# American Institute of Mathematical Sciences

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. Chen, Y. J. Cho, J. 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. Chen, L. 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. Chen, S. J. Li, Z. 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. Li, Y. 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. Mastroeni, B. Panicucci, M. 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. Wu, H. 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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##### 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. Chen, Y. J. Cho, J. 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. Chen, L. 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. Chen, S. J. Li, Z. 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. Li, Y. 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. Mastroeni, B. Panicucci, M. 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. Wu, H. 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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