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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.

[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.

[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.

[14]

J. M. Borwein and V. Jeyakumar, On convexlike Lagrangian and minimax theorems, Research Report 24, University of Waterloo, (1988).

[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.

[16]

A. Capătă and G. Kassay, On vector equilibrium problems and applications, Taiwanese J. Math., 15 (2011), 365-380.

[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.

[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.

[19]

K. Fan, Minimax theorems, Proc. National Acad. Sci. USA, 39 (1953), 42-47. doi: 10.1073/pnas.39.1.42.

[20]

K. Fan, A minimax inequality and applications, in "Inequality III" (ed. O. Shisha), Academic Press, New York, (1972), 103-113.

[21]

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

[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.

[23]

R. B. Holmes, "Geometric Functional Analysis and its Applications," Springer-Verlag, Berlin, 1975.

[24]

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

[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.

[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.

[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.

[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.

[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.

[30]

R. T. Rockafellar, "Conjugate Duality and Optimization," Society for Industrial and Applied Mathematics, Philadelphia, 1974.

[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.

[32]

C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002. doi: 10.1142/9789812777096.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.

[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.

[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.

[14]

J. M. Borwein and V. Jeyakumar, On convexlike Lagrangian and minimax theorems, Research Report 24, University of Waterloo, (1988).

[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.

[16]

A. Capătă and G. Kassay, On vector equilibrium problems and applications, Taiwanese J. Math., 15 (2011), 365-380.

[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.

[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.

[19]

K. Fan, Minimax theorems, Proc. National Acad. Sci. USA, 39 (1953), 42-47. doi: 10.1073/pnas.39.1.42.

[20]

K. Fan, A minimax inequality and applications, in "Inequality III" (ed. O. Shisha), Academic Press, New York, (1972), 103-113.

[21]

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

[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.

[23]

R. B. Holmes, "Geometric Functional Analysis and its Applications," Springer-Verlag, Berlin, 1975.

[24]

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

[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.

[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.

[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.

[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.

[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.

[30]

R. T. Rockafellar, "Conjugate Duality and Optimization," Society for Industrial and Applied Mathematics, Philadelphia, 1974.

[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.

[32]

C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific, Singapore, 2002. doi: 10.1142/9789812777096.

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