Skip to main content
SpringerLink
Log in

Limiting Subdifferential Calculus and Perturbed Distance Function in Riemannian Manifolds

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We provide definitions for Fréchet \(\varepsilon \)-subdifferential and Fréchet \(\varepsilon \)-normals set for functions and sets in the Riemannian manifolds. Then we generalize the notions of Mordukhovich sequential subdifferential and normal cone (limiting subdifferential and normal cone) and develop several calculus rules for subdifferentials and normal cones in this setting. Finally, as an application, the limiting subdifferential of perturbed distance function is investigated in the Riemannian manifolds setting.

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

Similar content being viewed by others

References

  1. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithm on Matrix Manifolds. Princeton University Press, Princeton (2008)

    Book  Google Scholar 

  2. Adler, R.L., Dedieu, J.P., Margulies, J.Y., Martens, M., Shub, M.: Newton’s method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal. 22, 359–390 (2002)

    Article  MathSciNet  Google Scholar 

  3. Azagra, D., Ferrera, J.: Applications of proximal calculus to fixed point theory on Riemannian manifolds. Nonlinear Anal. 67, 154–174 (2007)

    Article  MathSciNet  Google Scholar 

  4. Azagra, D., Ferrera, J., Lopez-Mesas, F.: Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220, 304–361 (2005)

    Article  MathSciNet  Google Scholar 

  5. Barani, A.: Subdifferentials of perturbed distance function in Riemannian manifolds. Optimization 67, 1849–1868 (2018)

    Article  MathSciNet  Google Scholar 

  6. Baranger, J., Temam, R.: Nonconvex optimization problems depending on a parameter. SIAM J. Control Optim. 13, 377–405 (1973)

    MATH  Google Scholar 

  7. Batista, E.E.A., Bento, G.C., Ferreira, O.P.: Enlargement of monotone vector fields and an inexact proximal point method for variational inequalities in Hadamard manifolds. J. Optim. Theory Appl. 170(3), 916–931 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Bidaut, M.F.: Existence theorems for usual and approximate solutions of optimal control problem. J. Optim. Theory Appl. 15, 393–411 (1975)

    Article  MathSciNet  Google Scholar 

  12. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, Hoboken (1983)

    MATH  Google Scholar 

  13. Cobza̧s, S.: Generic existence of solutions for some perturbed optimization problems. J. Math. Anal. Appl. 243, 344–356 (2000)

    Article  MathSciNet  Google Scholar 

  14. Da Cruz Neto, J.X., Oliveira, O.P., Lucambio Perez, L.R.: Convex and mono-tone transformable mathematical programming problems and a proximal like point method. J. Glob. Optim. 94, 53–69 (2006)

    Article  Google Scholar 

  15. Da Cruz Neto, J.X., Lima, L.L., Oliveira, P.R.: Geodesic algorithm in Riemannian manifolds. Balkan J. Geom. Appl. 2, 89–100 (1998)

    MATH  Google Scholar 

  16. Ekeland, I.: Sur les probléms variationnels. C. R. Acad. Sci. Paris 275, 1057–1059 (1972)

    MathSciNet  MATH  Google Scholar 

  17. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  Google Scholar 

  18. Ferreira, O.P.: Proximal subgradient and a characterization of Lipschitz functions on Riemannian manifolds. J. Math. Anal. Appl. 313, 587–597 (2006)

    Article  MathSciNet  Google Scholar 

  19. Grohs, P., Hosseini, S.: \(\varepsilon \)-subgradient algorithms for locally Lipschitz functions on Riemannian manifolds. Adv. Comput. Math. 42, 333–360 (2016)

    Article  MathSciNet  Google Scholar 

  20. Hosseini, S., Pouryayevali, M.R.: Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds. Nonlinear Anal. 74, 3884–3895 (2011)

