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An Inexact Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds

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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.

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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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Authors and Affiliations

  1. Universidade Federal de Goiás, Goiania, Brazil

    G. C. Bento

  2. Universidade Federal Piauí, Teresina, Brazil

    J. X. da Cruz Neto & P. S. M. Santos

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  1. G. C. Bento
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  2. J. X. da Cruz Neto
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  3. P. S. M. Santos
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Correspondence to P. S. M. Santos.

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Communicated by Alfredo Iusem.

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Bento, G.C., da 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). https://doi.org/10.1007/s10957-013-0305-9

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  • DOI: https://doi.org/10.1007/s10957-013-0305-9

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