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doi: 10.3934/jimo.2020176
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Perturbation of Image and conjugate duality for vector optimization

Manxue You 1,, and  Shengjie Li 2,

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 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.

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 & Management Optimization, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 42 (1984), 331-365.  doi: 10.1007/BF00935321.  Google Scholar

[8]

F. Giannessi, On the theory of Lagrangian duality, Optim. Lett., 1 (2007), 9-20.  doi: 10.1007/s11590-006-0013-6.  Google Scholar

[9]

F. Giannessi, Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions, vol. 1. Springer, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. Pappalardo, Image space approach to penalty methods, J. Optim. Theory Appl., 64 (1990), 141-152.  doi: 10.1007/BF00940028.  Google Scholar

[23]

T. Tanino, Conjugate duality in vector optimization, J. Math. Anal. Appl., 167 (1992), 84-97.  doi: 10.1016/0022-247X(92)90237-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 42 (1984), 331-365.  doi: 10.1007/BF00935321.  Google Scholar

[8]

F. Giannessi, On the theory of Lagrangian duality, Optim. Lett., 1 (2007), 9-20.  doi: 10.1007/s11590-006-0013-6.  Google Scholar

[9]

F. Giannessi, Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions, vol. 1. Springer, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. Pappalardo, Image space approach to penalty methods, J. Optim. Theory Appl., 64 (1990), 141-152.  doi: 10.1007/BF00940028.  Google Scholar

[23]

T. Tanino, Conjugate duality in vector optimization, J. Math. Anal. Appl., 167 (1992), 84-97.  doi: 10.1016/0022-247X(92)90237-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The red curve shows the set of objective function values
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