American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The set covering problem revisited: An empirical study of the value of dual information
  • JIMO Home
  • This Issue
  • Next Article
    On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality
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

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  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.
Keywords: generalized nearly subconvexity, quasi-relative interior, constraint qualification, optimality conditions., Separation theorem.
Mathematics Subject Classification: Primary: 90C46; Secondary: 49K1.
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), 147-152. 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), 527-542. 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), 567-583. 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), 2542-2553. 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), 15-48. 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), 217-233. 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), 67-76. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. D. Craven, Mathematical Programming and Control Theory, xi+163 pp., Chapman and Hall, London, 1978.  Google Scholar

[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.  Google Scholar

[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.  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), 12-31. 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), 305-314. 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), 1455-1466. 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., Springer-Verlag, New York, 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), 1305-1323. 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), 387-407. doi: 10.1007/BF00934353.  Google Scholar

[19]

R. B. Holmes, Geometric Functional Analysis and its Applications, Springer-Verlag, Berlin, 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), 143-161. 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), 117-125. 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), 717-728. 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), 1441-1455. 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), 309-316. 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), 317-332. 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), Art. ID 898213, 13 pp. 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), 783-790. doi: 10.3934/jimo.2009.5.783.  Google Scholar

[28]

R. T. Rockafellar, Conjugate Duality and Optimization, Society for Industrial and Applied Mathematics, Philadelphia, 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), 413-427. doi: 10.1023/A:1017535631418.  Google Scholar

[30]

C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore, 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), 147-152. 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), 527-542. 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), 567-583. 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), 2542-2553. 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), 15-48. 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), 217-233. 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), 67-76. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. D. Craven, Mathematical Programming and Control Theory, xi+163 pp., Chapman and Hall, London, 1978.  Google Scholar

[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.  Google Scholar

[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.  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), 12-31. 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), 305-314. 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), 1455-1466. 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., Springer-Verlag, New York, 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), 1305-1323. 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), 387-407. doi: 10.1007/BF00934353.  Google Scholar

[19]

R. B. Holmes, Geometric Functional Analysis and its Applications, Springer-Verlag, Berlin, 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), 143-161. 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), 117-125. 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), 717-728. 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), 1441-1455. 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), 309-316. 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), 317-332. 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), Art. ID 898213, 13 pp. 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), 783-790. doi: 10.3934/jimo.2009.5.783.  Google Scholar

[28]

R. T. Rockafellar, Conjugate Duality and Optimization, Society for Industrial and Applied Mathematics, Philadelphia, 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), 413-427. doi: 10.1023/A:1017535631418.  Google Scholar

[30]

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

[1]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559
[2]

Majid E. Abbasov. Generalized exhausters: Existence, construction, optimality conditions. Journal of Industrial & Management Optimization, 2015, 11 (1) : 217-230. doi: 10.3934/jimo.2015.11.217
[3]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
[4]

Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023
[5]

Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2020, 14 (3) : 413-422. doi: 10.3934/amc.2020063
[6]

Zhengchun Zhou, Xiaohu Tang. New nearly optimal codebooks from relative difference sets. Advances in Mathematics of Communications, 2011, 5 (3) : 521-527. doi: 10.3934/amc.2011.5.521
[7]

Xiuhong Chen, Zhihua Li. On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity. Journal of Industrial & Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081
[8]

Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333
[9]

Albert Fathi. An Urysohn-type theorem under a dynamical constraint. Journal of Modern Dynamics, 2016, 10: 331-338. doi: 10.3934/jmd.2016.10.331
[10]

Fuzhong Cong, Hongtian Li. Quasi-effective stability for a nearly integrable volume-preserving mapping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1959-1970. doi: 10.3934/dcdsb.2015.20.1959
[11]

Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67
[12]

B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, 2007, 2007 (Special) : 145-154. doi: 10.3934/proc.2007.2007.145
[13]

Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231
[14]

José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
[15]

Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial & Management Optimization, 2020, 16 (2) : 623-631. doi: 10.3934/jimo.2018170
[16]

Luong V. Nguyen. A note on optimality conditions for optimal exit time problems. Mathematical Control & Related Fields, 2015, 5 (2) : 291-303. doi: 10.3934/mcrf.2015.5.291
[17]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086
[18]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087
[19]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483
[20]

Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial & Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy