Article Contents

2015, Volume 11, Issue 2: 563-574. Doi: 10.3934/jimo.2015.11.563

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

Adela Capătă^1,

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

Received: October 2013

Revised: April 2014

Published: April 2015

Abstract / Introduction Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: Primary: 90C46; Secondary: 49K10.

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References

[1]	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.
[2]	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.
[3]	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.
[4]	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.
[5]	J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: Quasi relative interiors and duality theory, Math. Prog., 57 (1992), 15-48.doi: 10.1007/BF01581072.
[6]	R. I. Boţ, E. R. Csetnek and G. Wanka, Regularity conditions via quasi-relative interior in convex programming, SIAM J. Optim., 19 (2008), 217-233.doi: 10.1137/07068432X.
[7]	A. Capătă, Families of Henig dilating cones and proper efficiency in vector equilibrium problems, Autom. Comp. Appl. Math., 19 (2010), 67-76.
[8]	A. Capătă, Optimality conditions for vector equilibrium problems and their applications, J. Ind. Manag. Optim., 9 (2013), 659-669.doi: 10.3934/jimo.2013.9.659.
[9]	G.-Y. Chen and S. H. Hou, Existence of solutions for vector variational inequalities, in F. Giannessi (ed.), Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht, 38 (2000), 73-86.doi: 10.1007/978-1-4613-0299-5_5.
[10]	B. D. Craven, Mathematical Programming and Control Theory, xi+163 pp., Chapman and Hall, London, 1978.
[11]	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.
[12]	F. Flores-Bazán and G. Mastroeni, Strong duality in cone constrained nonconvex optimization, SIAM. J. Optim., 23 (2013), 153-169.doi: 10.1137/120861400.
[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), 12-31.doi: 10.1016/j.jmaa.2004.10.001.
[14]	X. H. Gong, Symmetric strong vector quasi-equilibrium problems, Math. Methods Oper. Res., 65 (2007), 305-314.doi: 10.1007/s00186-006-0114-0.
[15]	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.
[16]	A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variatonal Methods in Partially Ordered Spaces, xiv+350 pp., Springer-Verlag, New York, 2003.
[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), 1305-1323.doi: 10.1016/j.na.2011.03.015.
[18]	M. I. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl., 36 (1982), 387-407.doi: 10.1007/BF00934353.
[19]	R. B. Holmes, Geometric Functional Analysis and its Applications, Springer-Verlag, Berlin, 1975.
[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), 143-161.doi: 10.1007/BF00940705.
[21]	K. L. Lin, D. P. Yang and J. C. Yao, Generalized vector variational inequalities, J. Optim. Theory Appl., 92 (1997), 117-125.doi: 10.1023/A:1022640130410.
[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), 717-728.doi: 10.1007/s11590-010-0241-7.
[23]	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.
[24]	J. Morgan and M. Romaniello, Scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities, J. Optim. Theory Appl., 130 (2006), 309-316.doi: 10.1007/s10957-006-9104-x.
[25]	S. Paeck, Convexlike and concavelike conditions in alternative, minimax and minimization theorems, J. Optim. Theory Appl., 74 (1992), 317-332.doi: 10.1007/BF00940897.
[26]	Q. Qiu, Optimality conditions of globally efficient solutions for vector equilibrium problems with generalized convexity, J. Ineq. Appl., (2009), Art. ID 898213, 13 pp.doi: 10.1155/2009/898213.
[27]	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.
[28]	R. T. Rockafellar, Conjugate Duality and Optimization, Society for Industrial and Applied Mathematics, Philadelphia, 1974.
[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), 413-427.doi: 10.1023/A:1017535631418.
[30]	C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002.doi: 10.1142/9789812777096.