Figure 1.
The red curve shows the set of objective function values
-
Previous Article
Planning rolling stock maintenance: Optimization of train arrival dates at a maintenance center
- JIMO Home
- This Issue
-
Next Article
Electricity supply chain coordination with carbon abatement technology investment under the benchmarking mechanism
March
2022, 18(2): 731-745.
doi: 10.3934/jimo.2020176
Perturbation of Image and conjugate duality for vector optimization
| 1. | College of Mathematics and Information, China West Normal University, Nanchong 637009, Sichuan, China |
| 2. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
This paper aims at employing the image space approach to investigate the conjugate duality theory for general constrained vector optimization problems. We introduce the concepts of conjugate map and subdifferential by using two types of maximums. We also construct the conjugate duality problems via a perturbation method. Moreover, the separation condition is proposed by means of vector weak separation functions. Then, it is proved to be a new sufficient condition, which ensures the strong duality theorem. This separation condition is different from the classical regular conditions in the literature. Simultaneously, the application to a nonconvex multi-objective optimization problem is shown to verify our main results.
Keywords:
Vector optimization,
image space analysis,
conjugate duality,
separation condition,
weakly subdifferentiable.
Mathematics Subject Classification:
90C26;90C29;90C46.
Citation:
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
![]()
References:
| [1] |
R. I. Bot, Conjugate Duality in Convex Optimization, Lecture Notes in Economics and Mathematical Systems, 637. Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04900-2. |
| [2] |
R. I. Bot, S. M. Grad and G. Wanka,
New constraint qualification and conjugate duality for composed convex optimization problems, J. Optim. Theory Appl., 135 (2007), 241-255.
doi: 10.1007/s10957-007-9247-4. |
| [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. |
| [4] |
J. W. Chen, S. J. Li, Z. P. Wang 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] |
M. Chinaie and J. Zafarani,
Image space analysis and scalarization of multivalued optimization, J. Optim. Theory Appl., 142 (2009), 451-467.
doi: 10.1007/s10957-009-9531-6. |
| [6] |
P. H. Dien, G. Mastroeni, M. Pappalardo and P. H. Quang,
Regularity condition for constrained extreme problems via image space, J. Optim. Theory Appl., 80 (1994), 19-37.
doi: 10.1007/BF02196591. |
| [7] |
F. Giannessi,
Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 42 (1984), 331-365.
doi: 10.1007/BF00935321. |
| [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, Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions, vol. 1. Springer, Berlin, 2005. |
| [10] |
C. Gutiérrez, B. Jiménez and V. Novo,
On approximate solutions in vector optimization problems via scalarization, Comput. Optim. Appl., 35 (2006), 305-324.
doi: 10.1007/s10589-006-8718-0. |
| [11] |
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. |
| [12] |
F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities, Image space analysis and seperation, In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, Kluwer Academic, Dordrech, (2000), 153-215. |
| [13] |
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. |
| [14] |
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. |
| [15] |
G. Mastroeni, Optimality conditions and image space analysis for vector optimization problems, In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, Vector Optimization, Springer, Dordrecht, 1 (2012), 169-220.
doi: 10.1007/978-3-642-21114-0_6. |
| [16] |
G. Mastroeni,
On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation, J. Glob. Optim., 53 (2012), 203-214.
doi: 10.1007/s10898-011-9674-5. |
| [17] |
G. Mastroeni,
Some applications of the image space analysis to the duality theory for constrained extremum problems, J. Glob. Optim., 46 (2010), 603-614.
doi: 10.1007/s10898-009-9445-8. |
| [18] |
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. |
| [19] |
G. Mastroeni, M. Pappalardo and N. D. Yen,
Image of a parametric optimization problem and continuity of the perturbation function, J. Optim. Theory Appl., 81 (1994), 193-202.
doi: 10.1007/BF02190319. |
| [20] |
A. Moldovan and L. Pellegrini,
On regularity for constrained extremum problems. Part 1: Suffficient optimality conditions, J. Optim. Theory Appl., 142 (2009), 147-163.
