Skip to main content
SpringerLink
Log in

On Lipschitz continuity of solutions to equilibrium problems via the Hiriart-Urruty oriented distance function

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider strong vector equilibrium problems in normed spaces. Firstly, we study stability conditions for a scalar equilibrium problem without assuming boundedness and concavity properties of the constraint map and objective function, respectively. Next, we discuss properties of the Hiriart-Urruty oriented distance function in an ordered space, and then using these properties, relationships between the strong equilibrium problems and the scalar ones are formulated. Then after, based on these relationships, we address sufficient conditions for the Lipschitz continuity of approximate solution maps to vector equilibrium problems via the corresponding results of the scalar equilibrium problems. As an application, we apply the obtained results to express stability conditions for network equilibrium problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Anh LQ, Tam TN (2017) Sensitivity analysis for parametric vector equilibrium problems. J Nonlinear Convex Anal 18(9):1707–1716

    MathSciNet  MATH  Google Scholar 

  • Anh LQ, Khanh PQ, Tam TN (2012) On Hölder continuity of approximate solutions to parametric equilibrium problems. Nonlinear Anal 75(4):2293–2303

    Article  MathSciNet  Google Scholar 

  • Anh LQ, Duoc PT, Tam TN (2017a) Continuity of approximate solution maps to vector equilibrium problems. J Ind Manag Optim 13(4):1685–1699

    MathSciNet  MATH  Google Scholar 

  • Anh LQ, Nguyen KT, Tam TN (2017b) On Hölder continuity of approximate solution maps to vector equilibrium problems. Turk J Math 41(6):1591–1607

    MATH  Google Scholar 

  • Anh LQ, Hung NV, Tam VM (2018) Regularized gap functions and error bounds for generalized mixed strong vector quasiequilibrium problems. Comput Appl Math 37(5):5935–5950

    Article  MathSciNet  Google Scholar 

  • Anh LQ, Danh NH, Tam TN (2020) Continuity of solution maps to parametric set optimization problems via parametric equilibrium problems. Acta Math Vietnam 45(2):383–395

    Article  MathSciNet  Google Scholar 

  • Anh LQ, Tam TN, Danh NH (2021) On Lipschitz continuity of approximate solutions to set-valued equilibrium problems via nonlinear scalarization. Optimization. https://doi.org/10.1080/02331934.2021.1970753

    Article  Google Scholar 

  • Borwein JM, Fitzpatrick S (1989) Existence of nearest points in Banach spaces. Can J Math 41(4):702–720

    Article  MathSciNet  Google Scholar 

  • Chen CR, Li LL, Li MH (2016) Hölder continuity results for nonconvex parametric generalized vector quasiequilibrium problems via nonlinear scalarizing functions. Optimization 65(1):35–51

    Article  MathSciNet  Google Scholar 

  • Chen J, Ansari QH, Yao JC (2017) Characterizations of set order relations and constrained set optimization problems via oriented distance function. Optimization 66(11):1741–1754

    Article  MathSciNet  Google Scholar 

  • Crespi GP, Ginchev I, Rocca M (2006) First-order optimality conditions in set-valued optimization. Math Methods Oper Res 63(1):87–106

    Article  MathSciNet  Google Scholar 

  • Fang SC, Rajasekera JR, Tsao HSJ (2012) Entropy optimization and mathematical programming, vol 8. Springer Science, New York

    MATH  Google Scholar 

  • Farajzadeh A, Lee BS, Plubteing S (2016) On generalized quasi-vector equilibrium problems via scalarization method. J Optim Theory Appl 168(2):584–599

    Article  MathSciNet  Google Scholar 

  • Gerstewitz C (1983) Nichtkonvexe dualität in der vektoroptimierung. Wissensch Zeitschr TH Leuna-Merseburg 25(3):357–364

    MathSciNet  MATH  Google Scholar 

  • Gutiérrez C, Jiménez B, Miglierina E, Molho E (2015) Scalarization in set optimization with solid and nonsolid ordering cones. J Glob Optim 61(3):525–552

    Article  MathSciNet  Google Scholar 

  • Gutiérrez C, Novo V, Ródenas-Pedregosa JL (2018) A note on existence of weak efficient solutions for vector equilibrium problems. Optim Lett 12(3):615–623

    Article  MathSciNet  Google Scholar 

  • Ha TXD (2018) A Hausdorff-type distance, a directional derivative of a set-valued map and applications in set optimization. Optimization 67(7):1031–1050

    Article  MathSciNet  Google Scholar 

  • Han Y (2018) Lipschitz continuity of approximate solution mappings to parametric generalized vector equilibrium problems. J Optim Theory Appl 178(3):763–793

    Article  MathSciNet  Google Scholar 

  • Han Y (2019) Nonlinear scalarizing functions in set optimization problems. Optimization 68(9):1685–1718

    Article  MathSciNet  Google Scholar 

  • Hiriart-Urruty JB (1979) Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math Oper Res 4(1):79–97

