• 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

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 and 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.

[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.

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.

[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.

[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 and 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 and 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 and 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 and 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 and 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 and 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]

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

[11]

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

[12]

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

[13]

Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete and 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 and Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327

[15]

Kota Kumazaki, Akio Ito, Masahiro Kubo. Generalized solutions of a non-isothermal phase separation model. Conference Publications, 2009, 2009 (Special) : 476-485. doi: 10.3934/proc.2009.2009.476

[16]

Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial and Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174

[17]

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

[18]

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

[19]

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

[20]

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

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]