Journal of Optimization Theory and Applications

, Volume 158, Issue 2, pp 328–342 | Cite as

Nonsmooth Optimization Techniques on Riemannian Manifolds

Article

Abstract

We present the notion of weakly metrically regular functions on manifolds. Then, a sufficient condition for a real valued function defined on a complete Riemannian manifold to be weakly metrically regular is obtained, and two optimization problems on Riemannian manifolds are considered. Moreover, we present a generalization of the Palais–Smale condition for lower semicontinuous functions defined on manifolds. Then, we use this notion to obtain necessary conditions of optimality for a general minimization problem on complete Riemannian manifolds.

Keywords

Ekeland variational principle Contingent cone Metric regularity Generalized gradient Riemannian manifolds 

References

  1. 1.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 331. Springer, Berlin (2006) Google Scholar
  2. 2.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998) MATHCrossRefGoogle Scholar
  3. 3.
    Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 1: Sufficient optimality condition. J. Optim. Theory Appl. 142, 147–163 (2009) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chang, K.C.: Variational methods for non-differentiable functional and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Mirenchi, E., Salvatore, A.: A non-Smooth two points boundary value problem on Riemannian manifolds. Ann. Mat. Pura Appl. 166, 253–265 (1994) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002) MathSciNetMATHCrossRefGoogle Scholar
  8. 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) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ferreira, O.P.: Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds. Nonlinear Anal. 68, 1517–1528 (2008) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ferreira, O.P.: Proximal subgradient and a characterization of Lipschitz functions in Riemannian manifolds. J. Math. Anal. Appl. 313, 587–597 (2006) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71, 5695–5706 (2009) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Li, C., Mordukhovich, B.S., Wang, J.H., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21, 1523–1560 (2011) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Wang, J.H., López, G., Martín-Márquez, V., Li, C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim. Theory Appl. 146, 691–708 (2010) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Azagra, D., Ferrera, J., López-Mesas, F.: Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220, 304–361 (2005) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Azagra, D., Ferrera, J.: Proximal calculus on Riemannian manifolds, with applications to fixed point theory. Mediterr. J. Math. 2, 437–450 (2005) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Ledyaev, Yu.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687–3732 (2007) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Hosseini, S., Pouryayevali, M.R.: Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds. Nonlinear Anal. 74, 3884–3895 (2011) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Le, K., Motreanu, D.: Some properties of general minimization problems with constraints. Set-Valued Anal. 14, 413–424 (2009) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lang, S.: Fundamentals of Differential Geometry. Graduate Texts in Mathematics, vol. 191. Springer, New York (1999) MATHCrossRefGoogle Scholar
  21. 21.
    Clarke, F.H., Ledayaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics, vol. 178. Springer, New York (1998) MATHGoogle Scholar
  22. 22.
    Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization. CMS Books in Mathematics. Springer, New York (2000) MATHCrossRefGoogle Scholar
  23. 23.
    Barani, A., Hosseini, S., Pouryayevali, M.R.: On the metric projection onto φ-convex subsets of Riemannian manifolds. Rev. Mat. Complut. (to appear). doi:10.1007/s13163-011-0085-4

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

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