American Institute of Mathematical Sciences

Advanced
March  2020, 16(2): 707-724. doi: 10.3934/jimo.2018174

Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points

Jiawei Chen 1,2, , Shengjie Li 3,, and  Jen-Chih Yao 4,

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

* Corresponding author: Shengjie Li

Received  May 2017 Revised  May 2018 Published  December 2018

Fund Project: This research was supported by the Natural Science Foundation of China (Nos: 11171362, 11401487), the Basic and Advanced Research Project of Chongqing (cstc2016jcyjA0239), the China Postdoctoral Science Foundation (No: 2015M582512), National Scholarship under the Grant of China Scholarship Council, the Fundamental Research Funds for the Central Universities(XDJK2019C073) and the grant MOST 106-2923-E-039-001-MY3

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.

Keywords: Constrained vector optimization problem, image space analysis, nonlinear vector-valued separation function, saddle point, optimality condition.
Mathematics Subject Classification: Primary: 49J40; Secondary: 65K10.
Citation: Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial & Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

B. D. Craven, Control and Optimization, Chapman & Hall, 1995. doi: 10.1007/978-1-4899-7226-2.  Google Scholar

[6]

B. D. Craven and X. Q. Yang, A nonsmooth version of alterative theorem and nonsmooth multiobjective programming, Utilitas Math., 40 (1991), 117-128.   Google Scholar

[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.  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 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

[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.  Google Scholar

[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).  Google Scholar

[12]

F. Giannessi, Semidifferentiable functions and necessary optimality conditions, J. Optim. Theory Appl., 60 (1989), 191-241.  doi: 10.1007/BF00940005.  Google Scholar

[13]

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

[14]

F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 60 (1984), 331-365. doi: 10.1007/BF00935321.  Google Scholar

[15]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, Holland, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar

[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.  Google Scholar

[17]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, New York/Berlin, 2003.  Google Scholar

[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.  Google Scholar

[19]

J. Jahn, Vector Optimization Theory, Applications and Extensions, Springer, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. T. Luc, Theory of Vector Vptimization, Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

B. D. Craven, Control and Optimization, Chapman & Hall, 1995. doi: 10.1007/978-1-4899-7226-2.  Google Scholar

[6]

B. D. Craven and X. Q. Yang, A nonsmooth version of alterative theorem and nonsmooth multiobjective programming, Utilitas Math., 40 (1991), 117-128.   Google Scholar

[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.  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 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

[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.  Google Scholar

[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).  Google Scholar

[12]

F. Giannessi, Semidifferentiable functions and necessary optimality conditions, J. Optim. Theory Appl., 60 (1989), 191-241.  doi: 10.1007/BF00940005.  Google Scholar

[13]

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

[14]

F. Giannessi, Theorems of the alternative and optimality conditions, J. Optim. Theory Appl., 60 (1984), 331-365. doi: 10.1007/BF00935321.  Google Scholar

[15]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, Holland, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar

[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.  Google Scholar

[17]

A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, New York/Berlin, 2003.  Google Scholar

[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.  Google Scholar

[19]

J. Jahn, Vector Optimization Theory, Applications and Extensions, Springer, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. T. Luc, Theory of Vector Vptimization, Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

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
[2]

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
[3]

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
[4]

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
[5]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[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]

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
[8]

Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018
[9]

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
[10]

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
[11]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021
[12]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[13]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[14]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020027
[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]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[18]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[19]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[20]

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

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (168)
  • HTML views (862)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences