2018, 14(1): 65-79. doi: 10.3934/jimo.2017037

Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems

1. 

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

2. 

Center of Research and Development Duy Tan University K7/25, Quang Trung, Danang, VietNam

3. 

Department of Mathematics, Dong Thap University, Dong Thap, Viet Nam

* Corresponding author

Received  November 2015 Revised  February 2017 Published  April 2017

Fund Project: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2017.18

In this paper, we consider parametric strong vector quasiequilibrium problems in Hausdorff topological vector spaces. Firstly, we introduce parametric gap functions for these problems, and study the continuity property of these functions. Next, we present two key hypotheses related to the gap functions for the considered problems and also study characterizations of these hypotheses. Then, afterwards, we prove that these hypotheses are not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of solution mappings to these problems. Finally, as applications, we derive several results on Hausdorff (lower) continuity properties of the solution mappings in the special cases of variational inequalities of the Minty type and the Stampacchia type.

Citation: Lam Quoc Anh, Nguyen Van Hung. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems. Journal of Industrial & Management Optimization, 2018, 14 (1) : 65-79. doi: 10.3934/jimo.2017037
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.

[2]

M. Ait Mansour and L. Scrimali, Hölder continuity of solutions to elastic traffic networkmodels, J Global Optim., 40 (2008), 175-184. doi: 10.1007/s10898-007-9190-9.

[3]

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

[4]

L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems, J. Math. Anal. Appl., 321 (2006), 308-315. doi: 10.1016/j.jmaa.2005.08.018.

[5]

L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces, J. Global Optim., 37 (2007), 449-465. doi: 10.1007/s10898-006-9062-8.

[6]

L. Q. Anh and P. Q. Khanh, On the stability of the solution sets of general multivalued vector quasiequilibrium problems, J. Optim. Theory Appl., 135 (2007), 271-284. doi: 10.1007/s10957-007-9250-9.

[7]

L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions, J. Global Optim., 42 (2008), 515-531. doi: 10.1007/s10898-007-9268-4.

[8]

L. Q. Anh and P. Q. Khanh, Hölder continuity of the unique solution to quasiequilibrium problems in metric spaces, J. Optim. Theory Appl., 141 (2009), 37-54. doi: 10.1007/s10957-008-9508-x.

[9]

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

[10]

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.

[11]

L. Q. AnhA. Y. Kruger and N. H. Thao, On Hölder calmness of solution mappings in parametric equilibrium problems, TOP, 22 (2014), 331-342. doi: 10.1007/s11750-012-0259-3.

[12]

L. Q. Anh 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.

[13]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.

[14]

D. Aussel and J. Dutta, On gap functions for multivalued stampacchia variational inequalities, J. Optim. Theory Appl., 149 (2011), 513-527. doi: 10.1007/s10957-011-9801-y.

[15]

E. Bednarczuk, Strong pseudomonotonicity, sharp efficiency and stability for parametric vector equilibria, ESAIM Proc., 17 (2007), 9-18. doi: 10.1051/proc:071702.

[16]

M. Bianchi and R. Pini, A note on stability for parametric equilibrium problems, Oper. Res. Lett., 31 (2003), 445-450. doi: 10.1016/S0167-6377(03)00051-8.

[17]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria, Optimization, 55 (2006), 221-230. doi: 10.1080/02331930600662732.

[18]

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

[19]

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

[20]

C. R. Chen and S. J. Li, Semicontinuity of the solution set map to a set-valued weak vector variational inequality, J. Ind. Manag. Optim., 3 (2007), 519-528. doi: 10.3934/jimo.2007.3.519.

[21]

C. R. ChenS. J. Li and Z. M. Fang, On the solution semicontinuity to a parametric generalized vector quasivariational inequality, Comput. Math. Appl., 60 (2010), 2417-2425. doi: 10.1016/j.camwa.2010.08.036.

[22]

J. W. Chen and Z. P. Wan, Semicontinuity for parametric Minty vector quasivariational inequalities in Hausdorff topological vector spaces, Comput. Appl. Math., 33 (2014), 111-129. doi: 10.1007/s40314-013-0047-1.

[23]

Y. P. Fang and R. Hu, Parametric well-posedness for variational inequalities defined bifunctions, Comput. Math. Appl., 53 (2007), 1306-1316. doi: 10.1016/j.camwa.2006.09.009.

[24]

J. Y. Fu, Generalized vector quasiequilibrium problems, Math. Meth. Oper. Res., 52 (2000), 57-64. doi: 10.1007/s001860000058.

[25]

F. Giannessi, On Minty variational principle, New Trends in Mathematical Programming, Kluwer Academic Publishers, Dordrecht, Boston, London, 1 (1998), 93–99. doi: 10.1007/978-1-4757-2878-1_8.

[26]

Chr. (Tammer) Gerstewitz, Nichtkonvexe dualitat in der vektaroptimierung, Wissenschaftliche Zeitschrift der Technischen Hochschule Leuna-Mersebung, 25 (1983), 357-364.

[27]

N. X. Hai and P. Q. Khanh, Existence of solutions to general quasiequilibrium problems and applications, J. Optim. Theory Appl., 133 (2007), 317-327. doi: 10.1007/s10957-007-9170-8.

[28]

N. J. HuangJ. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Modelling, 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010.

[29]

A. N. Iusem and W. Sosa, New existence results for equilibrium problems, Nonlinear Anal., 52 (2003), 621-635. doi: 10.1016/S0362-546X(02)00154-2.

[30]

A. N. IusemG. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., 116 (2009), 259-273. doi: 10.1007/s10107-007-0125-5.

[31]

G. KassayJ. Kolumbán and Z. Páles, Factorization of Minty and Stampacchia variational inequality systems, European J. Oper. Res., 143 (2002), 377-389. doi: 10.1016/S0377-2217(02)00290-4.

[32]

B. T. Kien, On the lower semicontinuity of optimal solution sets, Optimization, 54 (2005), 123-130. doi: 10.1080/02331930412331330379.

[33]

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.

[34]

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.

[35]

S. Komlósi, On the Stampacchia and Minty Variational Inequalities, in: G. Giorgi, F. A. Rossi (Eds. ), Generalized Convexity and Optimization for Economic and Financial Decisions, Pitagora Editrice, Bologna, 1999.

[36]

I. Konnov and S. Schaible, Duality for equilibrium problems under generalized monotonicity, J. Optim. Theory Appl., 104 (2000), 395-408. doi: 10.1023/A:1004665830923.

[37]

C. S. Lalitha and G. Bhatia, Stability of parametric quasivariational inequality of the Minty type, J. Optim. Theory. Appl., 148 (2011), 281-300. doi: 10.1007/s10957-010-9755-5.

[38]

S. J. LiX. B. LiL. N. Wang and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems, European J. Oper. Res., 199 (2009), 334-338. doi: 10.1016/j.ejor.2008.12.024.

[39]

S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 149 (2011), 540-553. doi: 10.1007/s10957-011-9803-9.

[40]

S. J. LiK. L. TeoX. Q. Yang and S. Y. Wu, Gap functions and existence of solutions to generalized vector quasiequilibrium problems, J. Global Optim., 34 (2006), 427-440. doi: 10.1007/s10898-005-2193-5.

[41]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality problems, Nonlinear Anal., 70 (2009), 1528-1535. doi: 10.1016/j.na.2008.02.032.

[42]

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

[43]

X. J. LongN. J. Huang and K. L. Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problem, Math. Comput. Modelling, 47 (2008), 445-451. doi: 10.1016/j.mcm.2007.04.013.

[44]

D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economic and Mathematical Systems 319, Springer-verlag, Berlin, 1989. doi: 10.1007/978-3-642-50280-4.

[45]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences), 330. Springer, Berlin, 2006.

[46]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences), 331, Springer, Berlin, 2006.

[47]

Z. Y. PengX. M. Yang and J. W. Peng, On the Lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264. doi: 10.1007/s10957-011-9883-6.

[48]

Z. Y. PengY. Zhao and X. M. Yang, Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems, Numer. Funct. Anal. Optim., 36 (2015), 481-500. doi: 10.1080/01630563.2015.1013551.

