Scalarization of approximate solution for vector equilibrium problems

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	C. Gerth and P. Weidner, Nonconvex separation theorem and some applications in vector optimization. J. Optim. Theory Appl., 67 (1990), 297-320.
[7]	X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108 (2001), 139-154.
[8]	X. H. Gong, Connectedness of the set of efficient solution for generalized systems, J. Optim. Theory Appl., 138 (2008), 189-196.
[9]	X. H. Gong, Optimality conditions for vector equilibrium problems, J. Math. Anal. Appl., 342 (2008), 1455-1466.
[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.
[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.
[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.
[13]	J. Jahn, "Mathematical Vector Optimization in Partially Order Linear Spaces," Verlag Peter Lang, Frankfurt, 1986.
[14]	K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Mathematics, 12 (2008), 649-669.
[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.
[16]	Q. S. Qiu, Optimality conditions for vector equilibrium problems with constraints, J. Industrial Management Optim., 5 (2009), 783-790.
[17]	J. H. Qiu, Scalarization of Henig properly efficient points in locally convex spaces, J. Optim. Theory Appl., 147 (2010), 71-92.

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.
[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.
[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.
[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.
[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.
[6]	C. Gerth and P. Weidner, Nonconvex separation theorem and some applications in vector optimization. J. Optim. Theory Appl., 67 (1990), 297-320.
[7]	X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108 (2001), 139-154.
[8]	X. H. Gong, Connectedness of the set of efficient solution for generalized systems, J. Optim. Theory Appl., 138 (2008), 189-196.
[9]	X. H. Gong, Optimality conditions for vector equilibrium problems, J. Math. Anal. Appl., 342 (2008), 1455-1466.
[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.
[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.
[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.
[13]	J. Jahn, "Mathematical Vector Optimization in Partially Order Linear Spaces," Verlag Peter Lang, Frankfurt, 1986.
[14]	K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Mathematics, 12 (2008), 649-669.
[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.
[16]	Q. S. Qiu, Optimality conditions for vector equilibrium problems with constraints, J. Industrial Management Optim., 5 (2009), 783-790.
[17]	J. H. Qiu, Scalarization of Henig properly efficient points in locally convex spaces, J. Optim. Theory Appl., 147 (2010), 71-92.

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