Reduced Jacobian Method

  • Mounir El Maghri
  • Youssef Elboulqe


In this paper, we present the Wolfe’s reduced gradient method for multiobjective (multicriteria) optimization. We precisely deal with the problem of minimizing nonlinear objectives under linear constraints and propose a reduced Jacobian method, namely a reduced gradient-like method that does not scalarize those programs. As long as there are nondominated solutions, the principle is to determine a direction that decreases all goals at the same time to achieve one of them. Following the reduction strategy, only a reduced search direction is to be found. We show that this latter can be obtained by solving a simple differentiable and convex program at each iteration. Moreover, this method is conceived to recover both the discontinuous and continuous schemes of Wolfe for the single-objective programs. The resulting algorithm is proved to be (globally) convergent to a Pareto KKT-stationary (Pareto critical) point under classical hypotheses and a multiobjective Armijo line search condition. Finally, experiment results over test problems show a net performance of the proposed algorithm and its superiority against a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort.


Multiobjective optimization Nonlinear programming Pareto optima KKT-stationarity Descent direction Reduced gradient method 

Mathematics Subject Classification

90C29 90C30 90C52 



The authors are thankful for the valuable suggestions of the editor Dr. Franco Giannessi and the reviewers that helped improve the quality of the paper.


  1. 1.
    Collette, Y., Siarry, P.: Multiobjective Optimization. Principles and Case Studies. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Figueira, J., Greco, S., Ehrgott, M.: Multiple criteria decision analysis. In: State of the Art Surveys. Springer, New York (2005)Google Scholar
  3. 3.
    Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  4. 4.
    Deb, K.: Multi-objective Optimization Using Evolutionary Algorithm. Wiley, Chichester (2001)zbMATHGoogle Scholar
  5. 5.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)zbMATHGoogle Scholar
  6. 6.
    Fliege, J., Graña Drummond, L.M., Svaiter, B.F.: Newton’s method for multiobjective optimisation. SIAM J. Optim. 20, 602–626 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fliege, J.: An efficient interior-point method for convex multicriteria optimization problems. Math. Oper. Res. 31, 825–845 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eichfelder, G.: An adaptive scalarization method in multiobjective optimization. SIAM J. Optim. 19, 1694–1718 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Meth. Oper. Res. 51, 479–494 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Graña Drummond, L.M., Iusem, A.N.: A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28, 5–29 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    García-Palomares, U.M., Burguillo-Rial, J.C., González-Castaño, F.J.: Explicit gradient information in multiobjective optimization. Oper. Res. Lett. 36, 722–725 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15, 953–970 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wolfe, P.: Methods of nonlinear programming. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming, pp. 67–86. McGraw–Hill, New York (1963)Google Scholar
  14. 14.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New York (2006)CrossRefzbMATHGoogle Scholar
  15. 15.
    Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming, 3rd edn. Springer, New York (2008)zbMATHGoogle Scholar
  16. 16.
    El Maghri, M.: A free reduced gradient scheme. Asian J. Math. Comput. Res. 9, 228–239 (2016)Google Scholar
  17. 17.
    Goh, C.J., Yang, X.Q.: Duality in Optimization and Variational Inequalities. Taylor and Francis, London (2002)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ponstein, J.: Seven types of convexity. SIAM Rev. 9, 115–119 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bento, G.C., Cruz Neto, J.X., Oliveira, P.R., Soubeyran, A.: The self regulation problem as an inexact steepest descent method for multicriteria optimization. Eur. J. Oper. Res. 235, 494–502 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mangasarian, O.L.: Nonlinear Programming. SIAM, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  21. 21.
    Bento, G.C., Cruz Neto, J.X., Santos, P.S.M.: An inexact steepest descent method for multicriteria optimization on riemannian manifolds. J. Optim. Theory Appl. 159, 108–124 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Durea, M., Strugariu, R., Tammer, C.: Scalarization in geometric and functional vector optimization revisited. J. Optim. Theory Appl. 159, 635–655 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Huard, P.: Convergence of the reduced gradient method. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds.) Nonlinear Programming, pp. 29–59. Academic Press, New York (1975)Google Scholar
  24. 24.
    Armijo, L.: Minimization of functions having Lipschitz continuous first partial derivatives. Pac. J. Math. 16, 1–3 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput. 8, 173–195 (2000)CrossRefGoogle Scholar
  26. 26.
    Bolintinéanu, S., (new name: Bonnel, H.), El Maghri, M.: Pénalisation dans l’optimisation sur l’ensemble faiblement efficient. RAIRO Oper. Res. 31, 295–310 (1997)Google Scholar
  27. 27.
    Bolintinéanu, S., El Maghri, M.: Second-order efficiency conditions and sensitivity of efficient points. J. Optim. Theory Appl. 98, 569–592 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    El Maghri, M., Bernoussi, B.: Pareto optimizing and Kuhn–Tucker stationary sequences. Numer. Funct. Anal. Optim. 28, 287–305 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer, Faculty of Sciences Aïn ChockHassan II UniversityCasablancaMorocco

Personalised recommendations