Skip to main content
SpringerLink
Log in

Optimality conditions and duality for mathematical programming with equilibrium constraints including multiple interval-valued objective functions on Hadamard manifolds

  • Original Paper
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper investigates mathematical programming with equilibrium constraints including multiple interval-valued objective functions on Hadamard manifolds. In the first part, both necessary and sufficient optimality conditions for some types of efficient solutions are considered. After that, the Wolfe and Mond–Weir type dual problems are formulated and the duality relations under geodesic convexity assumptions are examined. Some examples are proposed to illustrate the results.

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

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)

    Book  Google Scholar 

  2. Antczak, T.: Optimality conditions and duality results for nonsmooth vector optimization problems with the multiple interval-valued objective function. Acta Math. Sci. 37, 1133–1150 (2017)

    Article  MathSciNet  Google Scholar 

  3. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)

    Google Scholar 

  4. Bao, T.Q., Gupta, P., Mordukhovich, B.S.: Necessary conditions in multiobjective optimization with equilibrium constraints. J. Optim. Theory Appl. 135, 179–203 (2007)

    Article  MathSciNet  Google Scholar 

  5. Bao, T.Q., Mordukhovich, B.S.: Sufficient optimality conditions for global Pareto solutions to multiobjective problems with equilibrium constraints. J. Nonlinear Convex Anal. 15, 105–127 (2014)

    MathSciNet  Google Scholar 

  6. Bergmann, R., Herzog, R.: Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds. SIAM J. Optim. 29, 2423–2444 (2019)

    Article  MathSciNet  Google Scholar 

  7. Ida, M.: Portfolio selection problem with interval coefficients. Appl. Math. Lett. 16(5), 709–713 (2003)

    Article  MathSciNet  Google Scholar 

  8. Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48(2), 219–225 (1990)

    Article  Google Scholar 

  9. Karaaslan, A., Gezen, M.: The evaluation of renewable energy resources in Turkey by integer multi-objective selection problem with interval coefficient. Renew. Energ. 182, 842–854 (2022)

    Article  Google Scholar 

  10. Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023)

    Book  Google Scholar 

  11. Chen, S.: The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds. Optimization 71(3), 613–632 (2022)

    Article  MathSciNet  Google Scholar 

  12. Devine, M.T., Siddiqui, S.: Strategic investment decisions in an oligopoly with a competitive fringe: an equilibrium problem with equilibrium constraints approach. Eur. J. Oper. Res. 306(3), 1473–1494 (2023)

    Article  MathSciNet  Google Scholar 

  13. Fernández, J., Pelegrín, B.: Using interval analysis for solving planar single-facility location problems: new discarding tests. J. Glob. Optim. 19, 61–81 (2001)

    Article  MathSciNet  Google Scholar 

  14. Flegel, M.L., Kanzow, C.: Abadie-type constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 124, 595–614 (2005)

    Article  MathSciNet  Google Scholar 

  15. Guo, Y., Ye, G., Liu, W., Zhao, D., Treanţă, S.: Solving nonsmooth interval optimization problems based on interval-valued symmetric invexity. Chaos Solit. Fract. 174, 113834 (2023)

    Article  MathSciNet  Google Scholar 

  16. Jayswal, A., Ahmad, I., Banerjee, J.: Nonsmooth interval-valued optimization and saddle-point optimality criteria. Bull. Malays. Math. Sci. Soc. 39, 1391–1441 (2016)

    Article  MathSciNet  Google Scholar 

  17. Kanzow, C., Schwartz, A.: Mathematical programs with equilibrium constraints: enhanced Fritz-John conditions, new constraint qualifications, and improved exact penalty results. SIAM J. Optim. 20, 2730–2753 (2010)

    Article  MathSciNet  Google Scholar 

  18. Karkhaneei, M.M., Mahdavi-Amiri, N.: Nonconvex weak sharp minima on Riemannian manifolds. J. Optim. Theory Appl. 183, 85–104 (2019)