    Article  MathSciNet  Google Scholar 

  • Iusem A, Lara F (2019) Optimality conditions for vector equilibrium problems with applications. J Optim Theory Appl 180(1):187–206

    Article  MathSciNet  Google Scholar 

  • Jafari S, Farajzadeh AP, Moradi S, Khanh PQ (2017) Existence results for \(\varphi \)-quasimonotone equilibrium problems in convex metric spaces. Optimization 66(3):293–310

    Article  MathSciNet  Google Scholar 

  • Jiménez B, Novo V, Vílchez A (2020a) Six scalarizations based on the oriented distance in set optimization: strict monotonicity and weak minimality. J Nonlinear Convex Anal 21(11):2433–2457

    MathSciNet  MATH  Google Scholar 

  • Jiménez B, Novo V, Vílchez A (2020b) Six set scalarizations based on the oriented distance: properties and application to set optimization. Optimization 69(3):437–470

    Article  MathSciNet  Google Scholar 

  • Kassay G, Rădulescu V (2018) Equilibrium problems and applications. Academic Press, Cambridge

    MATH  Google Scholar 

  • Khan AA, Tammer C, Zălinescu C (2015) Set-valued optimization. Springer, Berlin

    Book  Google Scholar 

  • Khanh PQ, Quan NH (2019) Versions of the Weierstrass theorem for bifunctions and solution existence in optimization. SIAM J Optim 29(2):1502–1523

    Article  MathSciNet  Google Scholar 

  • Kuroiwa D (1996) Convexity for set-valued maps. Appl Math Lett 9(2):97–101

    Article  MathSciNet  Google Scholar 

  • Kuroiwa D (1997) Some criteria in set-valued optimization. Surikaisekikenkyusho Kokyuroku (Kyoto Univ) 985:171–176

    MathSciNet  MATH  Google Scholar 

  • Lalitha C (2019) Scalarizations for a set optimization problem using generalized oriented distance function. Positivity 23(5):1195–1213

    Article  MathSciNet  Google Scholar 

  • Li XB, Li SJ, Chen CR (2012) Lipschitz continuity of an approximate solution mapping to equilibrium problems. Taiwan J Math 16(3):1027–1040

    Article  MathSciNet  Google Scholar 

  • Peng Z, Yang X, Teo KL (2015) On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan inequality. J Ind Manag Optim 11(2):549–562

    MathSciNet  MATH  Google Scholar 

  • Preechasilp P, Wangkeeree R (2017) Hölder continuity of solution maps to a parametric weak vector equilibrium problem. Bull Iran Math Soc 43(6):1751–1767

    MATH  Google Scholar 

  • Rockafellar RT, Wets RJB (2009) Variational analysis, vol 317. Springer, Berlin

    MATH  Google Scholar 

  • Sadeqi I, Alizadeh C (2011) Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces. Nonlinear Anal 74(6):2226–2234

    Article  MathSciNet  Google Scholar 

  • Sadeqi I, Salehi Paydar M (2016) Lipschitz continuity of an approximate solution mapping for parametric set-valued vector equilibrium problems. Optimization 65(5):1003–1021

    Article  MathSciNet  Google Scholar 

  • Smith MJ (1979) The existence, uniqueness and stability of traffic equilibria. Trans Res 13(4):295–304

    Article  MathSciNet  Google Scholar 

  • Wardrop JG (1952) Some theoretical aspects of road traffic research. Proc Inst Civ Eng 1(3):325–362

    Google Scholar 

  • Wei H, Chen C, Wu B (2021) Vector network equilibrium problems with uncertain demands and capacity constraints of arcs. Optim Lett 15(4):1113–1131

    Article  MathSciNet  Google Scholar 

  • Yen ND (1995) Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint. Math Oper Res 20(3):695–708

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This is a result of the project under Grant number B2022-TCT-02 supported by The Ministry of Education and Training of Viet Nam.

Funding

This study was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) (Grant no. 101.01-2020.11).

Author information

Authors and Affiliations

  1. Department of Mathematics, Teacher College, Cantho University, Can Tho, Vietnam

    Lam Quoc Anh

  2. University of Science, Ho Chi Minh City, Vietnam

    Pham Thanh Duoc

  3. Vietnam National University, Ho Chi Minh City, Vietnam

    Pham Thanh Duoc

  4. Department of Mathematics, Votruongtoan University, Chau Thanh, Hau Giang, Vietnam

    Pham Thanh Duoc

  5. Department of Mathematical Economic, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Nguyen Minh Tung

Authors
  1. Lam Quoc Anh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pham Thanh Duoc
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Minh Tung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pham Thanh Duoc.

Additional information

Communicated by Carlos Conca.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Anh, L.Q., Duoc, P.T. & Tung, N.M. On Lipschitz continuity of solutions to equilibrium problems via the Hiriart-Urruty oriented distance function. Comp. Appl. Math. 41, 57 (2022). https://doi.org/10.1007/s40314-022-01758-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01758-w

Keywords

Search

Navigation