A new reduced gradient method for solving linearly constrained multiobjective optimization problems | Computational Optimization and Applications Skip to main content
Springer Nature Link
Log in

A new reduced gradient method for solving linearly constrained multiobjective optimization problems

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

In this paper, we consider the linearly constrained multiobjective minimization, and we propose a new reduced gradient method for solving this problem. Our approach solves iteratively a convex quadratic optimization subproblem to calculate a suitable descent direction for all the objective functions, and then use a bisection algorithm to find an optimal stepsize along this direction. We prove, under natural assumptions, that the proposed algorithm is well-defined and converges globally to Pareto critical points of the problem. Finally, this algorithm is implemented in the MATLAB environment and comparative results of numerical experiments are reported.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Luc, D.T.: Theory of Vector Optimization. Lectures Notes in Economics and Mathematical Systems, vol. 319. Springer-Verlag, Berlin (1989). https://doi.org/10.1007/978-3-642-50280-4

    Book  Google Scholar 

  2. Miettinen, K.: Nonlinear Multiobjective Optimization, vol. 12. Springer, Berlin (2012)

    MATH  Google Scholar 

  3. Ehrgott, M.: Multicriteria Optimization, vol. 491. Springer, Berlin (2013)

    MATH  Google Scholar 

  4. Fliege, J., Drummond, L.M.G., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20(2), 602 (2009)

    Article  MathSciNet  Google Scholar 

  5. Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479 (2000)

    Article  MathSciNet  Google Scholar 

  6. Drummond, L.G., Iusem, A.N.: A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28(1), 5 (2004)

    Article  MathSciNet  Google Scholar 

  7. El Moudden, M., El Ghali, A.: Multiple reduced gradient method for multiobjective optimization problems. Numer. Algorithms (2018). https://doi.org/10.1007/s11075-018-0483-5

    Article  Google Scholar 

  8. Wolfe, P.: The reduced gradient method, Rand Document. Santa Monica, CA (1962)

  9. Huard, P.: Un algorithme général de gradient réduit. Bulletin de la Direction des Etudes et Recherches, EDF, Série C 2, 91 (1982)

  10. Luenberger, D., Ye, Y.: Linear and Nonlinear Programming, vol. 2. Springer, Berlin (1984)

    MATH  Google Scholar 

  11. Baushev, A., Morozova, E.: A Multidimensional Bisection Method for Minimizing Function Over Simplex. Lectures Notes in Engineering and Computer Science, vol. 2, pp. 801–803 (2007)

  12. García-Palomares, U.M., Burguillo-Rial, J.C., González-Castano, F.J.: Explicit gradient information in multiobjective optimization. Oper. Res. Lett. 36(6), 722 (2008)

    Article  MathSciNet  Google Scholar 

  13. Miglierina, E., Molho, E., Recchioni, M.: An interior point method for linearly constrained multiobjective optimization based on suitable descent directions. In Recent Developments on Mathematical Programming and Applications, (Pisa, 05-05 June 2009), Aracne editrice, Roma, pp. 89–102 (2009)

  14. Benson, H., Boger, G.: Outcome-space cutting-plane algorithm for linear multiplicative programming. J. Optim. Theory Appl. 104(2), 301 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Sciences, Moulay Ismail University, Meknes, Morocco

    Mustapha El Moudden & Ahmed El Ghali

Authors
  1. Mustapha El Moudden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmed El Ghali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mustapha El Moudden.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El Moudden, M., El Ghali, A. A new reduced gradient method for solving linearly constrained multiobjective optimization problems. Comput Optim Appl 71, 719–741 (2018). https://doi.org/10.1007/s10589-018-0023-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-018-0023-1

Keywords

Search

Navigation