    Article  MathSciNet  Google Scholar 

  19. Khanh, P.Q., Tung, L.T.: On optimality conditions and duality for multiobjective optimization with equilibrium constraints. Positivity 27(4), 1–27 (2023)

    Article  MathSciNet  Google Scholar 

  20. Lai, K.K., Mishra, S.K., Hassan, M., Bisht, J., Maurya, J.K.: Duality results for interval-valued semiinfinite optimization problems with equilibrium constraints using convexificators. J. Inequal. Appl. 2022(1), 128 (2022)

    Article  MathSciNet  Google Scholar 

  21. Lin, G.H., Zhang, D.L., Liang, Y.C.: Stochastic multiobjective problems with complementarity constraints and applications in healthcare management. Eur. J. Oper. Res. 226, 461–470 (2013)

    Article  MathSciNet  Google Scholar 

  22. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)

    Book  Google Scholar 

  23. Ma, T., Zhang, Y., Han, Z., Li, C.: Heterogeneous RAN slicing resource allocation using mathematical program with equilibrium constraints. IET Commun. 16(15), 1772–1786 (2022)

    Article  Google Scholar 

  24. Mangasarian, O.L.: Nonlinear Programming. McGraw Hill, New York (1969)

    Google Scholar 

  25. Mond, B., Weir, T.: Generalized concavity and duality. In: Schaible, S., Ziemba, W.T. (eds.) Generalized Concavity in Optimization and Economics, pp. 263–279. Academic Press, New York (1981)

    Google Scholar 

  26. Mordukhovich, B.S.: Equilibrium problems with equilibrium constraints via multiobjective optimization. Optim. Methods Softw. 19, 479–492 (2004)

    Article  MathSciNet  Google Scholar 

  27. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol I: Basic Theory. Springer, Berlin (2006)

    Book  Google Scholar 

  28. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol II. Applications. Springer, Berlin (2006)

    Google Scholar 

  29. Mordukhovich, B.S.: Multiobjective optimization problems with equilibrium constraints. Math. Program. 117, 331–354 (2009)

    Article  MathSciNet  Google Scholar 

  30. Movahedian, N.: Scaled constraint qualifications for generalized equation constrained problems and application to nonsmooth mathematical programs with equilibrium constraints. Positivity 24, 253–285 (2020)

    Article  MathSciNet  Google Scholar 

  31. Osuna-Gómez, R., Hernádez-Jiménez, B., Chalco-Cano, Y., Ruiz-Gazón, G.: New efficiency conditions for multiobjective interval-valued programming problems. Inform. Sci. 420, 235–248 (2017)

    Article  Google Scholar 

  32. Pandey, Y., Mishra, S.K.: Duality for nonsmooth optimization problems with equilibrium constraints, using convexificators. J. Optim. Theory Appl. 171, 694–707 (2016)

    Article  MathSciNet  Google Scholar 

  33. Pandey, Y., Mishra, S.K.: On strong KKT type sufficient optimality conditions for nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints. Oper. Res. Lett. 44, 148–151 (2016)

    Article  MathSciNet  Google Scholar 

  34. Rapcsák, T.: Smooth Nonlinear Optimization in \(\mathbb{R} ^n\). Kluwer Academic Publishers, Dordrecht (1997)

    Book  Google Scholar 

  35. Su, T.V.: Optimality and duality for nonsmooth mathematical programming problems with equilibrium constraints. J. Glob. Optim. 85(3), 663–685 (2023)

    Article  MathSciNet  Google Scholar 

  36. Treanţă, S.: LU-optimality conditions in optimization problems with mechanical work objective functionals. IEEE Trans. Neural Netw. Learn. Syst. 33(9), 4971–4978 (2021)

    Article  MathSciNet  Google Scholar 

  37. Treanţă, S.: On a class of constrained interval-valued optimization problems governed by mechanical work cost functionals. J. Optim. Theory Appl. 188(3), 913–924 (2021)

