• Previous Article
    Multi-period mean-variance portfolio selection with fixed and proportional transaction costs
  • JIMO Home
  • This Issue
  • Next Article
    Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces
July  2013, 9(3): 659-669. doi: 10.3934/jimo.2013.9.659

Optimality conditions for vector equilibrium problems and their applications

1. 

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

Received  December 2011 Revised  March 2013 Published  April 2013

The purpose of this paper is to establish necessary and sufficient conditions for a point to be solution of a vector equilibrium problem with cone and affine constraints. Using a separation theorem, which involves the quasi-interior of a convex set, we obtain optimality conditions for solutions of the vector equilibrium problem. Then, the main result is applied to vector optimization problems with cone and affine constraints and to duality theory.
Citation: 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
References:
[1]

L. Q. Anh, P. Q. Khanh, D. T. M. Van and J. C. Yao, Well-posedness for vector quasiequilibria,, Taiwanese J. Math., 13 (2009), 713.

[2]

Q. H. Ansari, I. V. Konnov and J. C. Yao, On generalized vector equilibrium problems,, Nonlinear Anal., 47 (2001), 543. doi: 10.1016/S0362-546X(01)00199-7.

[3]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems,, J. Optim. Theory Appl., 110 (2001), 481. doi: 10.1023/A:1017581009670.

[4]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Characterizations of solutions for vector equilibrium problems,, J. Optim. Theory Appl., 113 (2002), 435. doi: 10.1023/A:1015366419163.

[5]

Q. H. Ansari, W. Oettli and D. Schläger, A generalization of vectorial equilibria,, Math. Meth. Oper. Res., 46 (1997), 147. doi: 10.1007/BF01217687.

[6]

Q. H. Ansari, X. Q. Yang and J. C. Yao, Existence and duality of implicit vector variational problems,, Numer. Funct. Anal. Optim., 22 (2001), 815. doi: 10.1081/NFA-100108310.

[7]

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.

[8]

M. Bianchi, G. Kassay and R. Pini, Ekeland's principle for vector equilibrium problems,, Nonlinear Anal., 66 (2007), 1454. doi: 10.1016/j.na.2006.02.003.

[9]

M. Bianchi, G. Kassay and R. Pini, Well-posedness for vector equilibrium problems,, Math. Meth. Oper. Res., 70 (2009), 171. doi: 10.1007/s00186-008-0239-4.

[10]

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.

[11]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems,, Math. Stud., 63 (1994), 123.

[12]

J. M. Borwein and R. Goebel, Notions of relative interior in Banach spaces,, J. Math. Sci., 115 (2003), 2542. doi: 10.1023/A:1022988116044.

[13]

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.

[14]

J. M. Borwein and V. Jeyakumar, On convexlike Lagrangian and minimax theorems,, Research Report 24, (1988).

[15]

F. Cammaroto and B. Di Bella, Separation theorem based on the quasirelative interior and application to duality theory,, J. Optim. Theory Appl., 125 (2005), 223. doi: 10.1007/s10957-004-1724-4.

[16]

A. Capătă and G. Kassay, On vector equilibrium problems and applications,, Taiwanese J. Math., 15 (2011), 365.

[17]

A. Capătă, Optimality conditions for extended Ky Fan inequality with cone and affine constraints and their applications,, J. Optim. Theory. Appl., 152 (2012), 661. doi: 10.1007/s10957-011-9916-1.

[18]

P. Daniele, S. Giuffrè and A. Maugeri, Infinite dimensional duality and applications,, Math. Ann., 339 (2007), 221. doi: 10.1007/s00208-007-0118-y.

[19]

K. Fan, Minimax theorems,, Proc. National Acad. Sci. USA, 39 (1953), 42. doi: 10.1073/pnas.39.1.42.

[20]

K. Fan, A minimax inequality and applications,, in, (1972), 103.

[21]

F. Giannessi, "Vector Variational Inequalities and Vector Equilibria,", Kluwer Academic Publishers, (2000). doi: 10.1007/978-1-4613-0299-5.

[22]

X. H. Gong, Optimality conditions for vector equilibrium problems,, J. Math. Anal. Appl., 342 (2008), 1455. doi: 10.1016/j.jmaa.2008.01.026.

[23]

R. B. Holmes, "Geometric Functional Analysis and its Applications,", Springer-Verlag, (1975).

[24]

K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems,, Taiwanese J. Math., 12 (2008), 649.

[25]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems,, Taiwanese J. Math., 12 (2008), 2233.

[26]

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.

[27]

B. C. Ma and X. H. Gong, Optimality conditions for vector equilibrium problems in normed spaces,, Optimization, 60 (2011), 1441. doi: 10.1080/02331931003657709.

[28]

Q. Qiu, Optimality conditions of globally efficient solutions for vector equilibrium problems with generalized convexity,, J. Ineq. Appl., (2009). doi: 10.1155/2009/898213.

[29]

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.

[30]

R. T. Rockafellar, "Conjugate Duality and Optimization,", Society for Industrial and Applied Mathematics, (1974).

[31]

J. Salamon, Closedness and Hadamard well-posedness of the solution map for parametric vector equilibrium problems,, J. Global Optim., 47 (2010), 173. doi: 10.1007/s10898-009-9464-5.

[32]

C. Zălinescu, "Convex Analysis in General Vector Spaces,", World Scientific, (2002). doi: 10.1142/9789812777096.

show all references

References:
[1]

L. Q. Anh, P. Q. Khanh, D. T. M. Van and J. C. Yao, Well-posedness for vector quasiequilibria,, Taiwanese J. Math., 13 (2009), 713.

[2]

Q. H. Ansari, I. V. Konnov and J. C. Yao, On generalized vector equilibrium problems,, Nonlinear Anal., 47 (2001), 543. doi: 10.1016/S0362-546X(01)00199-7.

[3]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems,, J. Optim. Theory Appl., 110 (2001), 481. doi: 10.1023/A:1017581009670.

[4]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Characterizations of solutions for vector equilibrium problems,, J. Optim. Theory Appl., 113 (2002), 435. doi: 10.1023/A:1015366419163.

[5]

Q. H. Ansari, W. Oettli and D. Schläger, A generalization of vectorial equilibria,, Math. Meth. Oper. Res., 46 (1997), 147. doi: 10.1007/BF01217687.

[6]

Q. H. Ansari, X. Q. Yang and J. C. Yao, Existence and duality of implicit vector variational problems,, Numer. Funct. Anal. Optim., 22 (2001), 815. doi: 10.1081/NFA-100108310.

[7]

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.

[8]

M. Bianchi, G. Kassay and R. Pini, Ekeland's principle for vector equilibrium problems,, Nonlinear Anal., 66 (2007), 1454. doi: 10.1016/j.na.2006.02.003.

[9]

M. Bianchi, G. Kassay and R. Pini, Well-posedness for vector equilibrium problems,, Math. Meth. Oper. Res., 70 (2009), 171. doi: 10.1007/s00186-008-0239-4.

[10]

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.

[11]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems,, Math. Stud., 63 (1994), 123.

[12]

J. M. Borwein and R. Goebel, Notions of relative interior in Banach spaces,, J. Math. Sci., 115 (2003), 2542. doi: 10.1023/A:1022988116044.

[13]

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.

[14]

J. M. Borwein and V. Jeyakumar, On convexlike Lagrangian and minimax theorems,, Research Report 24, (1988).

[15]

F. Cammaroto and B. Di Bella, Separation theorem based on the quasirelative interior and application to duality theory,, J. Optim. Theory Appl., 125 (2005), 223. doi: 10.1007/s10957-004-1724-4.

[16]

A. Capătă and G. Kassay, On vector equilibrium problems and applications,, Taiwanese J. Math., 15 (2011), 365.

[17]

A. Capătă, Optimality conditions for extended Ky Fan inequality with cone and affine constraints and their applications,, J. Optim. Theory. Appl., 152 (2012), 661. doi: 10.1007/s10957-011-9916-1.

[18]

P. Daniele, S. Giuffrè and A. Maugeri, Infinite dimensional duality and applications,, Math. Ann., 339 (2007), 221. doi: 10.1007/s00208-007-0118-y.

[19]

K. Fan, Minimax theorems,, Proc. National Acad. Sci. USA, 39 (1953), 42. doi: 10.1073/pnas.39.1.42.

[20]

K. Fan, A minimax inequality and applications,, in, (1972), 103.

[21]

F. Giannessi, "Vector Variational Inequalities and Vector Equilibria,", Kluwer Academic Publishers, (2000). doi: 10.1007/978-1-4613-0299-5.

[22]

X. H. Gong, Optimality conditions for vector equilibrium problems,, J. Math. Anal. Appl., 342 (2008), 1455. doi: 10.1016/j.jmaa.2008.01.026.

[23]

R. B. Holmes, "Geometric Functional Analysis and its Applications,", Springer-Verlag, (1975).

[24]

K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems,, Taiwanese J. Math., 12 (2008), 649.

[25]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems,, Taiwanese J. Math., 12 (2008), 2233.

[26]

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.

[27]

B. C. Ma and X. H. Gong, Optimality conditions for vector equilibrium problems in normed spaces,, Optimization, 60 (2011), 1441. doi: 10.1080/02331931003657709.

[28]

Q. Qiu, Optimality conditions of globally efficient solutions for vector equilibrium problems with generalized convexity,, J. Ineq. Appl., (2009). doi: 10.1155/2009/898213.

[29]

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.

[30]

R. T. Rockafellar, "Conjugate Duality and Optimization,", Society for Industrial and Applied Mathematics, (1974).

[31]

J. Salamon, Closedness and Hadamard well-posedness of the solution map for parametric vector equilibrium problems,, J. Global Optim., 47 (2010), 173. doi: 10.1007/s10898-009-9464-5.

[32]

C. Zălinescu, "Convex Analysis in General Vector Spaces,", World Scientific, (2002). doi: 10.1142/9789812777096.

[1]

Qiu-Sheng Qiu. Optimality conditions for vector equilibrium problems with constraints. Journal of Industrial & Management Optimization, 2009, 5 (4) : 783-790. doi: 10.3934/jimo.2009.5.783

[2]

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

[3]

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

[4]

Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-18. doi: 10.3934/jimo.2018174

[5]

Nan-Jing Huang, Xian-Jun Long, Chang-Wen Zhao. Well-Posedness for vector quasi-equilibrium problems with applications. Journal of Industrial & Management Optimization, 2009, 5 (2) : 341-349. doi: 10.3934/jimo.2009.5.341

[6]

M. H. Li, S. J. Li, W. Y. Zhang. Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems. Journal of Industrial & Management Optimization, 2009, 5 (4) : 683-696. doi: 10.3934/jimo.2009.5.683

[7]

Kenji Kimura, Yeong-Cheng Liou, David S. Shyu, Jen-Chih Yao. Simultaneous system of vector equilibrium problems. Journal of Industrial & Management Optimization, 2009, 5 (1) : 161-174. doi: 10.3934/jimo.2009.5.161

[8]

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

[9]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47

[10]

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

[11]

Qiusheng Qiu, Xinmin Yang. Scalarization of approximate solution for vector equilibrium problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 143-151. doi: 10.3934/jimo.2013.9.143

[12]

Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013

[13]

Alain Chenciner. The angular momentum of a relative equilibrium. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1033-1047. doi: 10.3934/dcds.2013.33.1033

[14]

Miniak-Górecka Alicja, Nowakowski Andrzej. Sufficient optimality conditions for a class of epidemic problems with control on the boundary. Mathematical Biosciences & Engineering, 2017, 14 (1) : 263-275. doi: 10.3934/mbe.2017017

[15]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

[16]

Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial & Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329

[17]

Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67

[18]

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 - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

[19]

Ricardo Almeida. Optimality conditions for fractional variational problems with free terminal time. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 1-19. doi: 10.3934/dcdss.2018001

[20]

Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]