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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

 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.
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

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##### 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
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