-
Previous Article
Optimal pricing and inventory strategies for introducing a new product based on demand substitution effects
- JIMO Home
- This Issue
-
Next Article
Shipper collaboration in forward and reverse logistics
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 |
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.
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. |
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. 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. |
[1] |
Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2971-2989. doi: 10.3934/jimo.2019089 |
[2] |
G. Mastroeni, L. Pellegrini. On the image space analysis for vector variational inequalities. Journal of Industrial and Management Optimization, 2005, 1 (1) : 123-132. doi: 10.3934/jimo.2005.1.123 |
[3] |
Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011 |
[4] |
Xiaoqing Ou, Suliman Al-Homidan, Qamrul Hasan Ansari, Jiawei Chen. Image space analysis for uncertain multiobjective optimization problems: Robust optimality conditions. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021199 |
[5] |
Matteo Focardi. Vector-valued obstacle problems for non-local energies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 487-507. doi: 10.3934/dcdsb.2012.17.487 |
[6] |
Mohammad Safdari. The regularity of some vector-valued variational inequalities with gradient constraints. Communications on Pure and Applied Analysis, 2018, 17 (2) : 413-428. doi: 10.3934/cpaa.2018023 |
[7] |
Markus Kunze, Abdallah Maichine, Abdelaziz Rhandi. Vector-valued Schrödinger operators in Lp-spaces. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1529-1541. doi: 10.3934/dcdss.2020086 |
[8] |
Tadeusz Antczak. The $ F $-objective function method for differentiable interval-valued vector optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2761-2782. doi: 10.3934/jimo.2020093 |
[9] |
Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial and Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483 |
[10] |
Michele Campiti. Korovkin-type approximation of set-valued and vector-valued functions. Mathematical Foundations of Computing, 2022, 5 (3) : 231-239. doi: 10.3934/mfc.2021032 |
[11] |
Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial and Management Optimization, 2022, 18 (2) : 731-745. doi: 10.3934/jimo.2020176 |
[12] |
Luciano Abadías, Carlos Lizama, Pedro J. Miana, M. Pilar Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2679-2708. doi: 10.3934/dcds.2019112 |
[13] |
Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure and Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313 |
[14] |
Nikos Katzourakis. Corrigendum to the paper: Nonuniqueness in Vector-Valued Calculus of Variations in $ L^\infty $ and some Linear Elliptic Systems. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2197-2198. doi: 10.3934/cpaa.2019098 |
[15] |
Olaf Klein. On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2591-2614. doi: 10.3934/dcds.2015.35.2591 |
[16] |
Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108 |
[17] |
Emmanuel Hebey. The Lin-Ni's conjecture for vector-valued Schrödinger equations in the closed case. Communications on Pure and Applied Analysis, 2010, 9 (4) : 955-962. doi: 10.3934/cpaa.2010.9.955 |
[18] |
Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial and Management Optimization, 2020, 16 (1) : 55-70. doi: 10.3934/jimo.2018140 |
[19] |
Giovanni P. Crespi, Ivan Ginchev, Matteo Rocca. Two approaches toward constrained vector optimization and identity of the solutions. Journal of Industrial and Management Optimization, 2005, 1 (4) : 549-563. doi: 10.3934/jimo.2005.1.549 |
[20] |
M. M. Rao. Integration with vector valued measures. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5429-5440. doi: 10.3934/dcds.2013.33.5429 |
2020 Impact Factor: 1.801
Tools
Metrics
Other articles
by authors
[Back to Top]