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July  2013, 9(3): 659-669. doi: 10.3934/jimo.2013.9.659

## Optimality conditions for vector equilibrium problems and their applications

 1 Technical University of Cluj-Napoca, Department of Mathematics, Str. G. Bariţiu 25, 400027, Cluj-Napoca, Romania

Received  December 2011 Revised  March 2013 Published  April 2013

The purpose of this paper is to establish necessary and sufficient conditions for a point to be solution of a vector equilibrium problem with cone and affine constraints. Using a separation theorem, which involves the quasi-interior of a convex set, we obtain optimality conditions for solutions of the vector equilibrium problem. Then, the main result is applied to vector optimization problems with cone and affine constraints and to duality theory.
Citation: Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial & Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659
##### References:
 [1] L. Q. Anh, P. Q. Khanh, D. T. M. Van and J. C. Yao, Well-posedness for vector quasiequilibria, Taiwanese J. Math., 13 (2009), 713-737.  Google Scholar [2] Q. H. Ansari, I. V. Konnov and J. C. Yao, On generalized vector equilibrium problems, Nonlinear Anal., 47 (2001), 543-554. doi: 10.1016/S0362-546X(01)00199-7.  Google Scholar [3] Q. H. Ansari, I. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl., 110 (2001), 481-492. doi: 10.1023/A:1017581009670.  Google Scholar [4] 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 [5] Q. H. Ansari, W. Oettli and D. Schläger, A generalization of vectorial equilibria, Math. Meth. Oper. Res., 46 (1997), 147-152. doi: 10.1007/BF01217687.  Google Scholar [6] Q. H. Ansari, X. Q. Yang and J. C. Yao, Existence and duality of implicit vector variational problems, Numer. Funct. Anal. Optim., 22 (2001), 815-829. doi: 10.1081/NFA-100108310.  Google Scholar [7] M. Bianchi, N. Hadjisavvas 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 [8] M. Bianchi, G. Kassay and R. Pini, Ekeland's principle for vector equilibrium problems, Nonlinear Anal., 66 (2007), 1454-1464. doi: 10.1016/j.na.2006.02.003.  Google Scholar [9] M. Bianchi, G. Kassay and R. Pini, Well-posedness for vector equilibrium problems, Math. Meth. Oper. Res., 70 (2009), 171-182. doi: 10.1007/s00186-008-0239-4.  Google Scholar [10] G. Bigi, A. Capătă and G. Kassay, Existence results for strong vector equilibrium problems and their applications, Optimization, 61 (2012), 567-583. doi: 10.1080/02331934.2010.528761.  Google Scholar [11] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.  Google Scholar [12] J. M. Borwein and R. Goebel, Notions of relative interior in Banach spaces, J. Math. Sci., 115 (2003), 2542-2553. doi: 10.1023/A:1022988116044.  Google Scholar [13] J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: Quasi relative interiors and duality theory, Math. Prog., 57 (1992), 15-48. doi: 10.1007/BF01581072.  