[23]

D. T. Luc, Theory of Vector Vptimization, Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, 1989.

[24]

G. Mastroeni and L. Pellegrini, On the image space analysis for vector variational inequalities, J. Ind. Manag. Optim., 1 (2005), 123-132.  doi: 10.3934/jimo.2005.1.123.

[25]

G. MastroeniB. PanicucciM. Passacantando and J. C. Yao, A separation approach to vector quasiequilibrium problems: Saddle point and gap function, Taiwanese J. Math., 13 (2009), 657-673.  doi: 10.11650/twjm/1500405393.

[26]

G. Mastroeni, Nonlinear separation in the image space with applications to penalty methods, Appl. Anal., 91 (2012), 1901-1914.  doi: 10.1080/00036811.2011.614603.

[27]

Q. Wang and S. J. Li, Semicontinuity of approximate solution mappings to generalized vector equilibrium problems, J. Indust. Manag. Optim., 12 (2016), 1303-1309.  doi: 10.3934/jimo.2016.12.1303.

[28]

H. X. WuH. Z. Luo and J. F. Yang, Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming, J. Glob. Optim., 59 (2014), 695-727.  doi: 10.1007/s10898-013-0093-7.

[29]

Y. D. Xu and S. J. Li, Nonlinear separation functions and constrained extremum problems, Optim. Lett., 8 (2014), 1149-1160.  doi: 10.1007/s11590-013-0644-3.

[30]

Y. D. Xu, Nonlinear separation functions, optimality conditions and error bounds for Ky Fan quasi-inequalities, Optim. Lett., 10 (2016), 527-542.  doi: 10.1007/s11590-015-0879-2.

[31]

Y. D. Xu and S. J. Li, Gap functions and error bounds for weak vector variational inequalities, Optim., 63 (2014), 1339-1352.  doi: 10.1080/02331934.2012.721115.

[32]

K. Q. Zhao and X. M. Yang, Characterizations of efficient and weakly efficient points in nonconvex vector optimization, J. Glob. Optim., 61 (2015), 575-590.  doi: 10.1007/s10898-014-0191-1.

[33]

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.

[34]

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.

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