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Mathematical Programming

, Volume 159, Issue 1–2, pp 339–369 | Cite as

Trust region globalization strategy for the nonconvex unconstrained multiobjective optimization problem

  • Gabriel A. Carrizo
  • Pablo A. Lotito
  • María C. Maciel
Full Length Paper Series A

Abstract

A trust-region-based algorithm for the nonconvex unconstrained multiobjective optimization problem is considered. It is a generalization of the algorithm proposed by Fliege et al. (SIAM J Optim 20:602–626, 2009), for the convex problem. Similarly to the scalar case, at each iteration a subproblem is solved and the step needs to be evaluated. Therefore, the notions of decrease condition and of predicted reduction are adapted to the vectorial case. A rule to update the trust region radius is introduced. Under differentiability assumptions, the algorithm converges to points satisfying a necessary condition for Pareto points and, in the convex case, to a Pareto points satisfying necessary and sufficient conditions. Furthermore, it is proved that the algorithm displays a q-quadratic rate of convergence. The global behavior of the algorithm is shown in the numerical experience reported.

Keywords

Multiobjective optimization Trust region Newton method Convergence 

Mathematics Subject Classification

90C29 65K05 49M37 

Notes

Acknowledgments

The authors are grateful to the anonymous reviewers, whose comments improved this work.

References

  1. 1.
    Andreani, R., Birgin, E., Martínez, J., Schuverdt, M.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18(4), 1286–1309 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ashry, G.A.: On globally convergent multi-objective optimization. Appl. Math. Comput. 183, 209–216 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athenas Scientific, Belmont, MA (1999)zbMATHGoogle Scholar
  4. 4.
    Conn, A., Gould, N.I.M., Toint, P.: Trust-Region Methods. SIAM-MPS, Philadelphia, Pennsylvania (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Carrizo, G.A.: Estrategia de región de confianza para problemas multiobjetivo no convexos. Universidad Nacional del Sur, Argentina (2013)Google Scholar
  6. 6.
    Custodio, A.L., Madeira, J.F.A., Vaz, A.I.F., Vicente, L.N.: Direct multisearch for multiobjective optimization. SIAM J. Optim. 21, 1109–1140 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)CrossRefGoogle Scholar
  8. 8.
    Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Systems. Prentice-Hall, Englewood Cliffs, New Jersey (1983)zbMATHGoogle Scholar
  9. 9.
    Fliege, J., Drummond, L.M.Graña, Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20, 602–626 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fischer, A., Shukia, P.K.: A Levenberg–Marquardt aglorithm for uncostrained multicriteria optimization. Oper. Res. Lett. 175, 395–414 (2005)Google Scholar
  12. 12.
    Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Optim. Theory Appl. 22, 618–630 (1968)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Laumanns, M., Thiele, L., Deb, K., Zitzler, E.: Combining convergence and diversity in evolutionary multiobjective optimization. Evol. Comput. 10, 263–282 (2002)CrossRefGoogle Scholar
  14. 14.
    Lobato, F.S., Steffen, J.V.: Multi-objective optimization firefly algorithm applied to (bio)chemical engineering system design. Am. J. Appl. Math. Stat. 1, 110–116 (2013)CrossRefGoogle Scholar
  15. 15.
    Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)CrossRefGoogle Scholar
  16. 16.
    Qu, S., Goh, M., Lian, B.: Trust region methods for solving multiobjective optimisation. Optim. Methods Softw. 28(4), 796–811 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ravanbakhsh, A., Franchini, S.: Multiobjective optimization applied to structural sizing of low cost university-class microsatellite projects. Acta Astronaut. 79, 212–220 (2012)CrossRefGoogle Scholar
  18. 18.
    Steuer, R.E., Na, P.: Multiple criteria decision making combined with finance: a categorized bibliographic study. Eur. J. Oper. Res. 150(3), 496–515 (2003)CrossRefzbMATHGoogle Scholar
  19. 19.
    Villacorta, K.D., Oliveira, P.R., Souberyran, A.: A trust region method for unconstrained multipbjective problems with applications in satisficing processes. J. Optim. Theory Appl. 160, 865–889 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput. 8, 173–195 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  • Gabriel A. Carrizo
    • 1
  • Pablo A. Lotito
    • 2
  • María C. Maciel
    • 1
  1. 1.Departamento de MatemáticaUniversidad Nacional del SurBahía BlancaArgentina
  2. 2.Universidad Nacional del CentroTandilArgentina

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