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On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
B. D. Craven, Mathematical Programming and Control Theory,, xi+163 pp., (1978).
|
[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. |
[12] |
F. Flores-Bazán and G. Mastroeni, Strong duality in cone constrained nonconvex optimization,, SIAM. J. Optim., 23 (2013), 153.
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.
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.
doi: 10.1007/s00186-006-0114-0. |
[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. |
[16] |
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variatonal Methods in Partially Ordered Spaces,, xiv+350 pp., (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.
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.
doi: 10.1007/BF00934353. |
[19] |
R. B. Holmes, Geometric Functional Analysis and its Applications,, Springer-Verlag, (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.
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.
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.
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.
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.
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.
doi: 10.1007/BF00940897. |
[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. |
[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. |
[28] |
R. T. Rockafellar, Conjugate Duality and Optimization,, Society for Industrial and Applied Mathematics, (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.
doi: 10.1023/A:1017535631418. |
[30] |
C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (2002).
doi: 10.1142/9789812777096. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
B. D. Craven, Mathematical Programming and Control Theory,, xi+163 pp., (1978).
|
[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. |
[12] |
F. Flores-Bazán and G. Mastroeni, Strong duality in cone constrained nonconvex optimization,, SIAM. J. Optim., 23 (2013), 153.
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.
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.
doi: 10.1007/s00186-006-0114-0. |
[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. |
[16] |
A. Göpfert, H. Riahi, C. Tammer and C. Zălinescu, Variatonal Methods in Partially Ordered Spaces,, xiv+350 pp., (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.
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.
doi: 10.1007/BF00934353. |
[19] |
R. B. Holmes, Geometric Functional Analysis and its Applications,, Springer-Verlag, (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.
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.
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.
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.
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.
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.
doi: 10.1007/BF00940897. |
[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. |
[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. |
[28] |
R. T. Rockafellar, Conjugate Duality and Optimization,, Society for Industrial and Applied Mathematics, (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.
doi: 10.1023/A:1017535631418. |
[30] |
C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (2002).
doi: 10.1142/9789812777096. |
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