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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

* Corresponding author: Manxue You

Received  January 2019 Revised  October 2019 Published  March 2022 Early access  December 2020

Fund Project: This research was supported by the National Natural Science Foundation of China (Grant numbers: 12001438, 11971078, 11871059) and the Fund of China West Normal University (NO. 18Q059, 19B043)

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.

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.

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##### 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.
The red curve shows the set of objective function values
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