Google Scholar [14] J. M. Borwein and V. Jeyakumar, On convexlike Lagrangian and minimax theorems, Research Report 24, University of Waterloo, (1988). Google Scholar [15] F. Cammaroto and B. Di Bella, Separation theorem based on the quasirelative interior and application to duality theory, J. Optim. Theory Appl., 125 (2005), 223-229. doi: 10.1007/s10957-004-1724-4.  Google Scholar [16] A. Capătă and G. Kassay, On vector equilibrium problems and applications, Taiwanese J. Math., 15 (2011), 365-380.  Google Scholar [17] A. Capătă , Optimality conditions for extended Ky Fan inequality with cone and affine constraints and their applications, J. Optim. Theory. Appl., 152 (2012), 661-674. doi: 10.1007/s10957-011-9916-1.  Google Scholar [18] P. Daniele, S. Giuffrè and A. Maugeri, Infinite dimensional duality and applications, Math. Ann., 339 (2007), 221-239. doi: 10.1007/s00208-007-0118-y.  Google Scholar [19] K. Fan, Minimax theorems, Proc. National Acad. Sci. USA, 39 (1953), 42-47. doi: 10.1073/pnas.39.1.42.  Google Scholar [20] K. Fan, A minimax inequality and applications, in "Inequality III" (ed. O. Shisha), Academic Press, New York, (1972), 103-113.  Google Scholar [21] F. Giannessi, "Vector Variational Inequalities and Vector Equilibria," Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar [22] X. H. Gong, Optimality conditions for vector equilibrium problems, J. Math. Anal. Appl., 342 (2008), 1455-1466. doi: 10.1016/j.jmaa.2008.01.026.  Google Scholar [23] R. B. Holmes, "Geometric Functional Analysis and its Applications," Springer-Verlag, Berlin, 1975.  Google Scholar [24] K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Math., 12 (2008), 649-669.  Google Scholar [25] K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems, Taiwanese J. Math., 12 (2008), 2233-2268.  Google Scholar [26] M. A. Limber and R. K. Goodrich, Quasi interiors, Lagrange multipliers and $L^p$ spectral estimation with lattice bounds, J. Optim. Theory Appl., 78 (1993), 143-161. doi: 10.1007/BF00940705.  Google Scholar [27] B. C. Ma and X. H. Gong, Optimality conditions for vector equilibrium problems in normed spaces, Optimization, 60 (2011), 1441-1455. doi: 10.1080/02331931003657709.  Google Scholar [28] Q. Qiu, Optimality conditions of globally efficient solutions for vector equilibrium problems with generalized convexity, J. Ineq. Appl., (2009). doi: 10.1155/2009/898213.  Google Scholar [29] Q. S. Qiu, Optimality conditions for vector equilibrium problems with constraints, J. Ind. Manag. Optim., 5 (2009), 783-790. doi: 10.3934/jimo.2009.5.783.  Google Scholar [30] R. T. Rockafellar, "Conjugate Duality and Optimization," Society for Industrial and Applied Mathematics, Philadelphia, 1974.  Google Scholar [31] J. Salamon, Closedness and Hadamard well-posedness of the solution map for parametric vector equilibrium problems, J. Global Optim., 47 (2010), 173-183. doi: 10.1007/s10898-009-9464-5.  Google Scholar [32] C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002. doi: 10.1142/9789812777096.  Google Scholar

