Subgradient Method for Convex Feasibility on Riemannian Manifolds



In this paper, a subgradient type algorithm for solving convex feasibility problem on Riemannian manifold is proposed and analysed. The sequence generated by the algorithm converges to a solution of the problem, provided the sectional curvature of the manifold is non-negative. Moreover, assuming a Slater type qualification condition, we analyse a variant of the first algorithm, which generates a sequence with finite convergence property, i.e., a feasible point is obtained after a finite number of iterations. Some examples motivating the application of the algorithm for feasibility problems, nonconvex in the usual sense, are considered.


Nonsmooth analysis Feasibility problem General convexity Subgradient algorithm Riemannian manifolds 


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  1. 1.
    Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155–270 (1996) CrossRefGoogle Scholar
  2. 2.
    Censor, Y., Altschuler, M.D., Powlis, W.D.: On the use of Cimmino’s simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning. Inverse Probl. 4, 607–623 (1988) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Marks, L.D., Sinkler, W., Landree, E.: A feasible set approach to the crystallographic phase problem. Acta Crystallogr. A, Found. Crystallogr. 55, 601–612 (1999) CrossRefGoogle Scholar
  4. 4.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Butnariu, D., Censor, Y., Gurfil, P., Hadar, E.: On the behavior of subgradient projections methods for convex feasibility problems in Euclidean spaces. SIAM J. Optim. 19, 786–807 (2008) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Butnariu, D., Iusem, A., Burachik, R.: Iterative methods of solving stochastic convex feasibility problems and applications. Comput. Optim. Appl. 15, 269–307 (2000) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    O’Hara, J.G., Pillay, P., Xu, H.K.: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Anal. 64, 2022–2042 (2006) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Shor, N.Z.: Minimization Algorithms for Non-differentiable Function. Springer, Berlin (1985) CrossRefGoogle Scholar
  9. 9.
    Polyak, B.T.: Minimization of nonsmooth functionals. U.S.S.R. Comput. Math. Math. Phys. 9, 14–29 (1969) CrossRefGoogle Scholar
  10. 10.
    Alber, Ya.I., Iusem, A.N., Solodov, M.V.: On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math. Program. 81(1), 23–35 (1998) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bertsekas, D.P., Nedic, A.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 56(1), 109–138 (2001) MathSciNetGoogle Scholar
  12. 12.
    Burachik, R.S., Iusem, A.N., Melo, J.G.: A primal dual modified subgradient algorithm with sharp Lagrangian. J. Glob. Optim. 46(3), 347–361 (2010) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Censor, Y., Lent, A.: Cyclic subgradient projections. Math. Program. 24, 233–235 (1982) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Udriste, C.: Convex functions and optimization algorithms on Riemannian manifolds. In: Mathematics and Its Applications, vol. 297. Kluwer Academic, Dordrecht (1994) Google Scholar
  15. 15.
    Alvarez, F., Bolte, J., Munier, J.: A unifying local convergence result for Newton’s method in Riemannian manifolds. Found. Comput. Math. 8, 197–226 (2008) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Baker, C.G., Absil, P.-A., Gallivan, K.A.: An implicit trust-region method on Riemannian manifolds. IMA J. Numer. Anal. 28, 665–689 (2008) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    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
  19. 19.
    Ferreira, O.P.: Proximal subgradient and a characterization of Lipschitz function on Riemannian manifolds. J. Math. Anal. Appl. 313, 587–597 (2006) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Ferreira, O.P.: Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds. Nonlinear Anal. 68, 1517–1528 (2008) MathSciNetMATHGoogle Scholar
  21. 21.
    Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifold. Optimization 51, 257–270 (2002) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52, 1491–1498 (2003) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    da 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 algorithm. J. Glob. Optim. 35, 53–69 (2006) MATHCrossRefGoogle Scholar
  25. 25.
    Ledyaev, Yu.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687–3732 (2007) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Wang, J.H., Huang, S.C., Li, C.: Extended Newton’s Algorithm for mappings on Riemannian manifolds with values in a cone. Taiwan. J. Math. 13, 633–656 (2009) MathSciNetMATHGoogle Scholar
  27. 27.
    Wang, J.H., Lopez, G., Martin-Marquez, V., Li, C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim. Theory Appl. 146, 691–708 (2010) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Li, C., Wang, J.H.: Newton’s method for sections on Riemannian manifolds: generalized covariant α-theory. J. Complex. 24, 423–451 (2008) MATHCrossRefGoogle Scholar
  29. 29.
    Wang, J.H., Dedieu, J.P.: Newton’s method on Lie groups: Smale’s point estimate theory under the γ-condition. J. Complex. 25, 128–151 (2009) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Wang, J.H., Li, C.: Newton’s method on Lie groups with applications to optimization. IMA J. Numer. Anal. 31, 322–347 (2011) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Wang, J.H.: Convergence of Newton’s method for sections on Riemannian manifolds. J. Optim. Theory Appl. 148(1), 125–145 (2011) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Rapcsák, T.: Local convexity on smooth manifolds. J. Optim. Theory Appl. 127(1), 165–176 (2005) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Rapcsák, T.: Geodesic convexity in nonlinear optimization. J. Optim. Theory Appl. 69(1), 169–183 (1991) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Ferreira, O.P., Oliveira, P.R.: Subgradient algorithm on Riemannian manifolds. J. Optim. Theory Appl. 97, 93–104 (1998) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    da Cruz Neto, J.X., de Lima, L.L., Oliveira, P.R.: Geodesic algorithms in Riemannian geometry. Balk. J. Geom. Appl. 3, 89–100 (1998) MATHGoogle Scholar
  36. 36.
    do Carmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992) MATHGoogle Scholar
  37. 37.
    Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. Am. Math. Soc., Providence (1996) MATHGoogle Scholar
  38. 38.
    Rapcsák, T.: Smooth Nonlinear Optimization in ℝn. Kluwer Academic, Dordrecht (1997) Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.IME-Universidade Federal de GoiásGoiâniaBrazil

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