Duality for vector equilibrium problems with constraints

Document Type : Research Paper

Author

Department of Optimization and System Theory‎, ‎University of Science‎, ‎Vietnam National University Hochiminh City‎, ‎227 Nguyen Van Cu‎, ‎District 5‎, ‎Hochiminh City‎, ‎Vietnam.

Abstract

‎In the paper‎, ‎we study duality for vector equilibrium problems using a concept of generalized convexity in dealing with the quasi-relative interior‎. ‎Then‎, ‎their applications to optimality conditions for quasi-relative efficient solutions are obtained‎. ‎Our results are extensions of several existing ones in the literature when the ordering cones in both the objective space and the constraint space have possibly empty interior.

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References

N.L.H. Anh, Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality, Positivity 18 (2014) 449--473.
L.Q. Anh and P.Q. Khanh, Various kinds of semicontinuity and solution sets of parametric multivalued symmetric vector quasiequilibrium problems, J. Global Optim. 41 (2008) 539--558.
N.L.H. Anh and P.Q. Khanh, Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives, J. Global Optim. 56 (2013) 519--536.
L.Q. Anh, P.Q. Khanh and T.N. Tam, On Holder continuity of solution maps of parametric primal and dual Ky Fan inequalities, TOP 23 (2015) 151--167.
N.L.H. Anh, P.Q. Khanh and L.T. Tung, Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization, Nonlinear Anal. TMA 74 (2011) 7365--7379.
Q.H. Ansari and J.C. Yao, An existence result for the generalized vector equilibrium problem, Appl. Math. Lett. 12 (1999) 53--56.
J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.
T.Q. Bao and B.S. Mordukhovich, Relative Pareto minimizers for multiobjective problems: existence and optimality conditions, Math. Program. 122 (2010) 301--347.
E.M. Bednarczuk and W. Song, Contingent epiderivative and its applications to set-valued maps, Control Cybernet. 27 (1998) 375--386.
J.M. Borwein and R. Goebel, Notions of relative interior in Banach spaces, J. Math. Sci. (N.Y.) 115 (2003) 2542--2553.
J.M. Borwein and A.S. Lewis, Partially finite convex programming, Part I : Quasi relative interiors and duality theory, Math. Program. 57 (1992) 15--48.
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.
R.I. Boţ and S.M. Grad, Wolfe duality and Mond-Weir duality via perturbations, Non-linear Anal. 73 (2010) 374--384.
C.R. Chen, S.J. Li and K.L. Teo, Higher-order weak epiderivatives and applications to duality and optimality conditions, Comp. Math. Appl. 57 (2009) 1389--1399.
C.R. Chen, S.J. Li and K.L. Teo, Solutions semicontinuity of parametric generalized vector equilibrium problems, J. Global Optim. 45 (2009) 309--318.
P. Daniele, S. Giuffrè, G. Idone and A. Maugeri, Infinite dimensional duality and applications, Math. Ann. 339 (2007) 221--239.
H.T.H. Diem and P.Q. Khanh, Approximations of optimization-related problems in terms of variational convergence, Vietnam J. Math. 44 (2016) 399--417.
X.P. Ding, Y.C. Liou and C. Yao, Existence and algorithms for bilevel generalized mixed equilibrium problems in Banach spaces, J. Global Optim. 53 (2012) 331--346.
X.H. Gong, Optimality conditions for efficient solutions to the vector equilibrium problems with constraints, Taiwanese J. Math. 16 (2012), 1453--1473.
M.S. Gowda and M. Teboulle, A comparison of constraint qualifications in finite-dimensional convex programming, SIAM J. Control Optim. 28 (1990) 925--935.
S.M. Grad and E.L. Pop, Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone, Optimization 63 (2014) 21--37.
T.X.D. Ha, Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium, Nonlinear Anal. 75 (2012) 1305--1323.
J. Jahn and A.A. Khan, Generalized contingent epiderivatives in set valued optimization: optimality conditions, Numer. Funct. Anal. Optim. 23 (2002) 807--831.
P.Q. Khanh and N.M. Tung, Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints, Optimization 64 (2015) 1547--1575.
I.V. Konnov and S. Schaible, Duality for equilibrium problems under generalized monotonicity, J. Optim. Theory Appl. 104 (2000) 395--408.
Z.F. Li, Benson proper efficiency in vector optimization of set-valued maps, J. Optim. Theory Appl. 98 (1998) 623--649.
X.B. Li and S.J. Li, Existence of solutions for generalized vector quasi-equilibrium problems, Optim. Lett. 4 (2010) 17--28.
S.J. Li, K.L. Teo and X.Q. Yang, Higher-order Mond-Weir duality for set-valued optimization, J. Comp. Appl. Math. 217 (2008) 339--349.
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.
B.C. Ma and X.H. Gong, Optimality conditions for vector equilibrium problems in normed spaces, Optimization 60 (2011) 1441--1455.
P.S.M. Santos and S. Scheimberg, An outer approximation algorithm for equilibrium problems in Hilbert spaces, Optim. Method Softw. 30 (2015) 379--390.
Q.L. Wang, Z. Lin, J. Zeng, Z. Peng and X. Li, Second-order composed contingent epiderivatives and set-valued vector equilibrium problems, J. Inequal. Appl. 2014 (2014) no. 406, 17 pages.
Z.F. Wei and X.H. Gong, Kuhn-Tucker optimality conditions for vector equilibrium problems, J. Inequal. Appl. 2010 (2010), Article ID 842715, 15 pages.
C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 2002.
Bulletin of the Iranian Mathematical Society
Volume 43, Issue 6
November 2017
Pages 1679-1694

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History

  • Receive Date: 22 December 2015
  • Revise Date: 22 July 2016
  • Accept Date: 11 September 2016

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