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The proximal point algorithm in metric spaces

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Abstract

The proximal point algorithm, which is a well-known tool for finding minima of convex functions, is generalized from the classical Hilbert space framework into a nonlinear setting, namely, geodesic metric spaces of non-positive curvature. We prove that the sequence generated by the proximal point algorithm weakly converges to a minimizer, and also discuss a related question: convergence of the gradient flow.

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References

  1. M. Bačák, I. Searston and B. Sims, Alternating projections in CAT(0) spaces, Journal of Mathematical Analysis and Applications 385 (2012), 599–607.

    Article  MathSciNet  MATH  Google Scholar 

  2. J.-B. Baillon, Un exemple concernant le comportement asymptotique de la solution du problème du/dt + ∂ϖ(u) ∈ 0, Journal of Functional Analysis 28 (1978), 369–376.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. H. Bauschke, J. V. Burke, F. R. Deutsch, H. S. Hundal and J. D. Vanderwerff, A new proximal point iteration that converges weakly but not in norm, Proceedings of the American Mathematical Society 133 (2005), 1829–1835.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York, 2011.

    Book  MATH  Google Scholar 

  5. H. H. Bauschke, E. Matoušková and S. Reich, Projection and proximal point methods: convergence results and counterexamples, Nonlinear Analysis 56 (2004), 715–738.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. C. Bento, O. P. Ferreira and P.R. Oliveira, Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds, Nonlinear Analysis 73 (2010), 564–572.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Brézis and P.L. Lions, Produits infinis de résolvantes, Israel Journal of Mathematics 29 (1978), 329–345.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Springer-Verlag, Berlin, 1999.

    Book  MATH  Google Scholar 

  9. P. L. Combettes, Fejér monotonicity in convex optimization, in Encyclopedia of Optimization (C. A. Floudas and P. M. Pardalos, eds.), Kluwer, Boston, MA, 2001.

    Google Scholar 

  10. S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Analysis 65 (2006), 762–772.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Espínola and A. Fernández-León, CAT(k)-spaces, weak convergence and fixed points, Journal of Mathematical Analysis and Applications 353 (2009), 410–427.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization 29 (1991), 403–419.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Jost, Equilibrium maps between metric spaces, Calculus of Variations and Partial Differential 2 (1994), 173–204.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Commentarii Mathematici Helvetici 70 (1995), 659–673.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1997.

    Book  MATH  Google Scholar 

  17. J. Jost, Nonlinear Dirichlet forms, in New Directions in Dirichlet Forms, AMS/IP Studies in Advanced Mathematics, Vol. 8, American Mathematical Society, Providence, RI, 1998, pp. 1–47.

    Google Scholar 

  18. J. Jost, Riemannian Geometry and Geometric Analysis, Fifth ed., Universitext, Springer-Verlag, Berlin, 2008.

    MATH  Google Scholar 

  19. W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Analysis 68 (2008), 3689–3696.

    Article  MathSciNet  MATH  Google Scholar 

  20. C. Li, G. López and V. Martín-Márquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, Journal of the London Mathematical Society 79 (2009), 663–683.

    Article  MATH  Google Scholar 

  21. T. C. Lim, Remarks on some fixed point theorems, Proceedings of the American Mathematical Society 60 (1976), 179–182.

    Article  MathSciNet  Google Scholar 

  22. B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle 4 (1970), 154–158.

    MathSciNet  MATH  Google Scholar 

  23. U. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Communications in Analysis and Geometry 6 (1998), 199–253.

    MathSciNet  MATH  Google Scholar 

  24. E. A. Papa Quiroz and P. R. Oliveira, Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds, Journal of Convex Analysis 16 (2009), 49–69.

    MathSciNet  MATH  Google Scholar 

  25. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization 14 (1976), 877–898.

    Article  MathSciNet  MATH  Google Scholar 

  26. E. N. Sosov, On analogues of weak convergence in a special metric space, Russian Mathematics (Iz. VUZ) 48 (2004), 79–83.

    MathSciNet  MATH  Google Scholar 

  27. A. J. Zaslavski, Inexact proximal point methods in metric spaces, Set-Valued Analysis 19 (2011), 589–608.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Max Planck Institute, Inselstrasse 22, 04 103, Leipzig, Germany

    Miroslav Bačák

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  1. Miroslav Bačák
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Correspondence to Miroslav Bačák.

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Bačák, M. The proximal point algorithm in metric spaces. Isr. J. Math. 194, 689–701 (2013). https://doi.org/10.1007/s11856-012-0091-3

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  • DOI: https://doi.org/10.1007/s11856-012-0091-3

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