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Limiting Subdifferential Calculus and Perturbed Distance Function in Riemannian Manifolds


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.

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  1. 1.

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

  2. 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)

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 6.

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

  7. 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)

  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)

  9. 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)

  10. 10.

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

  11. 11.

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

  12. 12.

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

  13. 13.

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

  14. 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)

  15. 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)

  16. 16.

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

  17. 17.

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

  18. 18.

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

  19. 19.

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

  20. 20.

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

  21. 21.

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

  22. 22.

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

  23. 23.

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

  24. 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. 25.

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

  26. 26.

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

  27. 27.

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

  28. 28.

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

  29. 29.

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

  30. 30.

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

  31. 31.

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

  32. 32.

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

  33. 33.

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

  34. 34.

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

  35. 35.

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

  36. 36.

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

  37. 37.

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

  38. 38.

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

  39. 39.

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

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Correspondence to A. Barani.

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Farrokhiniya, M., Barani, A. Limiting Subdifferential Calculus and Perturbed Distance Function in Riemannian Manifolds. J Glob Optim (2020).

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  • Fréchet \(\varepsilon \)-subdifferential
  • Fréchet \(\varepsilon \)-normals set
  • Limiting subdifferential
  • Limiting normal cone
  • Perturbed distance function
  • Riemannian manifolds

Mathematics Subject Classification

  • 58C99
  • 58C05