    Article  MathSciNet  Google Scholar 

  38. Treanţă, S., Upadhyay, B.B., Ghosh, A., Nonlaopon, K.: Optimality conditions for multiobjective mathematical programming problems with equilibrium constraints on Hadamard manifolds. Mathematics 10(19), 3516 (2022)

    Article  Google Scholar 

  39. Treanţă, S., Saeed, T.: On weak variational control inequalities via interval analysis. Mathematics 11(9), 2177 (2023)

    Article  Google Scholar 

  40. Tung, L.T.: Karush–Kuhn–Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions. J. Appl. Math. Comput. 62(1–2), 67–91 (2020)

    Article  MathSciNet  Google Scholar 

  41. Tung, L.T., Tam, D.H.: Optimality conditions and duality for multiobjective semi-infinite programming on Hadamard manifolds. Bull. Iran. Math. Soc. 48, 2191–2219 (2022)

    Article  MathSciNet  Google Scholar 

  42. Tung, L.T.: Karush–Kuhn–Tucker optimality conditions and duality for multiobjective semi-infinite programming with equilibrium constraints. Yugosl. J. Oper. Res. 31(4), 429–453 (2023)

    MathSciNet  Google Scholar 

  43. Tung, L.T., Tam, D.H., Singh, V.: Characterization of solution sets of geodesic convex semi-infinite programming on Riemannian manifolds. Appl. Set-Val. Anal. Optim. 5(1), 1–18 (2023)

    Google Scholar 

  44. Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic Publishers, Dordrecht (1994)

    Book  Google Scholar 

  45. Upadhyay, B.B., Ghosh, A.: On constraint qualifications for mathematical programming problems with vanishing constraints on Hadamard manifolds. J. Optim. Theory Appl. 199, 1–35 (2023)

    Article  MathSciNet  Google Scholar 

  46. Upadhyay, B.B., Ghosh, A., Treanţă, S.: Optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems on Hadamard manifolds. Bull. Iran. Math. Soc. 49(4), 45 (2023)

    Article  MathSciNet  Google Scholar 

  47. Upadhyay, B.B., Ghosh, A., Treanţă, S.: Constraint qualifications and optimality criteria for nonsmooth multiobjective programming problems on Hadamard manifolds. J. Optim. Theory Appl. 1–26 (2023)

  48. Upadhyay, B.B., Ghosh, A., Treanţă, S.: Optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints on Hadamard manifolds. J. Math. Anal. Appl. 531(1), 127785 (2024)

    Article  MathSciNet  Google Scholar 

  49. Wolfe, P.: A duality theorem for non-linear programming. Q. Appl. Math. 19(3), 239–244 (1961)

    Article  MathSciNet  Google Scholar 

  50. Wu, H.C.: The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions. Eur. J. Oper. Res. 196, 49–60 (2009)

    Article  MathSciNet  Google Scholar 

  51. Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Editors for the help in the processing of the article. The authors are very grateful to the Anonymous Referees for the very valuable remarks, which helped to improve the paper. The first author was supported by The Ministry of Education and Training of Vietnam under Grant No. B2022-TCT-01.

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Natural Sciences, Can Tho University, Can Tho, 900000, Vietnam

    L. T. Tung

  2. Department of Mathematics, National Institute of Technology, Aizawl, Mizoram, 796012, India

    V. Singh

Authors
  1. L. T. Tung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. T. Tung.

Additional information

Publisher's Note

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

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tung, L.T., Singh, V. Optimality conditions and duality for mathematical programming with equilibrium constraints including multiple interval-valued objective functions on Hadamard manifolds. Japan J. Indust. Appl. Math. (2024). https://doi.org/10.1007/s13160-024-00646-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13160-024-00646-6

Keywords

Mathematics Subject Classification

Search

Navigation