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April  2015, 11(2): 563-574. doi: 10.3934/jimo.2015.11.563

## Optimality conditions for strong vector equilibrium problems under a weak constraint qualification

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

Received  October 2013 Revised  April 2014 Published  September 2014

The purpose of this paper is to present necessary and sufficient optimality conditions for a feasible solution to be weakly efficient or Henig weakly efficient solution of a nonconvex vector equilibrium problem with cone constraints. These theorems are based on the quasi-relative interior notion and a very recent separation theorem which involves this notion. Our results deal with some conditions where no previous results are applicable.
Citation: Adela Capătă. Optimality conditions for strong vector equilibrium problems under a weak constraint qualification. Journal of Industrial & Management Optimization, 2015, 11 (2) : 563-574. doi: 10.3934/jimo.2015.11.563
##### References:
 [1] Q. H. Ansari, W. Oettli and D. Schläger, A generalization of vectorial equilibria,, Math. Meth. Oper. Res., 46 (1997), 147. doi: 10.1007/BF01217687. Google Scholar [2] M. Bianchi, N. Hadjisavvas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions,, J. Optim. Theory Appl., 92 (1997), 527. doi: 10.1023/A:1022603406244. Google Scholar [3] G. Bigi, A. Capătă and G. Kassay, Existence results for strong vector equilibrium problems and their applications,, Optimization, 61 (2012), 567. doi: 10.1080/02331934.2010.528761. Google Scholar [4] J. M. Borwein and R. Goebel, Notions of relative interior in Banach spaces,, J. Math. Sci., 115 (2003), 2542. doi: 10.1023/A:1022988116044. Google Scholar [5] J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: Quasi relative interiors and duality theory,, Math. Prog., 57 (1992), 15. doi: 10.1007/BF01581072. Google Scholar [6] R. I. Boţ, E. R. Csetnek and G. Wanka, Regularity conditions via quasi-relative interior in convex programming,, SIAM J. Optim., 19 (2008), 217. doi: 10.1137/07068432X. Google Scholar [7] A. Capătă, Families of Henig dilating cones and proper efficiency in vector equilibrium problems,, Autom. Comp. Appl. Math., 19 (2010), 67. Google Scholar [8] A. Capătă, Optimality conditions for vector equilibrium problems and their applications,, J. Ind. Manag. Optim., 9 (2013), 659. doi: 10.3934/jimo.2013.9.659. Google Scholar [9] G.-Y. Chen and S. H. Hou, Existence of solutions for vector variational inequalities,, in F. Giannessi (ed.), 38 (2000), 73. doi: 10.1007/978-1-4613-0299-5_5. Google Scholar [10] B. D. Craven, Mathematical Programming and Control Theory,, xi+163 pp., (1978). Google Scholar [11] P. Daniele, S. Giuffrè and A. Maugeri, Infinite dimensional duality and applications,, Math. Ann., 339 (2007), 221. doi: 10.1007/s00208-007-0118-y. Google Scholar [12] F. Flores-Bazán and G. Mastroeni, Strong duality in cone constrained nonconvex optimization,, SIAM. J. Optim., 23 (2013), 153. doi: 10.1137/120861400. Google Scholar [13] X. H. Gong, Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior,, J. Math. Anal. Appl., 307 (2005), 12. doi: 10.1016/j.jmaa.2004.10.001. Google Scholar [14] X. H. Gong, Symmetric strong vector quasi-equilibrium problems,, Math. Methods Oper. Res., 65 (2007), 305. doi: 10.1007/s00186-006-0114-0. Google Scholar [15] X. H. Gong, Optimality conditions for vector equilibrium problems,, J. Math. Anal. Appl., 342 (2008), 1455. doi: 10.1016/j.jmaa.2008.01.026. Google Scholar [16] A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variatonal Methods in Partially Ordered Spaces,, xiv+350 pp., (2003). Google Scholar [17] T. X. D. Ha, Optimality conditions for various solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems,, Nonlinear Anal., 75 (2012), 1305. doi: 10.1016/j.na.2011.03.015. Google Scholar [18] M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387. doi: 10.1007/BF00934353. Google Scholar [19] R. B. Holmes, Geometric Functional Analysis and its Applications,, Springer-Verlag, (1975). Google Scholar [20] 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. doi: 10.1007/BF00940705. Google Scholar [21] K. L. Lin, D. P. Yang and J. C. Yao, Generalized vector variational inequalities,, J. Optim. Theory Appl., 92 (1997), 117. doi: 10.1023/A:1022640130410. Google Scholar [22] X. J. Long, Y. Q. Huang and Z. Y. Peng, Optimality conditions for the Henig efficient solution of vector equilibrium problems with constraints,, Optim. Lett., 5 (2011), 717. doi: 10.1007/s11590-010-0241-7. Google Scholar [23] B. C. Ma and X. H. Gong, Optimality conditions for vector equilibrium problems in normed spaces,, Optimization, 60 (2011), 1441. doi: 10.1080/02331931003657709. Google Scholar [24] J. Morgan and M. Romaniello, Scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities,, J. Optim. Theory Appl., 130 (2006), 309. doi: 10.1007/s10957-006-9104-x. Google Scholar [25] S. Paeck, Convexlike and concavelike conditions in alternative, minimax and minimization theorems,, J. Optim. Theory Appl., 74 (1992), 317. doi: 10.1007/BF00940897. Google Scholar [26] 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 [27] Q. S. Qiu, Optimality conditions for vector equilibrium problems with constraints,, J. Ind. Manag. Optim., 5 (2009), 783. doi: 10.3934/jimo.2009.5.783. Google Scholar [28] R. T. Rockafellar, Conjugate Duality and Optimization,, Society for Industrial and Applied Mathematics, (1974). Google Scholar [29] X. M. Yang, D. Li and X. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, J. Optim. Theory Appl., 110 (2001), 413. doi: 10.1023/A:1017535631418. Google Scholar [30] C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (2002). doi: 10.1142/9789812777096. Google Scholar

