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Convergence of the Gauss-Newton method for convex composite optimization problems under majorant condition on Riemannian manifolds

Published: 27 February 2024 Publication History

Abstract

In this paper, we consider convex composite optimization problems on Riemannian manifolds, and discuss the semi-local convergence of the Gauss-Newton method with quasi-regular initial point and under the majorant condition. As special cases, we also discuss the convergence of the sequence generated by the Gauss-Newton method under Lipschitz-type condition, or under γ-condition.

Highlights

Study of convex composite optimization problems on Riemannian manifolds.
Convergence of the Gauss-Newton method under the majorant condition.
Special cases are studied under
(i) Lipschitz-type condition,
(ii) γ-condition.

References

[1]
F. Alvarez, J. Bolte, J. Munier, A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math. 8 (2) (2008) 197–226.
[2]
Q.H. Ansari, F. Babu, Proximal point algorithm for inclusion problems in Hadamard manifolds with applications, Optim. Lett. 15 (3) (2021) 901–921.
[3]
Q.H. Ansari, F. Babu, X.-B. Li, Variational inclusion problems in Hadamard manifolds, J. Nonlinear Convex Anal. 19 (2) (2018) 219–237.
[4]
Q.H. Ansari, F. Babu, M. Uddin, Regularization methods for hierarchical variational inequality problems on Hadamard manifolds, Arab. J. Math. 12 (2023) 309–330.
[5]
Q.H. Ansari, F. Babu, J.C. Yao, Regularization of proximal point algorithms in Hadamard manifolds, J. Fixed Point Theory Appl. 21 (2019).
[6]
Q.H. Ansari, M. Uddin, J.C. Yao, Iterative algorithms for pseudomonotone variational inequalities and fixed point problems on Hadamard manifolds, J. Nonlinear Convex Anal. 24 (5) (2023) 1043–1063.
[7]
T. Bittencourt, O.P. Ferreira, Kantorovich's theorem on Newton's method under majorant condition in Riemannian manifolds, J. Glob. Optim. 68 (2) (2017) 387–411.
[8]
M.P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, Basel, Berlin, 1992.
[9]
J.-P. Dedieu, P. Priouret, G. Malajovich, Newton's method on Riemannian manifolds: convariant alpha theory, IMA J. Numer. Anal. 23 (3) (2003) 395–419.
[10]
O.P. Ferreira, M.L.N. Gonçalves, P.R. Oliveira, Local convergence analysis of the Gauss-Newton method under a majorant condition, J. Complex. 27 (1) (2011) 111–125.
[11]
O.P. Ferreira, M.L.N. Gonçalves, P.R. Oliveira, Local convergence analysis of inexact Gauss-Newton like methods under majorant condition, J. Comput. Appl. Math. 2736 (2012) 2487–2498.
[12]
O.P. Ferreira, M.L.N. Gonçalves, P.R. Oliveira, Convergence of the Gauss-Newton method for convex composite optimization under a majorant condition, SIAM J. Optim. 23 (3) (2013) 1757–1783.
[13]
O.P. Ferreira, P.R. Oliveira, Proximal point algorithm on Riemannian manifolds, Optimization 51 (2) (2002) 257–270.
[14]
O.P. Ferreira, R.C.M. Silva, Local convergence of Newton's method under a majorant condition in Riemannian manifolds, IMA J. Numer. Anal. 32 (4) (2012) 1696–1713.
[15]
O.P. Ferreira, B.F. Svaiter, Kantorovich's theorem on Newton's method in Riemannian manifolds, J. Complex. 18 (1) (2002) 304–329.
[16]
O.P. Ferreira, B.F. Svaiter, Kantorovich's majorants principle for Newton's method, Comput. Optim. Appl. 42 (2) (2009) 213–219.
[17]
J-B. Hiriat-Urruty, C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer-Verlag, Berlin, 1993.
[18]
C. Li, J. Wang, Newton's method on Riemannian manifolds: Smale's point estimate theory under the γ-condition, IMA, J. Numer. Anal. 26 (2) (2006) 228–251.
[19]
C. Li, J. Wang, Newton's method for sections on Riemannian manifolds: generalized covariant α-theory, J. Complex. 24 (3) (2008) 423–451.
[20]
C. Li, X. Wang, On convergence of the Gauss–Newton method for convex composite optimization, Math. Program. 91 (2) (2002) 349–356.
[21]
C. Li, K.F. Ng, Majorizing functions and convergence of the Gauss-Newton method for convex composite optimization, SIAM J. Optim. 18 (2) (2007) 613–642.
[22]
C. Li, K.F. Ng, Convergence analysis of the Gauss-Newton method for convex inclusion and convex-composite optimization problems, J. Math. Anal. Appl. 389 (1) (2012) 469–485.
[23]
X.B. Li, N.J. Huang, Q.H. Ansari, J.C. Yao, Convergence rate of descent method with new inexact line-search on Riemannian manifolds, J. Optim. Theory Appl. 180 (3) (2019) 830–854.
[24]
J.X. da Cruz Neto, O.P. Ferreira, L.R. Lucambio Pérez, S.Z. Németh, Convex- and monotone-transformable mathematical programming problems and a proximal-like point method, J. Glob. Optim. 35 (2006) 53–69.
[25]
R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
[26]
S.M. Robinson, Extension of Newton's method to nonlinear functions with values in a cone, Numer. Math. 19 (1972) 341–347.
[27]
T. Sakai, Riemannian Geometry, Translations of Mathematical Monographs, Amer. Math. Soc., Providence, RI, 1996.
[28]
C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds, Kluwer Academic Publishers, Dordrecht, Boston, London, 1994.
[29]
J.H. Wang, Convergence of Newton's method for sections on Riemannian manifolds, J. Optim. Theory Appl. 148 (1) (2011) 125–145.
[30]
J.H. Wang, S. Huang, C. Li, Extended Newton's method for mappings on Riemannian manifolds with values in a cone, Taiwan. J. Math. 13 (2) (2009) 633–656.
[31]
J.H. Wang, J.C. Yao, C. Li, Gauss–Newton method for convex composite optimizations on Riemannian manifolds, J. Glob. Optim. 53 (2012) 1–24.

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Information & Contributors

Information

Published In

cover image Journal of Complexity
Journal of Complexity  Volume 80, Issue C
Feb 2024
156 pages

Publisher

Academic Press, Inc.

United States

Publication History

Published: 27 February 2024

Author Tags

  1. 49M15
  2. 90C53
  3. 65K05
  4. 65H05

Author Tags

  1. Convex composite optimization problems
  2. Gauss-Newton's method
  3. Majorant condition
  4. Semilocal convergence
  5. Riemannian manifolds

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