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Unconstrained Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. IME-Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    G. C. Bento & O. P. Ferreira

  2. COPPE/Sistemas-Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21945-970, Brazil

    P. R. Oliveira

Authors
  1. G. C. Bento
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  2. O. P. Ferreira
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  3. P. R. Oliveira
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Correspondence to G. C. Bento.

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Communicated by Dinh The Luc.

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Bento, G.C., Ferreira, O.P. & Oliveira, P.R. Unconstrained Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds. J Optim Theory Appl 154, 88–107 (2012). https://doi.org/10.1007/s10957-011-9984-2

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