Skip to main content
SpringerLink
Log in

On Minty Variational Principle for Nonsmooth Interval-Valued Multiobjective Programming Problems

  • Conference paper
  • First Online:
Optimization, Variational Analysis and Applications (IFSOVAA 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 355))

Abstract

In this chapter, we consider a class of nonsmooth interval-valued multiobjective programming problems and a class of approximate Minty and Stampacchia vector variational inequalities. Under generalized approximate LU-convexity hypotheses, we establish the relations between the solutions of approximate Minty and Stampacchia vector variational inequalities and the approximate LU-efficient solutions of the nonsmooth interval-valued multiobjective programming problem. The results of this chapter extend and unify the corresponding results of [14, 22, 23, 30, 33] for nonsmooth interval-valued multiobjective programming problems.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Article  MathSciNet  Google Scholar 

  2. Antczak, T.: Exactness property of the exact absolute value penalty function method for solving convex nondifferentiable interval-valued optimization problems. J. Optim. Theory Appl. 176, 205–224 (2018)

    Article  MathSciNet  Google Scholar 

  3. Bhatia, D., Gupta, A., Arora, P.: Optimality via generalized approximate convexity and quasiefficiency. Optim. Lett. 7, 127–135 (2013)

    Article  MathSciNet  Google Scholar 

  4. Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)

    Article  MathSciNet  Google Scholar 

  5. Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, A.: Pareto Optimality. Game Theory and Equilibria. Springer, Berlin (2008)

    Book  Google Scholar 

  6. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  7. Dafermos, S.: Exchange price equilibria and variational inequalities. Math. Program. 46, 391–402 (1990)

    Article  MathSciNet  Google Scholar 

  8. Ghosh, D.: Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions. J. Appl. Math. Comput. 53, 709–731 (2017)

    Article  MathSciNet  Google Scholar 

  9. Ghosh, D., Chauhan, R.S., Mesiar, R., Debnath, A.K.: Generalized Hukuhara Gâteaux and Fréchet derivatives of interval-valued functions and their application in optimization with interval-valued functions. Inform. Sci. 510, 317–340 (2020)

    Article  MathSciNet  Google Scholar 

  10. Giannessi, F.: Theorems of the alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, Chichester (1980)

    Google Scholar 

  11. Giannessi, F.: On Minty variational principle. In: Giannessi, F., Komlósi, S., Rapcśak, T. (eds.) New Trends in Mathematical Programming, pp. 93–99. Kluwer Academic Publishers, Dordrecht (1997)

    Google Scholar 

  12. Gupta, D., Mehra, A.: Two types of approximate saddle points. Numer. Funct. Anal. Optim. 29, 532–550 (2008)

    Article  MathSciNet  Google Scholar 

  13. Gupta, A., Mehra, A., Bhatia, D.: Approximate convexity in vector optimisation. Bull. Aust. Math. Soc. 74, 207–218 (2006)

    Article  MathSciNet  Google Scholar 

  14. Gupta, P., Mishra, S.K.: On Minty variational principle for nonsmooth vector optimization problems with generalized approximate convexity. Optimization 67, 1157–1167 (2018)

    Article  MathSciNet  Google Scholar 

  15. Hanson, M.A., Mond, B.: Necessary and sufficient conditions in constrained optimization. Math. Program. 37, 51–58 (1987)

    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, 1391–1411 (2016)

    Article  MathSciNet  Google Scholar 

  17. Jayswal, A., Stancu-Minasian, I., Banerjee, J.: Optimality conditions and duality for interval-valued optimization problems using convexifactors. Rend. Circ. Mat. Palermo 65, 17–32 (2016)

    Article  MathSciNet  Google Scholar 

  18. Jeyakumar, V., Mond, B.: On generalised convex mathematical programming. J. Austral. Math. Soc. Ser. B. 34, 43–53 (1992)

    Article  MathSciNet  Google Scholar 

  19. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, London (1980)