[49]

I. Sadeqi and C. Alizadeh, Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces, Nonlinear Anal., 74 (2011), 2226-2234. doi: 10.1016/j.na.2010.11.027.

[50]

J. Zhao, The lower semicontinuity of optimal solution sets, J. Math. Anal. Appl., 207 (1997), 240-254. doi: 10.1006/jmaa.1997.5288.

[51]

R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vetor variational inequalities in reflexive Banach spaces, J. Optim. Theory Appl., 150 (2011), 317-326. doi: 10.1007/s10957-011-9843-1.

[52]

R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems, Comput. Math. Appl., 63 (2012), 807-815. doi: 10.1016/j.camwa.2011.11.046.

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.

[2]

M. Ait Mansour and L. Scrimali, Hölder continuity of solutions to elastic traffic networkmodels, J Global Optim., 40 (2008), 175-184. doi: 10.1007/s10898-007-9190-9.

[3]

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

[4]

L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems, J. Math. Anal. Appl., 321 (2006), 308-315. doi: 10.1016/j.jmaa.2005.08.018.

[5]

L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces, J. Global Optim., 37 (2007), 449-465. doi: 10.1007/s10898-006-9062-8.

[6]

L. Q. Anh and P. Q. Khanh, On the stability of the solution sets of general multivalued vector quasiequilibrium problems, J. Optim. Theory Appl., 135 (2007), 271-284. doi: 10.1007/s10957-007-9250-9.

[7]

L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions, J. Global Optim., 42 (2008), 515-531. doi: 10.1007/s10898-007-9268-4.

[8]

L. Q. Anh and P. Q. Khanh, Hölder continuity of the unique solution to quasiequilibrium problems in metric spaces, J. Optim. Theory Appl., 141 (2009), 37-54. doi: 10.1007/s10957-008-9508-x.

[9]

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

[10]

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.

[11]

L. Q. AnhA. Y. Kruger and N. H. Thao, On Hölder calmness of solution mappings in parametric equilibrium problems, TOP, 22 (2014), 331-342. doi: 10.1007/s11750-012-0259-3.

[12]

L. Q. Anh 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.

[13]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.

[15]

E. Bednarczuk, Strong pseudomonotonicity, sharp efficiency and stability for parametric vector equilibria, ESAIM Proc., 17 (2007), 9-18. doi: 10.1051/proc:071702.

[16]

M. Bianchi and R. Pini, A note on stability for parametric equilibrium problems, Oper. Res. Lett., 31 (2003), 445-450. doi: 10.1016/S0167-6377(03)00051-8.

[17]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria, Optimization, 55 (2006), 221-230. doi: 10.1080/02331930600662732.

[18]

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

[19]

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

[20]

C. R. Chen and S. J. Li, Semicontinuity of the solution set map to a set-valued weak vector variational inequality, J. Ind. Manag. Optim., 3 (2007), 519-528. doi: 10.3934/jimo.2007.3.519.

[21]

C. R. ChenS. J. Li and Z. M. Fang, On the solution semicontinuity to a parametric generalized vector quasivariational inequality, Comput. Math. Appl., 60 (2010), 2417-2425. doi: 10.1016/j.camwa.2010.08.036.

[22]

J. W. Chen and Z. P. Wan, Semicontinuity for parametric Minty vector quasivariational inequalities in Hausdorff topological vector spaces, Comput. Appl. Math., 33 (2014), 111-129. doi: 10.1007/s40314-013-0047-1.

[23]

Y. P. Fang and R. Hu, Parametric well-posedness for variational inequalities defined bifunctions, Comput. Math. Appl., 53 (2007), 1306-1316. doi: 10.1016/j.camwa.2006.09.009.

[24]

J. Y. Fu, Generalized vector quasiequilibrium problems, Math. Meth. Oper. Res., 52 (2000), 57-64. doi: 10.1007/s001860000058.

[25]

F. Giannessi, On Minty variational principle, New Trends in Mathematical Programming, Kluwer Academic Publishers, Dordrecht, Boston, London, 1 (1998), 93–99. doi: 10.1007/978-1-4757-2878-1_8.

