American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Proximal point algorithm for nonlinear complementarity problem based on the generalized Fischer-Burmeister merit function
  • JIMO Home
  • This Issue
  • Next Article
    Optimality conditions and duality in nondifferentiable interval-valued programming
January  2013, 9(1): 143-151. doi: 10.3934/jimo.2013.9.143

Scalarization of approximate solution for vector equilibrium problems

Qiusheng Qiu 1, and  Xinmin Yang 2,

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China
2. 

Department of Mathematics, Chongqing Normal University, Chongqing, 400047

Received  November 2011 Revised  June 2012 Published  December 2012

In this paper, some scalar characterizations of approximate weakly efficient solutions and approximate Henig efficient solutions for vector equilibrium problems are derived without imposing any convexity assumption on objective functions and feasible set. Meanwhile, the linear scalar characterization of approximate weakly efficient solutions is also established under the conditions of generalized convexity. As an application of the results in this paper, scalar characterizations of weakly efficient solution, Henig efficient solution and super efficient solution for vector equilibrium problems are obtained.
Keywords: approximate weakly efficient solution, Vector equilibrium problems, scalarization, approximate Henig efficient solution..
Mathematics Subject Classification: 90C26, 91B5.
Citation: Qiusheng Qiu, Xinmin Yang. Scalarization of approximate solution for vector equilibrium problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 143-151. doi: 10.3934/jimo.2013.9.143
References:
[1]

Q. H. Ansari, W. Oettli and D. Schager, A generalization of vectorial equilibria, Mathematical Methods of Operations Research, 46 (1997), 147-152.  Google Scholar

[2]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Characterizations of solutions for vector equilibrium problems, J. Optim. Theory Appl., 113 (2002), 435-447. doi: 10.1023/A:1015366419163.  Google Scholar

[3]

M. Bianchi, N. Hadjisawas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 92 (1997), 527-542. doi: 10.1023/A:1022603406244.  Google Scholar

[4]

J. P. Dauer and R. J. Gallagher, Positive proper efficient points and related cone results in vector optimization theory, SIAM J. Control Optim., 28 (1990), 158-172. doi: 10.1137/0328008.  Google Scholar

[5]

J. Fu, Simultaneous vector variational inequalities and vector implicit complementarity problems, J. Optim. Theory Appl., 93 (1997), 141-151. doi: 10.1023/A:1022653918733.  Google Scholar

[6]

C. Gerth and P. Weidner, Nonconvex separation theorem and some applications in vector optimization. J. Optim. Theory Appl., 67 (1990), 297-320.  Google Scholar

[7]

X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108 (2001), 139-154.  Google Scholar

[8]

X. H. Gong, Connectedness of the set of efficient solution for generalized systems, J. Optim. Theory Appl., 138 (2008), 189-196.  Google Scholar

[9]

X. H. Gong, Optimality conditions for vector equilibrium problems, J. Math. Anal. Appl., 342 (2008), 1455-1466.  Google Scholar

[10]

X. H. Gong, W. T. Fu and W. Liu, Super efficiency for a vector equilibrium in locally convex topological vector spaces, in "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories" (eds. F. Giannessi), Kluwer Academic Publishers, (2000), 233-252.  Google Scholar

[11]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205. doi: 10.1007/s10957-008-9379-1.  Google Scholar

[12]

N. Hadjisawas and S. Schaile, From scalar to vector equilibrium problems in the quasimonotone case, J. Optim. Theory Appl., 96 (1998), 297-305. doi: 10.1023/A:1022666014055.  Google Scholar

[13]

J. Jahn, "Mathematical Vector Optimization in Partially Order Linear Spaces," Verlag Peter Lang, Frankfurt, 1986.  Google Scholar

[14]

K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Mathematics, 12 (2008), 649-669.  Google Scholar

[15]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems, J. Global Optim., 41 (2008), 187-202. doi: 10.1007/s10898-007-9210-9.  Google Scholar

[16]

Q. S. Qiu, Optimality conditions for vector equilibrium problems with constraints, J. Industrial Management Optim., 5 (2009), 783-790.  Google Scholar

[17]

J. H. Qiu, Scalarization of Henig properly efficient points in locally convex spaces, J. Optim. Theory Appl., 147 (2010), 71-92.  Google Scholar

show all references

References:
[1]

Q. H. Ansari, W. Oettli and D. Schager, A generalization of vectorial equilibria, Mathematical Methods of Operations Research, 46 (1997), 147-152.  Google Scholar

[2]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Characterizations of solutions for vector equilibrium problems, J. Optim. Theory Appl., 113 (2002), 435-447. doi: 10.1023/A:1015366419163.  Google Scholar

[3]

