article

Convergence analysis of inexact proximal point algorithms on Hadamard manifolds

Authors:

Jen-Chih YaoAuthors Info & Claims

Journal of Global Optimization, Volume 61, Issue 3

Pages 553 - 573

https://doi.org/10.1007/s10898-014-0182-2

Published: 01 March 2015 Publication History

Abstract

Inexact proximal point methods are extended to find singular points for multivalued vector fields on Hadamard manifolds. Convergence criteria are established under some mild conditions. In particular, in the case of proximal point algorithm, that is, $$\varepsilon _n=0$$ n = 0 for each $$n$$ n , our results improve sharply the corresponding results in Li et al. ( 2009 ). Applications to optimization problems, variational inequality problems and gradient methods are also given.

References

[1]

Adler, R., Dedieu, J.P., Margulies, J., Martens, M., Shub, M.: Newton's method on Riemannian manifolds and a geometric model for human spine. IMA J. Numer. Anal. 22, 359---390 (2002)

[2]

Auslender, A.: Problèmes de Minimax via l'Analyse Convexe et les Inequalities Variationelles: Theories et algorithmes, Lectures Notes in Eco. and Math. Systems, 77, Springer Verlag (1972)

[3]

Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Prog. Ser. B 116, 5---16 (2009)

Digital Library

[4]

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

[5]

Bauschke, H.H., Combettes, P.L.: Construction of best Bregman approximations in reflexive Banach spaces. Proc. Am. Math. Soc. 131, 3757---3766 (2003)

[6]

Burachik, R.S., Lopes, J.O., Da Silva, G.J.P.: An inexact interior point proximal method for the variational inequality problem. Comput. Appl. Math. 28, 15---36 (2009)

[7]

Browder, F.E.: Multi-valued monotone nonlinear mappings and duality mappings in Banach spaces. Trans. Am. Math. Soc. 118, 338---351 (1965)

[8]

Dedieu, J.P., Priouret, P., Malajovich, G.: Newton's method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal. 23, 395---419 (2003)

[9]

Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Monotone point-to-set vector fields. Balkan J. Geom. Appl. 5, 69---79 (2000)

[10]

Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: A proximal regularization of the steepest descent method in Riemannian manifold. Balkan J. Geom. Appl. 4(2), 1---8 (1999)

[11]

Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Contributions to the study of monotone vector fields. Acta. Math. Hungarica 94(4), 307---320 (2002)

[12]

DoCarmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992)

[13]

Ferreira, O.P., Lucambio Pérez, L.R., Németh, S.Z.: Singularities of monotone vector fields and an extragradient-type algorithm. J. Global Optim. 31, 133---151 (2005)

Digital Library

[14]

Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51(2), 257---270 (2002)

[15]

Ferreira, O.P., Svaiter, B.F.: Kantorovich's theorem on Newton's method in Riemannian manifolds. J. Complex. 18, 304---329 (2002)

Digital Library

[16]

Greene, R., Wu, H.: On the subharmonicity and plurisubharmonicity of geodesically convex functions. Indiana Univ. Math. J. 22(7), 641---653 (1973)

[17]

Güler, O.: New proximal point algorithms for convex minimization. SIAM J. Optim. 2, 649---664 (1992)

Digital Library

[18]

Helmke, U., Huper, K., Moore, J.B.: Quadratically convergent algorithms for optimal dexterous hand grasping. IEEE Trans. Robot. Automat. 18(2), 138---146 (2002)

[19]

Helmke, U., Moore, J.B.: Optimization and Dynamical Systems. Springer-Verlag, New York (1994)

[20]

Krasnoselskii, M.A.: Two observations about the method of successive approximations. Uspehi Mat. Nauk 10, 123---127 (1955)

[21]

Kryanev, A.V.: The solution of incorrectly posed problems by methods of successive approximations. Soviet Math. 14, 673---676 (1973)

[22]

Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687---3732 (2007)

[23]

Lemaire, B.: The proximal algorithm. In: Penot, J.P. (ed.) New Methods of Optimization and Their Industrial Use. International Series of Numerical Mathematics 87, pp. 73---87. Birkhauser, Basel (1989)

[24]

Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79(2), 663---683 (2009)

[25]

Li, C., Mordukhovich, B.S., Wang, J.H., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21(4), 1523---1560 (2011)

Digital Library

[26]

Li, C., Yao, J.C.: Variational inequalities for set-valued vector fields on Riemannian manifolds: convexity of the solution set and the proximal point algorithm. SIAM J. Control Optim. 50(4), 2486---2514 (2012)

Digital Library

[27]

Li, C., Wang, J.H.: Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds. Sci. China Ser. A. 48, 1465---1478 (2005)

[28]