[26]

Chr. (Tammer) Gerstewitz, Nichtkonvexe dualitat in der vektaroptimierung, Wissenschaftliche Zeitschrift der Technischen Hochschule Leuna-Mersebung, 25 (1983), 357-364.

[27]

N. X. Hai and P. Q. Khanh, Existence of solutions to general quasiequilibrium problems and applications, J. Optim. Theory Appl., 133 (2007), 317-327. doi: 10.1007/s10957-007-9170-8.

[28]

N. J. HuangJ. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Modelling, 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010.

[29]

A. N. Iusem and W. Sosa, New existence results for equilibrium problems, Nonlinear Anal., 52 (2003), 621-635. doi: 10.1016/S0362-546X(02)00154-2.

[30]

A. N. IusemG. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., 116 (2009), 259-273. doi: 10.1007/s10107-007-0125-5.

[31]

G. KassayJ. Kolumbán and Z. Páles, Factorization of Minty and Stampacchia variational inequality systems, European J. Oper. Res., 143 (2002), 377-389. doi: 10.1016/S0377-2217(02)00290-4.

[32]

B. T. Kien, On the lower semicontinuity of optimal solution sets, Optimization, 54 (2005), 123-130. doi: 10.1080/02331930412331330379.

[33]

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.

[34]

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.

[35]

S. Komlósi, On the Stampacchia and Minty Variational Inequalities, in: G. Giorgi, F. A. Rossi (Eds. ), Generalized Convexity and Optimization for Economic and Financial Decisions, Pitagora Editrice, Bologna, 1999.

[36]

I. Konnov and S. Schaible, Duality for equilibrium problems under generalized monotonicity, J. Optim. Theory Appl., 104 (2000), 395-408. doi: 10.1023/A:1004665830923.

[37]

C. S. Lalitha and G. Bhatia, Stability of parametric quasivariational inequality of the Minty type, J. Optim. Theory. Appl., 148 (2011), 281-300. doi: 10.1007/s10957-010-9755-5.

[38]

S. J. LiX. B. LiL. N. Wang and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems, European J. Oper. Res., 199 (2009), 334-338. doi: 10.1016/j.ejor.2008.12.024.

[39]

S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 149 (2011), 540-553. doi: 10.1007/s10957-011-9803-9.

[40]

S. J. LiK. L. TeoX. Q. Yang and S. Y. Wu, Gap functions and existence of solutions to generalized vector quasiequilibrium problems, J. Global Optim., 34 (2006), 427-440. doi: 10.1007/s10898-005-2193-5.

[41]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality problems, Nonlinear Anal., 70 (2009), 1528-1535. doi: 10.1016/j.na.2008.02.032.

[42]

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

[43]

X. J. LongN. J. Huang and K. L. Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problem, Math. Comput. Modelling, 47 (2008), 445-451. doi: 10.1016/j.mcm.2007.04.013.

[44]

D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economic and Mathematical Systems 319, Springer-verlag, Berlin, 1989. doi: 10.1007/978-3-642-50280-4.

[45]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences), 330. Springer, Berlin, 2006.

[46]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences), 331, Springer, Berlin, 2006.

[47]

Z. Y. PengX. M. Yang and J. W. Peng, On the Lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264. doi: 10.1007/s10957-011-9883-6.

[48]

Z. Y. PengY. Zhao and X. M. Yang, Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems, Numer. Funct. Anal. Optim., 36 (2015), 481-500. doi: 10.1080/01630563.2015.1013551.

[49]

I. Sadeqi and C. Alizadeh, Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces, Nonlinear Anal., 74 (2011), 2226-2234. doi: 10.1016/j.na.2010.11.027.

[50]

J. Zhao, The lower semicontinuity of optimal solution sets, J. Math. Anal. Appl., 207 (1997), 240-254. doi: 10.1006/jmaa.1997.5288.

[51]

R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vetor variational inequalities in reflexive Banach spaces, J. Optim. Theory Appl., 150 (2011), 317-326. doi: 10.1007/s10957-011-9843-1.

[52]

R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems, Comput. Math. Appl., 63 (2012), 807-815. doi: 10.1016/j.camwa.2011.11.046.

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