    MATH  Google Scholar 

  20. Lee, G.M., Lee, K.B.: Vector variational inequalities for nondifferential convex vector optimization problems. J. Glob. Optim. 32, 597–612 (2005)

    Article  Google Scholar 

  21. Loridan, P.: \(\epsilon \)-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984)

    Google Scholar 

  22. Mishra, S.K., Laha, V.: On Minty variational principle for nonsmooth vector optimization problems with approximate convexity. Optim. Lett. 10, 577–589 (2016)

    Article  MathSciNet  Google Scholar 

  23. Mishra, S.K., Upadhyay, B.B.: Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity 17, 1071–1083 (2013)

    Article  MathSciNet  Google Scholar 

  24. Mishra, S.K., Upadhyay, B.B.: Pseudolinear Functions and Optimization. Chapman and Hall, CRC Press (2014)

    Google Scholar 

  25. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs, New Jersey (1966)

    MATH  Google Scholar 

  26. Moore, R.E.: Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics, Philadelphia (1979)

    Book  Google Scholar 

  27. Upadhyay, B.B., Mishra, P.: On generalized Minty and Stampacchia vector variational-like inequalities and nonsmooth vector optimization problem involving higher order strong invexity. J. Sci. Res. 64, 182–191 (2020)

    Google Scholar 

  28. Upadhyay, B.B., Mishra, P.: On vector variational inequalities and vector optimization problems. In: Soft Computing: Theories and Applications, pp. 257–267. Springer, Singapore (2020)

    Google Scholar 

  29. Upadhyay, B.B., Mishra, P., Mohapatra, R.N., Mishra, S.K.: On the applications of nonsmooth vector optimization problems to solve generalized vector variational inequalities using convexificators. Adv. Intell. Sys. Comput. https://doi.org/10.1007/978-3-030-21803-4_66

  30. Upadhyay, B.B., Mohapatra, R.N., Mishra, S.K.: On relationships between vector variational inequality and nonsmooth vector optimization problems via strict minimizers. Adv. Nonlinear Var. Inequal. 20, 1–12 (2017)

    MathSciNet  Google Scholar 

  31. Wu, H.-C.: The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur. J. Oper. Res. 176, 46–59 (2007)

    Google Scholar 

  32. Yang, X.Q.: Vector variational inequality and vector pseudolinear optimization. J. Optim. Theory Appl. 95, 729–734 (1997)

    Article  MathSciNet  Google Scholar 

  33. Zhang, J., Zheng, Q., Ma, X., Li, L.: Relationships between interval-valued vector optimization problems and variational inequalities. Fuzzy Optim. Decis. Mak. 15, 33–55 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author is supported by the Science and Engineering Research Board, Department of Science and Technology, Government of India, through grant number “ECR/2016/001961. The authors would like to express their thanks to the anonymous reviewers for their very valuable comments and suggestions to improve the quality of the chapter.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Patna, Patna, 801103, India

    Balendu Bhooshan Upadhyay & Priyanka Mishra

Authors
  1. Balendu Bhooshan Upadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Priyanka Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Balendu Bhooshan Upadhyay .

Editor information

Editors and Affiliations

  1. Institute of Science, Banaras Hindu University, Varanasi, Uttar Pradesh, India

    Vivek Laha

  2. Institut de Mathématiques de Toulouse, Paul Sabatier University, Toulouse, France

    Pierre Maréchal

  3. Institute of Science, Banaras Hindu University, Varanasi, Uttar Pradesh, India

    S. K. Mishra

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Upadhyay, B.B., Mishra, P. (2021). On Minty Variational Principle for Nonsmooth Interval-Valued Multiobjective Programming Problems. In: Laha, V., Maréchal, P., Mishra, S.K. (eds) Optimization, Variational Analysis and Applications. IFSOVAA 2020. Springer Proceedings in Mathematics & Statistics, vol 355. Springer, Singapore. https://doi.org/10.1007/978-981-16-1819-2_12

Download citation

Publish with us

Policies and ethics

Search

Navigation