Unconstrained Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds

  • G. C. Bento
  • O. P. Ferreira
  • P. R. Oliveira


In this paper, we present a steepest descent method with Armijo’s rule for multicriteria optimization in the Riemannian context. The sequence generated by the method is guaranteed to be well defined. Under mild assumptions on the multicriteria function, we prove that each accumulation point (if any) satisfies first-order necessary conditions for Pareto optimality. Moreover, assuming quasiconvexity of the multicriteria function and nonnegative curvature of the Riemannian manifold, we prove full convergence of the sequence to a critical Pareto point.


Steepest descent Pareto optimality Vector optimization Quasi-Fejér convergence Quasiconvexity Riemannian manifolds 



G.C. Bento was supported in part by CNPq Grant 473756/2009-9 and PROCAD/NF. O.P. Ferreira was supported in part by CNPq Grant 302618/2005-8, PRONEX–Optimization (FAPERJ/CNPq) and FUNAPE/UFG. P.R. Oliveira was supported in part by CNPq.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • G. C. Bento
    • 1
  • O. P. Ferreira
    • 1
  • P. R. Oliveira
    • 2
  1. 1.IME-Universidade Federal de GoiásGoiâniaBrazil
  2. 2.COPPE/Sistemas-Universidade Federal do Rio de JaneiroRio de JaneiroBrazil

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