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On the Convergence Rate of an Inexact Proximal Point Algorithm for Quasiconvex Minimization on Hadamard Manifolds

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Abstract

In this paper, we present an analysis about the rate of convergence of an inexact proximal point algorithm to solve minimization problems for quasiconvex objective functions on Hadamard manifolds. We prove that under natural assumptions the sequence generated by the algorithm converges linearly or superlinearly to a critical point of the problem.

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References

  1. Martinet, B.: Regularisation d’inéquations variationelles par approximations successives. Revue Française Autom. Inf. Rech. Opérationnelle 4, 154–159 (1970)

    MATH  Google Scholar 

  2. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29(2), 403–419 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Da Cruz Neto, J.X., Ferreira, O.P., Lucâmbio Pérez, L.R.: Contribution to the study of monotone vector fields. Acta Math. Hung. 94(4), 307–320 (2002)

  5. Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Perez, L.R., Németh, S.Z.: Convex and monotone transformable mathematical programming and a proximal-like point method. J. Glob. Optim. 35, 53–69 (2006)

  6. Ferreira, O.P.: Lucambio Prez, L.R., Németh, S.Z.: Singularities of monotone vector fields and an extragradient-type algorithm. J. Glob. Optim. 31, 133151 (2005)

    Article  Google Scholar 

  7. Rapcsák, T.: Smooth Nonlinear Optimization in \(\mathbb{R}^n\). Kluwer Academic Publishers, New York (1997)

    Book  MATH  Google Scholar 

  8. Udriste, C.: Convex Function and Optimization Methods on Riemannian Manifolds. Kluwer, New York (1994)

    Book  MATH  Google Scholar 

  9. Papa Quiroz, E.A., Oliveira, P.R.: Full convergence of the proximal point method for quasiconvex function on Hadamard manifolds. ESAIM Control Optim. Calc. Var. 18, 483–500 (2012)

  10. Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51(2), 257–270 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ahmadi, P., Khatibzadeh, H.: On the convergence of inexact proximal point algorithm on Hadamard manifolds. Taiwan. J. Math. 18(2), 419–433 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bento, G.C., Ferreira, O.P., Oliveira, P.R.: Proximal point method for a special class of nonconvex functions on Hadamard manifolds. Optimization 64(2), 289–319 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Tang, G.J., Zhou, L.W., Huang, N.J.: The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds. Optim. Lett. 7(4), 779–790 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tang, G.J., Huang, N.J.: Rate of convergence for proximal point algorithms on Hadamard manifolds. Oper. Res. Lett. 42, 383–387 (2014)

    Article  MathSciNet  Google Scholar 

  15. Luque, F.J.: Asymptotic convegence analysis of the proximal point algorithm. SIAM J. Control Optim. 22, 277–293 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Baygorrea, N., Papa Quiroz, E.A., Maculan, N.: Inexact proximal point methods for quasiconvex minimization on Hadamard manifolds. J. Oper. Res. Soc. China. (2016). doi:10.1007/s40305-016-0133-3

  17. Papa Quiroz, E.A., Oliveira, P.R.: Proximal point method for minimizing quasiconvex locally lipschitz functions on Hadamard manifolds. Nonlinear Anal. 75, 5924–5932 (2012)

  18. Papa Quiroz, E.A., Oliveira, P.R.: Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16(1), 49–69 (2009)

  19. Papa Quiroz, E.A., Mallma Ramirez L. and Oliveira, P.R.: An inexact proximal method for quasiconvex minimizations. http://www.optimization-online.org/DB_FILE/2013/08/3982.pdf, accepted for publication in EJOR (2015)

  20. Do Carmo, M.P.: Riemannian Geometry. Bikhauser, Boston (1992)

  21. Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1996)

  22. Aussel, D., Corvellec, J.N., Lassonde, M.: Mean-value property and subdifferential criteria for lower semicontinuous functions. Trans. Am. Math. Soc. 347, 4147–4161 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Federal University of Rio de Janeiro, PESC-COPPE-UFRJ, PO Box 68511, Rio de Janeiro, CEP 21941-972, Brazil

    Nancy Baygorrea, Erik Alex Papa Quiroz & Nelson Maculan

Authors
  1. Nancy Baygorrea
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  2. Erik Alex Papa Quiroz
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  3. Nelson Maculan
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Corresponding author

Correspondence to Nelson Maculan.

Additional information

This work was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior of the Federal University of Rio de Janeiro (UFRJ), Brazil.

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Baygorrea, N., Papa Quiroz, E.A. & Maculan, N. On the Convergence Rate of an Inexact Proximal Point Algorithm for Quasiconvex Minimization on Hadamard Manifolds. J. Oper. Res. Soc. China 5, 457–467 (2017). https://doi.org/10.1007/s40305-016-0129-z

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  • DOI: https://doi.org/10.1007/s40305-016-0129-z

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