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Journal of Optimization Theory and Applications

, Volume 159, Issue 1, pp 108–124 | Cite as

An Inexact Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds

  • G. C. Bento
  • J. X. da Cruz Neto
  • P. S. M. Santos
Article

Abstract

In this paper, we present an inexact version of the steepest descent method with Armijo’s rule for multicriteria optimization in the Riemannian context given in Bento et al. (J. Optim. Theory Appl., 154: 88–107, 2012). 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 that the multicriteria function is quasi-convex and the Riemannian manifold has nonnegative curvature, we show full convergence of any sequence generated by the method to a Pareto critical point.

Keywords

Steepest descent Pareto optimality Multicriteria optimization Quasi-Fejér convergence Quasi-convexity Riemannian manifolds 

Notes

Acknowledgements

The authors would like to extend their gratitude toward anonymous referees whose suggestions helped us to improve the presentation of this paper. The first author was partially supported by CNPq Grant 471815/2012-8, Project CAPES-MES-CUBA 226/2012, PROCAD-nf-UFG/UnB/IMPA, and FAPEG/CNPq. The second author was partially supported by CNPq GRANT 301625-2008 and PRONEX-Optimization (FAPERJ/CNPq).

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • G. C. Bento
    • 1
  • J. X. da Cruz Neto
    • 2
  • P. S. M. Santos
    • 2
  1. 1.Universidade Federal de GoiásGoianiaBrazil
  2. 2.Universidade Federal PiauíTeresinaBrazil

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