show all references

##### References:
 [1] L. Q. Anh, P. Q. Khanh, D. T. M. Van and J. C. Yao, Well-posedness for vector quasiequilibria, Taiwanese J. Math., 13 (2009), 713-737.  Google Scholar [2] Q. H. Ansari, I. V. Konnov and J. C. Yao, On generalized vector equilibrium problems, Nonlinear Anal., 47 (2001), 543-554. doi: 10.1016/S0362-546X(01)00199-7.  Google Scholar [3] Q. H. Ansari, I. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl., 110 (2001), 481-492. doi: 10.1023/A:1017581009670.  Google Scholar [4] 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 [5] Q. H. Ansari, W. Oettli and D. Schläger, A generalization of vectorial equilibria, Math. Meth. Oper. Res., 46 (1997), 147-152. doi: 10.1007/BF01217687.  Google Scholar [6] Q. H. Ansari, X. Q. Yang and J. C. Yao, Existence and duality of implicit vector variational problems, Numer. Funct. Anal. Optim., 22 (2001), 815-829. doi: 10.1081/NFA-100108310.  Google Scholar [7] M. Bianchi, N. Hadjisavvas 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 [8] M. Bianchi, G. Kassay and R. Pini, Ekeland's principle for vector equilibrium problems, Nonlinear Anal., 66 (2007), 1454-1464. doi: 10.1016/j.na.2006.02.003.  Google Scholar [9] M. Bianchi, G. Kassay and R. Pini, Well-posedness for vector equilibrium problems, Math. Meth. Oper. Res., 70 (2009), 171-182. doi: 10.1007/s00186-008-0239-4.  Google Scholar [10] G. Bigi, A. Capătă and G. Kassay, Existence results for strong vector equilibrium problems and their applications, Optimization, 61 (2012), 567-583. doi: 10.1080/02331934.2010.528761.  Google Scholar [11] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.  Google Scholar [12] J. M. Borwein and R. Goebel, Notions of relative interior in Banach spaces, J. Math. Sci., 115 (2003), 2542-2553. doi: 10.1023/A:1022988116044.  Google Scholar [13] J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: Quasi relative interiors and duality theory, Math. Prog., 57 (1992), 15-48. doi: 10.1007/BF01581072.  Google Scholar [14] J. M. Borwein and V. Jeyakumar, On convexlike Lagrangian and minimax theorems, Research Report 24, University of Waterloo, (1988). Google Scholar [15] F. Cammaroto and B. Di Bella, Separation theorem based on the quasirelative interior and application to duality theory, J. Optim. Theory Appl., 125 (2005), 223-229. doi: 10.1007/s10957-004-1724-4.  Google Scholar [16] A. Capătă and G. Kassay, On vector equilibrium problems and applications, Taiwanese J. Math., 15 (2011), 365-380.  Google Scholar [17] A. Capătă , Optimality conditions for extended Ky Fan inequality with cone and affine constraints and their applications, J. Optim. Theory. Appl., 152 (2012), 661-674. doi: 10.1007/s10957-011-9916-1.  Google Scholar [18] P. Daniele, S. Giuffrè and A. Maugeri, Infinite dimensional duality and applications, Math. Ann., 339 (2007), 221-239. doi: 10.1007/s00208-007-0118-y.  Google Scholar [19] K. Fan, Minimax theorems, Proc. National Acad. Sci. USA, 39 (1953), 42-47. doi: 10.1073/pnas.39.1.42.  Google Scholar [20] K. Fan, A minimax inequality and applications, in "Inequality III" (ed. O. Shisha), Academic Press, New York, (1972), 103-113.  Google Scholar [21] F. Giannessi, "Vector Variational Inequalities and Vector Equilibria," Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar [22] X. H. Gong, Optimality conditions for vector equilibrium problems, J. Math. Anal. Appl., 342 (2008), 1455-1466. doi: 10.1016/j.jmaa.2008.01.026.  Google Scholar [23] R. B. Holmes, "Geometric Functional Analysis and its Applications," Springer-Verlag, Berlin, 1975.  Google Scholar [24] K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Math., 12 (2008), 649-669.  Google Scholar [25] K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems, Taiwanese J. Math., 12 (2008), 2233-2268.  Google Scholar [26] M. A. Limber and R. K. Goodrich, Quasi interiors, Lagrange multipliers and $L^p$ spectral estimation with lattice bounds, J. Optim. Theory Appl., 78 (1993), 143-161. doi: 10.1007/BF00940705.  Google Scholar [27] B. C. Ma and X. H. Gong, Optimality conditions for vector equilibrium problems in normed spaces, Optimization, 60 (2011), 1441-1455. doi: 10.1080/02331931003657709.  Google Scholar [28] Q. Qiu, Optimality conditions of globally efficient solutions for vector equilibrium problems with generalized convexity, J. Ineq. Appl., (2009). doi: 10.1155/2009/898213.  Google Scholar [29] Q. S. Qiu, Optimality conditions for vector equilibrium problems with constraints, J. Ind. Manag. Optim., 5 (2009), 783-790. doi: 10.3934/jimo.2009.5.783.  Google Scholar [30] R. T. Rockafellar, "Conjugate Duality and Optimization," Society for Industrial and Applied Mathematics, Philadelphia, 1974.  Google Scholar [31] J. Salamon, Closedness and Hadamard well-posedness of the solution map for parametric vector equilibrium problems, J. Global Optim., 47 (2010), 173-183. doi: 10.1007/s10898-009-9464-5.  Google Scholar [32] C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002. doi: 10.1142/9789812777096.  Google Scholar
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