    Article  MathSciNet  Google Scholar 

  21. Jofré, A., Luc, D.T., Théra, M.: Extensions of Fréchet \(\varepsilon \)-subdifferential calculus and applications. J. Math. Anal. Appl. 268, 266–290 (2202)

    Google Scholar 

  22. Kristály, A.: Location of Nash equilibria: a Riemannian geometrical approach. Proc. Am. Math. Soc. 138, 1803–1810 (2010)

    Article  MathSciNet  Google Scholar 

  23. Kruger, A.Y., Mordukhovich, B.S.: Extremal points and the Euler equation in nonsmooth optimization. Dokl. Akad. Nauk BSSR 24, 684–687 (1980)

    MathSciNet  MATH  Google Scholar 

  24. Kruger, A.Y., Mordukhovich, B.S.: Generalized normals and derivatives, and necessery optimality coditions in nondifferentiable programming. Part I. Depon. VINITI, No. 408-80; Part II, Depon. VINITI, No. 494-80, Moscow (1980)

  25. Lang, S.: Fundamentals of Differential Geometry. Graduate Texts in Mathematics, vol. 191. Springer, New York (1999)

    Book  Google Scholar 

  26. Lee, P.Y.: Geometric optimization for computer vision. Ph.D. thesis, Australian National University (2005)

  27. Li, C., Peng, L.H.: Porosity of perturbed optimization problems in Banach spaces. J. Math. Anal. Appl. 324, 751–761 (2006)

    Article  MathSciNet  Google Scholar 

  28. Meng, Li, Chong, Li, Yao, Jen-Chih: Limiting subdifferentials of perturbed distance functions in Banach spaces. Nonlinear Anal. 75, 1483–1495 (2012)

    Article  MathSciNet  Google Scholar 

  29. Mordukhovich, B.S., Shao, Y.: Nonsmooth sequentialy analysis in Asplund spaces. Trans. Amer. Math. Soc. 348(4), 1235–1280 (1996)

    Article  MathSciNet  Google Scholar 

  30. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330. Springer, Berlin (2006)

    Book  Google Scholar 

  31. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, New York (2018)

    Book  Google Scholar 

  32. Ni, R.: Generic solutions for some perturbed optimization problem in non-reflexive Banach spaces. J. Math. Anal. Appl. 302, 417–424 (2005)

    Article  MathSciNet  Google Scholar 

  33. Rapscák, T.: Smooth Nonlinear Optimization in \({{\mathbb{R}}}^n\). Kluwer Academic Publishers, Dordrecht (1997)

    Google Scholar 

  34. Rockafellar, R.T.: Directioonally Lipschitzian functions and subdifferential calculus. Proc. Lond. Math. Soc. 39, 331–355 (1979)

    Article  Google Scholar 

  35. Rockafellar, R.T.: Extensions of subgradiant calculus with applications to optimization. Nonlinear Anal. 9, 665–698 (1985)

    Article  MathSciNet  Google Scholar 

  36. Sakai, T.: Riemannian Geometry. American Mathematic Society, Philadelphia (1992)

    Google Scholar 

  37. Sepahvand, A., Barani, A.: On the regularity of sets in Riemannian manifolds. J. Aust. Math. Soc. (2020) (Accepted)

  38. Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Acadmic Publishers, Dordrecht (1994)

    Book  Google Scholar 

  39. Wang, J.H., Li, C., Xu, H.K.: Subdifferentials of perturbed distance functions in Banach spaces. J. Glob. Optim. 46, 489–501 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Lorestan University, P. O. Box 465, Khoramabad, Iran

    M. Farrokhiniya & A. Barani

Authors
  1. M. Farrokhiniya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Barani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Barani.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Farrokhiniya, M., Barani, A. Limiting Subdifferential Calculus and Perturbed Distance Function in Riemannian Manifolds. J Glob Optim 77, 661–685 (2020). https://doi.org/10.1007/s10898-020-00889-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-020-00889-w

Keywords

Mathematics Subject Classification

Search

Navigation