M. Bianchi, N. Hadjisawas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 92 (1997), 527-542. doi: 10.1023/A:1022603406244.  Google Scholar

[4]

J. P. Dauer and R. J. Gallagher, Positive proper efficient points and related cone results in vector optimization theory, SIAM J. Control Optim., 28 (1990), 158-172. doi: 10.1137/0328008.  Google Scholar

[5]

J. Fu, Simultaneous vector variational inequalities and vector implicit complementarity problems, J. Optim. Theory Appl., 93 (1997), 141-151. doi: 10.1023/A:1022653918733.  Google Scholar

[6]

C. Gerth and P. Weidner, Nonconvex separation theorem and some applications in vector optimization. J. Optim. Theory Appl., 67 (1990), 297-320.  Google Scholar

[7]

X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108 (2001), 139-154.  Google Scholar

[8]

X. H. Gong, Connectedness of the set of efficient solution for generalized systems, J. Optim. Theory Appl., 138 (2008), 189-196.  Google Scholar

[9]

X. H. Gong, Optimality conditions for vector equilibrium problems, J. Math. Anal. Appl., 342 (2008), 1455-1466.  Google Scholar

[10]

X. H. Gong, W. T. Fu and W. Liu, Super efficiency for a vector equilibrium in locally convex topological vector spaces, in "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories" (eds. F. Giannessi), Kluwer Academic Publishers, (2000), 233-252.  Google Scholar

[11]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205. doi: 10.1007/s10957-008-9379-1.  Google Scholar

[12]

N. Hadjisawas and S. Schaile, From scalar to vector equilibrium problems in the quasimonotone case, J. Optim. Theory Appl., 96 (1998), 297-305. doi: 10.1023/A:1022666014055.  Google Scholar

[13]

J. Jahn, "Mathematical Vector Optimization in Partially Order Linear Spaces," Verlag Peter Lang, Frankfurt, 1986.  Google Scholar

[14]

K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Mathematics, 12 (2008), 649-669.  Google Scholar

[15]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems, J. Global Optim., 41 (2008), 187-202. doi: 10.1007/s10898-007-9210-9.  Google Scholar

[16]

Q. S. Qiu, Optimality conditions for vector equilibrium problems with constraints, J. Industrial Management Optim., 5 (2009), 783-790.  Google Scholar

[17]

J. H. Qiu, Scalarization of Henig properly efficient points in locally convex spaces, J. Optim. Theory Appl., 147 (2010), 71-92.  Google Scholar

[1]

Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013
[2]

Nguyen Ba Minh, Pham Huu Sach. Strong vector equilibrium problems with LSC approximate solution mappings. Journal of Industrial & Management Optimization, 2020, 16 (2) : 511-529. doi: 10.3934/jimo.2018165
[3]

Qilin Wang, Shengji Li. Semicontinuity of approximate solution mappings to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1303-1309. doi: 10.3934/jimo.2016.12.1303
[4]

Yu Han, Nan-Jing Huang. Some characterizations of the approximate solutions to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1135-1151. doi: 10.3934/jimo.2016.12.1135
[5]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309
[6]

Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661
[7]

Kenji Kimura, Jen-Chih Yao. Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems. Journal of Industrial & Management Optimization, 2008, 4 (1) : 167-181. doi: 10.3934/jimo.2008.4.167
[8]

Xin Zuo, Chun-Rong Chen, Hong-Zhi Wei. Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings. Journal of Industrial & Management Optimization, 2017, 13 (1) : 477-488. doi: 10.3934/jimo.2016027
[9]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483
[10]

Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429
[11]

Savin Treanţă. Characterization of efficient solutions for a class of PDE-constrained vector control problems. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 93-106. doi: 10.3934/naco.2019035
[12]

Lili Li, Chunrong Chen. Nonlinear scalarization with applications to Hölder continuity of approximate solutions. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 295-307. doi: 10.3934/naco.2014.4.295
[13]

Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225
[14]

Guoshan Zhang, Shiwei Wang, Yiming Wang, Wanquan Liu. LS-SVM approximate solution for affine nonlinear systems with partially unknown functions. Journal of Industrial & Management Optimization, 2014, 10 (2) : 621-636. doi: 10.3934/jimo.2014.10.621
[15]

Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235
[16]

Zaiyun Peng, Xinmin Yang, Kok Lay Teo. On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality. Journal of Industrial & Management Optimization, 2015, 11 (2) : 549-562. doi: 10.3934/jimo.2015.11.549
[17]

Wenxia Chen, Ping Yang, Weiwei Gao, Lixin Tian. The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium. Communications on Pure & Applied Analysis, 2018, 17 (3) : 823-848. doi: 10.3934/cpaa.2018042
[18]

Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075
[19]

Editorial Office. WITHDRAWN: Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2020173
[20]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy