Skip to main content
SpringerLink
Log in

On the Relationship Between the Kurdyka–Łojasiewicz Property and Error Bounds on Hadamard Manifolds

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper studies the interplay between the concepts of error bounds and the Kurdyka–Łojasiewicz (KL) inequality on Hadamard manifolds. To this end, we extend some properties and existence results of a solution for differential inclusions on Hadamard manifolds. As a second contribution, we show how the KL inequality can be used to obtain the convergence of the gradient method for solving convex feasibility problems on Hadamard manifolds. The convergence results of the alternating projection method are also established for cyclic and random projections on Hadamard manifolds and, more generally, CAT(0) spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Amemiya, I., Andô, T.: Convergence of random products of contractions in Hilbert space. Acta Sci. Math. 26, 239–244 (1965)

    MathSciNet  Google Scholar 

  2. Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting and regularized Gauss-Seidel methods. Math. Program. 137, 91–129 (2013)

    Article  MathSciNet  Google Scholar 

  3. Bačák, M.: Convex Analysis and Optimization in Hadamard Spaces. De Gruyter Series in Nonlinear Analysis and Applications, vol. 22. Walter de Gruyter, Berlin (2014)

    Book  Google Scholar 

  4. Bačák, M.: Computing medians and means in Hadamard spaces. SIAM J. Optim. 24, 1542–1566 (2014)

    Article  MathSciNet  Google Scholar 

  5. Barani, A., Hosseini, S.: Characterization of solution sets of convex optimization problems in Riemannian manifolds. Arch. Math. 114, 215–225 (2020)

    Article  MathSciNet  Google Scholar 

  6. Bačák, M., Searston, I., Sims, B.: Alternating projections in CAT(0) spaces. J. Math. Anal. Appl. 385, 599–607 (2012)

    Article  MathSciNet  Google Scholar 

  7. Bauschke, H.H.: A norm convergence result on random products of relaxed projections in Hilbert space. Trans. Am. Math. Soc. 347, 1365–1373 (1995)

    Article  MathSciNet  Google Scholar 

  8. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    Article  MathSciNet  Google Scholar 

  9. Bauschke, H.H., Borwein, J.M.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1, 185–212 (1993)

    Article  MathSciNet  Google Scholar 

  10. Beck, A., Teboulle, M.: Convergence rate analysis and error bounds for projection algorithms in convex feasibility problem. Optim. Methods Softw. 18, 377–394 (2003)

    Article  MathSciNet  Google Scholar 

  11. Bento, G.C., Bitar, S.D.B., Cruz Neto, J.X., Oliveira, P.R., Souza, J.C.O.: Computing Riemannian center of mass on Hadamard manifolds. J. Optim. Theory Appl. 183, 977–992 (2019)

    Article  MathSciNet  Google Scholar 

  12. Bento, G.C., Cruz Neto, J.X., Oliveira, P.R.: A New approach to the proximal point method: convergence on general Riemannian manifolds. J. Optim. Theory Appl. 168, 743–755 (2016)

    Article  MathSciNet  Google Scholar 

  13. Bento, G.C., Melo, J.G.: Subgradient method for convex feasibility on Riemannian manifolds. J. Optim. Theory Appl. 152, 773–785 (2012)

    Article  MathSciNet  Google Scholar 

  14. Bergmann, R., Persch Steidl, G.: A parallel Douglas–Rachford algorithm for minimizing ROF-like functionals on images with values in symmetric Hadamard manifolds. SIAM J. Imaging Sci. 9(3), 901–937 (2016)

    Article  MathSciNet  Google Scholar 

  15. Bolte, J., Daniilidis, A., Lewis, A.: The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205–1223 (2007)

    Article  Google Scholar 

  16. Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2), 556–572 (2007)

    Article  MathSciNet  Google Scholar 

  17. Bolte, J., Daniilidis, A., Ley, O., Mazet, L.: Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Am. Math. Soc. 362, 3319–3363 (2010)