doi: 10.1007/s10957-009-9518-3. |
| [21] |
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. |
| [22] |
M. Pappalardo,
Image space approach to penalty methods, J. Optim. Theory Appl., 64 (1990), 141-152.
doi: 10.1007/BF00940028. |
| [23] |
T. Tanino,
Conjugate duality in vector optimization, J. Math. Anal. Appl., 167 (1992), 84-97.
doi: 10.1016/0022-247X(92)90237-8. |
| [24] |
F. Tardella,
On the image of a constrained extremum problem and some applications to existence of a minimum, J. Optim. Theory Appl., 60 (1989), 93-104.
doi: 10.1007/BF00938802. |
| [25] |
Y. D. Xu and S. J. Li,
Gap functions and error bounds for weak vector variational inequalities, Optimization, 63 (2014), 1339-1352.
doi: 10.1080/02331934.2012.721115. |
| [26] |
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. |
| [27] |
S. K. Zhu and S. J. Li,
United 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. |
| [28] |
S. K. Zhu and S. J. Li,
United 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] |
R. I. Bot, Conjugate Duality in Convex Optimization, Lecture Notes in Economics and Mathematical Systems, 637. Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04900-2. |
| [2] |
R. I. Bot, S. M. Grad and G. Wanka,
New constraint qualification and conjugate duality for composed convex optimization problems, J. Optim. Theory Appl., 135 (2007), 241-255.
doi: 10.1007/s10957-007-9247-4. |
| [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. |
| [4] |
J. W. Chen, S. J. Li, Z. P. Wang 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] |
M. Chinaie and J. Zafarani,
Image space analysis and scalarization of multivalued optimization, J. Optim. Theory Appl., 142 (2009), 451-467.
doi: 10.1007/s10957-009-9531-6. |
| [6] |
P. H. Dien, G. Mastroeni, M. Pappalardo and P. H. Quang,
Regularity condition for constrained extreme problems via image space, J. Optim. Theory Appl., 80 (1994), 19-37.
doi: 10.1007/BF02196591. |
| [7] |
F. Giannessi,
Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 42 (1984), 331-365.
doi: 10.1007/BF00935321. |
| [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, Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions, vol. 1. Springer, Berlin, 2005. |
| [10] |
C. Gutiérrez, B. Jiménez and V. Novo,
On approximate solutions in vector optimization problems via scalarization, Comput. Optim. Appl., 35 (2006), 305-324.
doi: 10.1007/s10589-006-8718-0. |
| [11] |
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. |
| [12] |
F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities, Image space analysis and seperation, In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, Kluwer Academic, Dordrech, (2000), 153-215. |
| [13] |
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. |
| [14] |
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. |
| [15] |
G. Mastroeni, Optimality conditions and image space analysis for vector optimization problems, In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, Vector Optimization, Springer, Dordrecht, 1 (2012), 169-220.
doi: 10.1007/978-3-642-21114-0_6. |
| [16] |
G. Mastroeni,
On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation, J. Glob. Optim., 53 (2012), 203-214.
doi: 10.1007/s10898-011-9674-5. |
| [17] |
G. Mastroeni,
Some applications of the image space analysis to the duality theory for constrained extremum problems, J. Glob. Optim., 46 (2010), 603-614.
doi: 10.1007/s10898-009-9445-8. |
| [18] |
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. |
| [19] |
G. Mastroeni, M. Pappalardo and N. D. Yen,
Image of a parametric optimization problem and continuity of the perturbation function, J. Optim. Theory Appl., 81 (1994), 193-202.
doi: 10.1007/BF02190319. |
| [20] |
A. Moldovan and L. Pellegrini,
On regularity for constrained extremum problems. Part 1: Suffficient optimality conditions, J. Optim. Theory Appl., 142 (2009), 147-163.
doi: 10.1007/s10957-009-9518-3. |
| [21] |
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. |
| [22] |
M. Pappalardo,
Image space approach to penalty methods, J. Optim. Theory Appl., 64 (1990), 141-152.
doi: 10.1007/BF00940028. |
| [23] |
T. Tanino,
Conjugate duality in vector optimization, J. Math. Anal. Appl., 167 (1992), 84-97.