show all references

##### References:
 [1] Q. H. Ansari, W. Oettli and D. Schläger, A generalization of vectorial equilibria,, Math. Meth. Oper. Res., 46 (1997), 147. doi: 10.1007/BF01217687. Google Scholar [2] M. Bianchi, N. Hadjisavvas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions,, J. Optim. Theory Appl., 92 (1997), 527. doi: 10.1023/A:1022603406244. Google Scholar [3] G. Bigi, A. Capătă and G. Kassay, Existence results for strong vector equilibrium problems and their applications,, Optimization, 61 (2012), 567. doi: 10.1080/02331934.2010.528761. Google Scholar [4] J. M. Borwein and R. Goebel, Notions of relative interior in Banach spaces,, J. Math. Sci., 115 (2003), 2542. doi: 10.1023/A:1022988116044. Google Scholar [5] J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: Quasi relative interiors and duality theory,, Math. Prog., 57 (1992), 15. doi: 10.1007/BF01581072. Google Scholar [6] R. I. Boţ, E. R. Csetnek and G. Wanka, Regularity conditions via quasi-relative interior in convex programming,, SIAM J. Optim., 19 (2008), 217. doi: 10.1137/07068432X. Google Scholar [7] A. Capătă, Families of Henig dilating cones and proper efficiency in vector equilibrium problems,, Autom. Comp. Appl. Math., 19 (2010), 67. Google Scholar [8] A. Capătă, Optimality conditions for vector equilibrium problems and their applications,, J. Ind. Manag. Optim., 9 (2013), 659. doi: 10.3934/jimo.2013.9.659. Google Scholar [9] G.-Y. Chen and S. H. Hou, Existence of solutions for vector variational inequalities,, in F. Giannessi (ed.), 38 (2000), 73. doi: 10.1007/978-1-4613-0299-5_5. Google Scholar [10] B. D. Craven, Mathematical Programming and Control Theory,, xi+163 pp., (1978). Google Scholar [11] P. Daniele, S. Giuffrè and A. Maugeri, Infinite dimensional duality and applications,, Math. Ann., 339 (2007), 221. doi: 10.1007/s00208-007-0118-y. Google Scholar [12] F. Flores-Bazán and G. Mastroeni, Strong duality in cone constrained nonconvex optimization,, SIAM. J. Optim., 23 (2013), 153. doi: 10.1137/120861400. Google Scholar [13] X. H. Gong, Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior,, J. Math. Anal. Appl., 307 (2005), 12. doi: 10.1016/j.jmaa.2004.10.001. Google Scholar [14] X. H. Gong, Symmetric strong vector quasi-equilibrium problems,, Math. Methods Oper. Res., 65 (2007), 305. doi: 10.1007/s00186-006-0114-0. Google Scholar [15] X. H. Gong, Optimality conditions for vector equilibrium problems,, J. Math. Anal. Appl., 342 (2008), 1455. doi: 10.1016/j.jmaa.2008.01.026. Google Scholar [16] A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variatonal Methods in Partially Ordered Spaces,, xiv+350 pp., (2003). Google Scholar [17] T. X. D. Ha, Optimality conditions for various solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems,, Nonlinear Anal., 75 (2012), 1305. doi: 10.1016/j.na.2011.03.015. Google Scholar [18] M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387. doi: 10.1007/BF00934353. Google Scholar [19] R. B. Holmes, Geometric Functional Analysis and its Applications,, Springer-Verlag, (1975). Google Scholar [20] 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. doi: 10.1007/BF00940705. Google Scholar [21] K. L. Lin, D. P. Yang and J. C. Yao, Generalized vector variational inequalities,, J. Optim. Theory Appl., 92 (1997), 117. doi: 10.1023/A:1022640130410. Google Scholar [22] X. J. Long, Y. Q. Huang and Z. Y. Peng, Optimality conditions for the Henig efficient solution of vector equilibrium problems with constraints,, Optim. Lett., 5 (2011), 717. doi: 10.1007/s11590-010-0241-7. Google Scholar [23] B. C. Ma and X. H. Gong, Optimality conditions for vector equilibrium problems in normed spaces,, Optimization, 60 (2011), 1441. doi: 10.1080/02331931003657709. Google Scholar [24] J. Morgan and M. Romaniello, Scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities,, J. Optim. Theory Appl., 130 (2006), 309. doi: 10.1007/s10957-006-9104-x. Google Scholar [25] S. Paeck, Convexlike and concavelike conditions in alternative, minimax and minimization theorems,, J. Optim. Theory Appl., 74 (1992), 317. doi: 10.1007/BF00940897. Google Scholar [26] 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 [27] Q. S. Qiu, Optimality conditions for vector equilibrium problems with constraints,, J. Ind. Manag. Optim., 5 (2009), 783. doi: 10.3934/jimo.2009.5.783. Google Scholar [28] R. T. Rockafellar, Conjugate Duality and Optimization,, Society for Industrial and Applied Mathematics, (1974). Google Scholar [29] X. M. Yang, D. Li and X. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, J. Optim. Theory Appl., 110 (2001), 413. doi: 10.1023/A:1017535631418. Google Scholar [30] C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (2002). doi: 10.1142/9789812777096. Google Scholar
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