    Article  Google Scholar 

  18. Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.W.: From error bounds to the complexity of first-order descent methods for convex functions. Math. Program. 165, 471–507 (2017)

    Article  MathSciNet  Google Scholar 

  19. Brézis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espace Hilbert. North-Holland Mathematics Studies, vol. 5. North-Holland, Amsterdam (1973)

    Google Scholar 

  20. Bridson, M., Haefliger, A.: Metric Spaces of Nonpositive Curvature. Springer, Berlin (1999)

    Book  Google Scholar 

  21. Bruck, R.E.: Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal. 18, 15–26 (1975)

    Article  MathSciNet  Google Scholar 

  22. Bruck, R.E.: Random products of contractions in metric and Banach Spaces. J. Math. Anal. Appl. 88, 319–322 (1982)

    Article  MathSciNet  Google Scholar 

  23. Burtscher, A.Y.: Length structures on manifolds with continuous Riemannian metrics. New York J. Math. 21, 273–296 (2015)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Cruz Neto, J.X., Ferreira, O.P., Lucâmbio Pérez, L.R., Németh, S.Z.: Convex- and monotone-transformable mathematical programming problems and a proximal-like point method. J. Global Optim. 35(1), 53–69 (2006)

    Article  MathSciNet  Google Scholar 

  26. Cruz Neto, J.X., Oliveira, P.R., Soares, P.A., Jr., Soubeyran, A.: A Learning how to play Nash, potential games and alternating minimization method for structured non-convex problems on Riemannian manifolds. J. Convex Anal. 20, 395–438 (2013)

    MathSciNet  Google Scholar 

  27. Cruz Neto, J.X., Lima, L.L., Oliveira, P.R.: Geodesic algorithms in Riemannian geometry. Balkan J. Geom. Appl. 3, 89–100 (1998)

    MathSciNet  Google Scholar 

  28. Cruz Neto, J.X., Melo, I.D., Sousa, P.A., Silva, J.P.: A note on the paper “Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds’’. J. Convex Anal. 24, 679–684 (2017)

    MathSciNet  Google Scholar 

  29. Ferreira, O.P., Louzeiro, M.S., Prudente, L.F.: Gradient method for optimization on Riemannian manifolds with lower bounded curvature. SIAM J. Optim. 29(4), 2517–2541 (2019)

    Article  MathSciNet  Google Scholar 

  30. Gubin, L.C., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. USSR Comput. Math. Math. Phys. 7, 1–24 (1967)

    Article  Google Scholar 

  31. Halperin, I.: The product of projection operators. Acta Sci. Math. 23, 96–99 (1962)

    MathSciNet  Google Scholar 

  32. Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Natl. Bur. Stand. 49(4), 263–265 (1952)

    Article  MathSciNet  Google Scholar 

  33. Hosseini, S., Uschmajew, A.: A Riemannian gradient sampling algorithm for nonsmooth optimization on manifolds. SIAM J. Optim. 27(1), 173–189 (2017)

    Article  MathSciNet  Google Scholar 

  34. Iusem, A.N., Svaiter, B.F., Cruz Neto, J.X.: Central paths, generalized proximal point methods and Cauchy trajectories in Riemannian manifolds. SIAM J. Control. Optim. 37, 566–588 (1999)

    Article  MathSciNet  Google Scholar 

  35. Jost, J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2(2), 173–204 (1994)

    Article  MathSciNet  Google Scholar 

  36. Kristály, A., Li, C., López-Acedo, G., Nicolae, A.: What do “convexity’’ imply on Hadamard manifolds? J. Optim. Theory Appl. 170, 1068–1074 (2016)

    Article  MathSciNet  Google Scholar 

  37. Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48, 769–783 (1998)

    Article  MathSciNet  Google Scholar 

  38. Lewis, A.S., Malick, J.: Alternating projection on manifolds. Math. Oper. Res. 33(1), 216–234 (2008)

    Article  MathSciNet  Google Scholar 

  39. Łojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels, pp. 87–89. Les Équations aux Dérivées Partielles, Éditions du centre National de la Recherche Scientifique (1963)