doi: 10.1016/0022-247X(92)90237-8. |
| [24] |
F. Tardella,
On the image of a constrained extremum problem and some applications to existence of a minimum, J. Optim. Theory Appl., 60 (1989), 93-104.
doi: 10.1007/BF00938802. |
| [25] |
Y. D. Xu and S. J. Li,
Gap functions and error bounds for weak vector variational inequalities, Optimization, 63 (2014), 1339-1352.
doi: 10.1080/02331934.2012.721115. |
| [26] |
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. |
| [27] |
S. K. Zhu and S. J. Li,
United 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. |
| [28] |
S. K. Zhu and S. J. Li,
United 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] |
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 |
| [2] |
Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial and Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415 |
| [3] |
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 |
| [4] |
Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial and Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523 |
| [5] |
Radu Ioan Boţ, Sorin-Mihai Grad. On linear vector optimization duality in infinite-dimensional spaces. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 407-415. doi: 10.3934/naco.2011.1.407 |
| [6] |
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 |
| [7] |
Shouhong Yang. Semidefinite programming via image space analysis. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1187-1197. doi: 10.3934/jimo.2016.12.1187 |
| [8] |
Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial and Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381 |
| [9] |
Yubo Yuan. Canonical duality solution for alternating support vector machine. Journal of Industrial and Management Optimization, 2012, 8 (3) : 611-621. doi: 10.3934/jimo.2012.8.611 |
| [10] |
Nam-Yong Lee, Bradley J. Lucier. Preconditioned conjugate gradient method for boundary artifact-free image deblurring. Inverse Problems and Imaging, 2016, 10 (1) : 195-225. doi: 10.3934/ipi.2016.10.195 |
| [11] |
Shishun Li, Zhengda Huang. Guaranteed descent conjugate gradient methods with modified secant condition. Journal of Industrial and Management Optimization, 2008, 4 (4) : 739-755. doi: 10.3934/jimo.2008.4.739 |
| [12] |
Gérard Cohen, Alexander Vardy. Duality between packings and coverings of the Hamming space. Advances in Mathematics of Communications, 2007, 1 (1) : 93-97. doi: 10.3934/amc.2007.1.93 |
| [13] |
Tingting Wu, Yufei Yang, Huichao Jing. Two-step methods for image zooming using duality strategies. Numerical Algebra, Control and Optimization, 2014, 4 (3) : 209-225. doi: 10.3934/naco.2014.4.209 |
| [14] |
Guanghui Zhou, Qin Ni, Meilan Zeng. A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 595-608. doi: 10.3934/jimo.2016034 |
| [15] |
El-Sayed M.E. Mostafa. A nonlinear conjugate gradient method for a special class of matrix optimization problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 883-903. doi: 10.3934/jimo.2014.10.883 |
| [16] |
Wataru Nakamura, Yasushi Narushima, Hiroshi Yabe. Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization. Journal of Industrial and Management Optimization, 2013, 9 (3) : 595-619. doi: 10.3934/jimo.2013.9.595 |
| [17] |
Abdulkarim Hassan Ibrahim, Jitsupa Deepho, Auwal Bala Abubakar, Kazeem Olalekan Aremu. A modified Liu-Storey-Conjugate descent hybrid projection method for convex constrained nonlinear equations and image restoration. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 569-582. doi: 10.3934/naco.2021022 |
| [18] |
David Grant, Mahesh K. Varanasi. Duality theory for space-time codes over finite fields. Advances in Mathematics of Communications, 2008, 2 (1) : 35-54. doi: 10.3934/amc.2008.2.35 |
| [19] |
Charles Curry, Stephen Marsland, Robert I McLachlan. Principal symmetric space analysis. Journal of Computational Dynamics, 2019, 6 (2) : 251-276. doi: 10.3934/jcd.2019013 |
| [20] |
Xin Li, Ziguan Cui, Linhui Sun, Guanming Lu, Debnath Narayan. Research on iterative repair algorithm of Hyperchaotic image based on support vector machine. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1199-1218. doi: 10.3934/dcdss.2019083 |
2020 Impact Factor: 1.801
Tools
Metrics
Other articles
by authors
[Back to Top]