American Institute of Mathematical Sciences

Advanced
  • Previous Article
    First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits
  • JIMO Home
  • This Issue
  • Next Article
    Continuity, differentiability and semismoothness of generalized tensor functions
doi: 10.3934/jimo.2020176

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 Published  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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Israa Mohammed Khudher, Yahya Ismail Ibrahim, Suhaib Abduljabbar Altamir. Individual biometrics pattern based artificial image analysis techniques. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2020056
[2]

Andrea Scapin. Electrocommunication for weakly electric fish. Inverse Problems & Imaging, 2020, 14 (1) : 97-115. doi: 10.3934/ipi.2019065
[3]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313
[4]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[5]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909
[6]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017
[7]

Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321
[8]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[9]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[10]

Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008
[11]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[12]

Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051
[13]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[14]

Sohana Jahan. Discriminant analysis of regularized multidimensional scaling. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 255-267. doi: 10.3934/naco.2020024
[15]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
[16]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
[17]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027
[18]

Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013
[19]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023
[20]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (23)
  • HTML views (89)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences