• Previous Article
    Artificial intelligence combined with nonlinear optimization techniques and their application for yield curve optimization
  • JIMO Home
  • This Issue
  • Next Article
    Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit
October  2017, 13(4): 1685-1699. doi: 10.3934/jimo.2017013

Continuity of approximate solution maps to vector equilibrium problems

1. 

Department of Mathematics, Teacher College, Can Tho University, Can Tho, 900000, Viet Nam

2. 

Department of Mathematics, Vo Truong Toan University, Hau Giang, Viet Nam

3. 

Department of Mathematics, Nam Can Tho University, Can Tho, 900000, Viet Nam

* Corresponding author

Received  October 2015 Revised  October 2016 Published  December 2016

Fund Project: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.44.

This paper considers the parametric primal and dual vector equilibrium problems in locally convex Hausdorff topological vector spaces. Based on linear scalarization technique, we establish sufficient conditions for the continuity of approximate solution maps to these problems. As applications, some new results for vector optimization problem and vector variational inequality are derived. Our results are new and improve the existing ones in the literature.

Citation: 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
References:
[1]

M. Ait Mansour and H. Riahi, Sensitivity analysis for abstract equilibrium problems, J. Math. Anal. Appl., 306 (2005), 684-691.  doi: 10.1016/j.jmaa.2004.10.011.  Google Scholar

[2]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699-711.  doi: 10.1016/j.jmaa.2004.03.014.  Google Scholar

[3]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim., 29 (2008), 24-42.  doi: 10.1080/01630560701873068.  Google Scholar

[4]

L. Q. Anh and P. Q. Khanh, Various kinds of semicontinuity and solution sets of parametric multivalued symmetric vector quasiequilibrium problems, J. Glob. Optim., 41 (2008), 539-558.  doi: 10.1007/s10898-007-9264-8.  Google Scholar

[5]

L. Q. Anh and P. Q. Khanh, Continuity of solution maps of parametric quasiequilibrium problems, J. Glob. Optim., 46 (2010), 247-259.  doi: 10.1007/s10898-009-9422-2.  Google Scholar

[6]

L. Q. AnhP. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems, Nonlinear Anal., 75 (2012), 2293-2303.  doi: 10.1016/j.na.2011.10.029.  Google Scholar

[7]

L. Q. AnhP. Q. Khanh and T. N. Tam, Hausdorff continuity of approximate solution maps to parametric primal and dual equilibrium problems, TOP, 24 (2016), 242-258.  doi: 10.1007/s11750-015-0390-z.  Google Scholar

[8]

Q. H. AnsariI. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl., 110 (2001), 481-492.  doi: 10.1023/A:1017581009670.  Google Scholar

[9]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar

[10]

C. Berge, Topological Spaces, Oliver and Boyd, London, 1963. Google Scholar

[11]

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31-43.  doi: 10.1007/BF02192244.  Google Scholar

[12]

M. BianchiN. Hadjisavas and S. Schaible, Equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 92 (1997), 527-542.  doi: 10.1023/A:1022603406244.  Google Scholar

[13]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student., 63 (1994), 123-145.   Google Scholar

[14]

C. R. ChenS. L. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, J. Glob. Optim., 45 (2009), 309-318.  doi: 10.1007/s10898-008-9376-9.  Google Scholar

[15]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550.  doi: 10.1007/s10898-004-2692-9.  Google Scholar

[16]

K. Fan, A minimax inequality and applications, In: Shisha O (ed) Inequality III Academic Press, New York, (1972), 103–113.  Google Scholar

[17] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, London, 1983.   Google Scholar
[18]

X. H. Gong, Continuity of the solution set to parametric vector equilibrium problem, J. Optim. Theory Appl., 139 (2008), 35-46.  doi: 10.1007/s10957-008-9429-8.  Google Scholar

[19]

N. X. HaiP. Q. Khanh and N. H. Quan, On the existence of solutions to quasivariational inclusion problems, J. Glob. Optim., 45 (2009), 565-581.  doi: 10.1007/s10898-008-9390-y.  Google Scholar

[20]

J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt, 1986.  Google Scholar

[21]

P. Q. Khanh and V. S. T. Long, Invariant-point theorems and existence of solutions to optimization-related problems, J. Global. Optim., 58 (2014), 545-564.  doi: 10.1007/s10898-013-0065-y.  Google Scholar

[22]

W. K. KimS. Kum and K. H. Lee, Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems, Nonlinear Anal., 71 (2009), 2182-2187.  doi: 10.1016/j.na.2009.04.036.  Google Scholar

[23]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443.  doi: 10.1007/s10957-008-9386-2.  Google Scholar

[24]

X. B. Li and S. J. Li, Continuity of approximate solution mapping for parametric equilibrium problems, J. Glob. Optim., 51 (2011), 541-548.  doi: 10.1007/s10898-010-9641-6.  Google Scholar

[25]

S. J. LiH. M. Liu and C. R. Chen, Lower semicontinuity of parametric generalized weak vector equilibrium problems, Bull. Aust. Math. Soc., 81 (2010), 85-95.  doi: 10.1017/S0004972709000628.  Google Scholar

[26]

