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: |
[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.![]() ![]() ![]() |