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Convergence of Newton’s method for sections on Riemannian manifolds. (English) Zbl 1228.90155
The author explores the local behavior of Newton’s method for sections on Riemannian manifolds. Under the assumption that the covariant derivatives of the sections satisfy one kind of Lipschitz condition with L-average, new estimates of the radii of convergence balls of Newton’s method and the radii of uniqueness balls of singular points of sections on Riemannian manifolds are given. In particular, the estimates here are completely independent of the sectional curvature of the underlying Riemannian manifold and improve the corresponding results due to J.-P. Dedieu, P. Priouret and G. Malajovich [IMA J. Numer. Anal. 23, No. 3, 395–419 (2003; Zbl 1047.65037)] as well as the ones by C. Li and J. Wang [Sci. China, Ser. A 48, No. 11, 1465–1478 (2005; Zbl 1116.53024)]. Applications to special cases, which include the Kantorovich condition and the γ-condition, as well as Smale’s γ-theory for sections on Riemannian manifolds, are provided, which consequently improve the corresponding results in [Dedieu et al., loc. cit.].
MSC:
90C53Methods of quasi-Newton type
References:
[1]Dedieu, J.P., Priouret, P., Malajovich, G.: Newton’s method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal. 23, 395–419 (2003) · Zbl 1047.65037 · doi:10.1093/imanum/23.3.395
[2]Li, C., Wang, J.H.: Convergence of Newton’s method and uniqueness of zeros of vector fields on Riemannian manifolds. Sci. China Ser. A 48(11), 1465–1478 (2005) · Zbl 1116.53024 · doi:10.1360/04ys0147
[3]Adler, R., Dedieu, J.P., Margulies, J., Martens, M., Shub, M.: Newton method on Riemannian manifolds and a geometric model for human spine. IMA J. Numer. Anal. 22, 1–32 (2002) · Zbl 1056.92002 · doi:10.1093/imanum/22.3.359
[4]Burke, J.V., Lewis, A., Overton, M.: Optimal stability and eigenvalue multiplicity. Found. Comput. Math. 1, 205–225 (2001) · Zbl 0994.15022 · doi:10.1007/s102080010008
[5]Mahony, R.E.: The constrained Newton method on a Lie group and the symmetric eigenvalue problem. Linear Algebra Appl. 248, 67–89 (1996) · Zbl 0864.65032 · doi:10.1016/0024-3795(95)00171-9
[6]Smith, S.T.: Optimization techniques on Riemannian manifolds. In: Fields Institute Communications, vol. 3, pp. 113–146. Am. Math. Soc., Providence (1994)
[7]Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Monotone point-to-set vector fields. Balk. J. Geom. Appl. 5, 69–79 (2000)
[8]Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002) · Zbl 1013.49024 · doi:10.1080/02331930290019413
[9]Ferreira, O.P., Lucambio Pérez, L.R., Nemeth, S.Z.: Singularities of monotone vector fields and an extragradient-type algorithm. J. Glob. Optim 31, 133–151 (2005) · Zbl 1229.58007 · doi:10.1007/s10898-003-3780-y
[10]Rapcsk, T.: Smooth Nonlinear Optimization in n . Nonconvex Optimization and Its Applications, vol. 19. Kluwer Academic, Dordrecht (1997)
[11]Martín-Márquez, V.: Nonexpansive mappings and monotone vector fields in Hadamard manifold. Commun. Appl. Anal. 13, 633–646 (2009)
[12]Edelman, A., Arias, T.A., Smith, T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20, 303–353 (1998) · Zbl 0928.65050 · doi:10.1137/S0895479895290954
[13]Smith, S.T.: Geometric optimization method for adaptive filtering. Ph.D. thesis, Harvard University, Cambridge, MA (1993)
[14]Absil, P.A., Baker, C.G., Gallivan, K.A.: Trust-region methods on Riemannian manifolds. Found. Comput. Math. 7, 303–330 (2007) · Zbl 1129.65045 · doi:10.1007/s10208-005-0179-9
[15]Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications, vol. 297. Kluwer Academic, Dordrecht (1994)
[16]Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71(11), 5695–5706 (2009) · Zbl 1180.58012 · doi:10.1016/j.na.2009.04.048
[17]Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687–3732 (2007) · Zbl 1157.49021 · doi:10.1090/S0002-9947-07-04075-5
[18]Wang, J.H., Huang, S., Li, C.: Extended Newton’s method for mappings on Riemannian manifolds with values in a cone. Taiwan. J. Math. 13, 633–656 (2009)
[19]Wang, J.H., Li, C.: Convergence of the family of Euler-Halley type methods on Riemannian manifolds under the γ-condition. Taiwan. J. Math. 13(2), 585–606 (2009)
[20]Wang, J.H., Li, C.: Uniqueness of the singular point of vector field on Riemannian manifold under the γ-condition. J. Complex. 22, 533–548 (2006) · Zbl 1102.65064 · doi:10.1016/j.jco.2005.11.004
[21]Li, C., Wang, J.H., Dedieu, J.P.: Smale’s point estimate theory for Newton’s method on Lie groups. J. Complex. 25, 128–151 (2009) · Zbl 1170.65040 · doi:10.1016/j.jco.2008.11.001
[22]Azagra, D., Ferrera, J., López-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220, 304–361 (2005) · Zbl 1067.49010 · doi:10.1016/j.jfa.2004.10.008
[23]Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79(3), 663–683 (2009) · Zbl 1171.58001 · doi:10.1112/jlms/jdn087
[24]Németh, S.Z.: Monotone vector fields. Publ. Math. (Debr.) 54(3–4), 437–449 (1999)
[25]Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Oxford, Pergamon (1982)
[26]Smale, S.: Newton’s method estimates from data at one point. In: Ewing, R., Gross, K., Martin, C. (eds.) The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics, pp. 185–196. Springer, New York (1986)
[27]Ferreira, O.P., Svaiter, B.F.: Kantorovich’s Theorem on Newton’s method in Riemannian manifolds. J. Complex. 18, 304–329 (2002) · Zbl 1003.65057 · doi:10.1006/jcom.2001.0582
[28]Li, C., Wang, J.H.: Newton’s method on Riemannian manifolds: Smale’s point estimate theory under the γ-condition. IMA J. Numer. Anal. 26, 228–251 (2006) · Zbl 1094.65052 · doi:10.1093/imanum/dri039
[29]Wang, X.H., Han, D.F.: Criterion α and Newton’s method. Chin. J. Numer. Appl. Math. 19, 96–105 (1997)
[30]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) · Zbl 1147.58008 · doi:10.1007/s10208-006-0221-6
[31]Li, C., Wang, J.H.: Newton’s method for sections on Riemannian manifolds: generalized covariant α-theory. J. Complex. 24, 423–451 (2008) · Zbl 1153.65059 · doi:10.1016/j.jco.2007.12.003
[32]DoCarmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992)
[33]Chern, S.S.: Vector bundle with connection. In: Selected Papers, vol. 4. pp. 245–268. Springer, New York (1989)
[34]Wells, R.O.: Differential Analysis on Complex Manifolds. GTM, vol. 65. Springer, New York (1980)
[35]Wang, X.H.: Convergence of Newton’s method and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal. 20(1), 123–134 (2000) · Zbl 0942.65057 · doi:10.1093/imanum/20.1.123
[36]Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1997)