S. J. LiH. M. LiuY. Zhang and Z. M. Fang, Continuity of solution mappings to parametric generalized strong vector equilibrium problems, J. Global Optim., 55 (2013), 597-610.  doi: 10.1007/s10898-012-9985-1.  Google Scholar

[27]

H. Nikaido and K. Isoda, Note on non-copperative convex games, Pacific J. Math., 5 (1955), 807-815.  doi: 10.2140/pjm.1955.5.807.  Google Scholar

show all references

References:
[1]

M. Ait Mansour and H. Riahi, Sensitivity analysis for abstract equilibrium problems, J. Math. Anal. Appl., 306 (2005), 684-691.  doi: 10.1016/j.jmaa.2004.10.011.  Google Scholar

[2]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699-711.  doi: 10.1016/j.jmaa.2004.03.014.  Google Scholar

[3]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim., 29 (2008), 24-42.  doi: 10.1080/01630560701873068.  Google Scholar

[4]

L. Q. Anh and P. Q. Khanh, Various kinds of semicontinuity and solution sets of parametric multivalued symmetric vector quasiequilibrium problems, J. Glob. Optim., 41 (2008), 539-558.  doi: 10.1007/s10898-007-9264-8.  Google Scholar

[5]

L. Q. Anh and P. Q. Khanh, Continuity of solution maps of parametric quasiequilibrium problems, J. Glob. Optim., 46 (2010), 247-259.  doi: 10.1007/s10898-009-9422-2.  Google Scholar

[6]

L. Q. AnhP. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems, Nonlinear Anal., 75 (2012), 2293-2303.  doi: 10.1016/j.na.2011.10.029.  Google Scholar

[7]

L. Q. AnhP. Q. Khanh and T. N. Tam, Hausdorff continuity of approximate solution maps to parametric primal and dual equilibrium problems, TOP, 24 (2016), 242-258.  doi: 10.1007/s11750-015-0390-z.  Google Scholar

[8]

Q. H. AnsariI. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl., 110 (2001), 481-492.  doi: 10.1023/A:1017581009670.  Google Scholar

[9]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar

[10]

C. Berge, Topological Spaces, Oliver and Boyd, London, 1963. Google Scholar

[11]

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31-43.  doi: 10.1007/BF02192244.  Google Scholar

[12]

M. BianchiN. Hadjisavas and S. Schaible, Equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 92 (1997), 527-542.  doi: 10.1023/A:1022603406244.  Google Scholar

[13]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student., 63 (1994), 123-145.   Google Scholar

[14]

C. R. ChenS. L. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, J. Glob. Optim., 45 (2009), 309-318.  doi: 10.1007/s10898-008-9376-9.  Google Scholar

[15]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550.  doi: 10.1007/s10898-004-2692-9.  Google Scholar

[16]

K. Fan, A minimax inequality and applications, In: Shisha O (ed) Inequality III Academic Press, New York, (1972), 103–113.  Google Scholar

[17] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, London, 1983.   Google Scholar
[18]

X. H. Gong, Continuity of the solution set to parametric vector equilibrium problem, J. Optim. Theory Appl., 139 (2008), 35-46.  doi: 10.1007/s10957-008-9429-8.  Google Scholar

[19]

N. X. HaiP. Q. Khanh and N. H. Quan, On the existence of solutions to quasivariational inclusion problems, J. Glob. Optim., 45 (2009), 565-581.  doi: 10.1007/s10898-008-9390-y.  Google Scholar

[20]

J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt, 1986.  Google Scholar

[21]

P. Q. Khanh and V. S. T. Long, Invariant-point theorems and existence of solutions to optimization-related problems, J. Global. Optim., 58 (2014), 545-564.  doi: 10.1007/s10898-013-0065-y.  Google Scholar

[22]

W. K. KimS. Kum and K. H. Lee, Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems, Nonlinear Anal., 71 (2009), 2182-2187.  doi: 10.1016/j.na.2009.04.036.  Google Scholar

[23]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443.  doi: 10.1007/s10957-008-9386-2.  Google Scholar

[24]

X. B. Li and S. J. Li, Continuity of approximate solution mapping for parametric equilibrium problems, J. Glob. Optim., 51 (2011), 541-548.  doi: 10.1007/s10898-010-9641-6.  Google Scholar

[25]

S. J. LiH. M. Liu and C. R. Chen, Lower semicontinuity of parametric generalized weak vector equilibrium problems, Bull. Aust. Math. Soc., 81 (2010), 85-95.  doi: 10.1017/S0004972709000628.  Google Scholar

[26]

S. J. LiH. M. LiuY. Zhang and Z. M. Fang, Continuity of solution mappings to parametric generalized strong vector equilibrium problems, J. Global Optim., 55 (2013), 597-610.  doi: 10.1007/s10898-012-9985-1.  Google Scholar

[27]

H. Nikaido and K. Isoda, Note on non-copperative convex games, Pacific J. Math., 5 (1955), 807-815.  doi: 10.2140/pjm.1955.5.807.  Google Scholar

[1]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[2]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[3]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[4]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[5]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119

[6]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[7]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385

[8]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[9]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[10]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452

[11]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018

[12]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

[13]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[14]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[15]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[16]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[17]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[18]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

[19]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[20]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (107)
  • HTML views (379)
  • Cited by (0)

Other articles
by authors

[Back to Top]