  40. Luenberger, D.G.: The gradient projection method along geodesics. Manag. Sci. 18, 620–631 (1972)

    Article  MathSciNet  Google Scholar 

  41. Luke, D.R., Sabach, S., Teboulle, M.: Optimization on spheres: models and proximal algorithms with computational performance comparisons. SIAM J. Math. Data Sci. 1, 408–445 (2019)

    Article  MathSciNet  Google Scholar 

  42. Palis, J., De Melo, W.: Geometric Theory of Dynamical Systems. An Introduction. Translated from the Portuguese by A.K. Manning,. Springer, Berlin (1982)

    Book  Google Scholar 

  43. Poly, J.B., Raby, G.: Fonction distance et singularités. Bull. Sci. Math. 108, 187–195 (1984)

    MathSciNet  Google Scholar 

  44. Sakai, M.: Strong convergence of infinite products of orthogonal projections in Hilbert space. Appl. Anal. 59, 109–120 (1995)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  46. Udriste, C.: Convex Functions and Optimization Algorithms on Riemannian Manifolds. Mathematics and Its Applications, vol. 297. Kluwer Academic, Dordrecht (1994)

    Google Scholar 

  47. von Neumann, J.: Functional Operators. II, The Geometry of Orthogonal Spaces. Annals of Mathematics Studies, vol. 22. Princeton University Press, Princeton (1950)

    Google Scholar 

  48. Wang, X.M., Li, C., Yao, J.C.: Subgradient projection algorithms for convex feasibility on Riemannian manifolds with lower bounded curvatures. J. Optim. Theory Appl. 164, 202–217 (2015)

    Article  MathSciNet  Google Scholar 

  49. Wang, X.M., Li, C., Wang, J., Yao, J.C.: Linear convergence of subgradient algorithm for convex feasibility on Riemannian manifolds. SIAM J. Optim. 25, 2334–2358 (2015)

    Article  MathSciNet  Google Scholar 

  50. Wang, J., Wang, X.M., Li, C., Wang, J., Yao, J.C.: Convergence analysis of gradient algorithms on Riemannian manifolds without curvature constraints and application to Riemannian mass. SIAM J. Optim. 31, 172–199 (2021)

    Article  MathSciNet  Google Scholar 

  51. Wang, X.M., Li, C., Yao, J.C.: On some basic results related to affine functions on Riemannian manifolds. J. Optim. Theory Appl. 170, 783–803 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  52. Wolter, F.E.: Distance function and cut loci on a complete Riemannian manifold. Arch. Math. 32, 92–96 (1979)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors wish to express their gratitude to the associate editor and referees for their valuable comments and advice. The authors were partially supported by the National Council for Scientific and Technological Development of Brazil - CNPq (Grants 302156/2022-4 and 315937/2023-8).

Author information

Authors and Affiliations

  1. Department of Mathematics, Federal University of Piauí, Teresina, PI, Brazil

    João Xavier da Cruz Neto, Ítalo Dowell Lira Melo, Paulo Alexandre Sousa & João Carlos de Oliveira Souza

Authors
  1. João Xavier da Cruz Neto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ítalo Dowell Lira Melo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paulo Alexandre Sousa
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. João Carlos de Oliveira Souza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to João Carlos de Oliveira Souza.

Additional information

Communicated by Nicolas Hadjisavvas.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

da Cruz Neto, J.X., Melo, Í.D.L., Sousa, P.A. et al. On the Relationship Between the Kurdyka–Łojasiewicz Property and Error Bounds on Hadamard Manifolds. J Optim Theory Appl 200, 1255–1285 (2024). https://doi.org/10.1007/s10957-024-02386-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-024-02386-6

Keywords

Mathematics Subject Classification

Search

Navigation