Li, C., Wang, J.H.: Newton's method on Riemannian manifolds: smale's point estimate theory under the $$\gamma $$¿-condition. IMA J. Numer. Anal. 26, 228---251 (2006)

[29]

Li, C., Wang, J.H.: Newton's method for sections on Riemannian manifolds: generalized covariant $$\alpha $$¿-theory. J. Complex. 24, 423---451 (2008)

Digital Library

[30]

Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlin. Anal. 71(11), 5695---5706 (2009)

[31]

Luque, F.J.: Asymptotic convergence analysis of the proximal point algorithm. SIAM J. Control Optim. 22, 277---293 (1984)

Digital Library

[32]

Lippert, R., Edelman, A.: Nonlinear eigenvalue problems with orthogonality constraints (Section 9.4). In: Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.) Templates for the Solution of Algebraic Eigenvalue Problems, pp. 290---314. SIAM, Philadelphia (2000)

[33]

Ma, Y., Kosecka, J., Sastry, S.S.: Optimization criteria and geometric algorithms for motion and structure estimation. Internat. J. Comput. Vis. 44, 219---249 (2001)

Digital Library

[34]

Martinet, B.: Régularisation, d'inéquations variationelles par approximations successives, Revue Française d'Informatique et de Recherche Operationelle, pp. 154---159 (1970)

[35]

Minty, G.J.: On the monotonicity of the gradient of a convex function. Pacific J. Math. 14, 243---247 (1964)

[36]

Németh, S.Z.: Monotone vector fields. Publicationes Mathematicae Debrecen 54(3---4), 437---449 (1999)

[37]

Németh, S.Z.: Variational inqualities on Hadamard manifolds. Nonlin. Anal. 52, 1491---1498 (2003)

Digital Library

[38]

Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold. Neurocomputing 67, 106---135 (2005)

Digital Library

[39]

Passty, G.B.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274---276 (1979)

[40]

Petersen, P.: Riemannian Geometry, GTM 171, 2nd edn. Springer, Berlin (2006)

[41]

Polyak, B.T.: Introduction to Optimization. Optimization sofware Inc, New York (1987)

[42]

Rapcsák, T.: Smooth Nonlinear Optimization in $$\mathbb{R}^n$$Rn, Nonconvex Optimization and Applications, 19. Kluwer Academic Publisher, Dordrecht (1997)

[43]

Reich, S.: Constructive techniques for accretive and monotone operators. In: Lakshmikantham, V. (ed.) Applied Nonlinear Analysis, pp. 335---345. Academic Press, New York (1979)

[44]

Robinson, S.M.: Stability theory for systems of inequalities, part I: linear systems. SIAM J. Numer. Anal. 12, 754---769 (1975)

[45]

Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75---88 (1970)

[46]

Rockafellar, R.T.: Monotone operators associated with saddle-functions and minimax problems, Nonlinear Functional Analysis, Part 1, F.E. Browder ed., Symp. in Pure Math., Amer. Math. Soc. Prov., R.I. 18, 397---407 (1970)

[47]

Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

[48]

Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877---898 (1976)

Digital Library

[49]

Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs 149. American Mathematical Society, Providence (1996)

[50]

Singer, I.: The Theory of Best Approximation and Functional Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, 13. SIAM, Philadelphia, (1974)

[51]

Smith, S.T.: Covariance, subspace, and intrinsic Cramér-Rao bounds. IEEE Trans. Signal Process. 53, 1610---1630 (2005)

Digital Library

[52]

Smith, S.T.: Optimization techniques on Riemannian manifolds. In: Bloch, A. (ed.) Fields Institute Communications, vol. 3, pp. 113---146. American Mathematical Society, Providence, RI (1994)

[53]

Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Prog. 87, 189---202 (2000)

[54]

Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications, vol. 297. Kluwer Academic, Dordrecht (1994)

[55]

Wang, J.H., Li, C.: Uniqueness of the singular points of vector fields on Riemannian manifolds under the $$\gamma $$¿-condition. J. Complex. 22, 533---548 (2006)

Digital Library

[56]

Wang, J.H., López, G., Martín-Márquez, V., Li, C.: Monotone and Accretive operators on Riemannian manifolds. J. Optim. Theory Appl. 146, 691---708 (2010)

Digital Library

[57]

Xavier, J., Barroso, V.: Intrinsic variance lower bound (IVLB): An extension of the Cramér-Rao bound to Riemannian manifolds. In: Proceedings of the 2005 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2005). Philadelphia, PA (2005)

[58]

Yan, W.Y., Lam, J.: An approximate approach to H2 optimal model reduction. IEEE Trans. Automat. Control. 44(7), 1341---1358 (1999)

[59]

Zeidler, E.: Nonlinear Functional Analysis and Applications II B, Nonlinear Monotone Operators. Springer, Berlin (1990)

Cited By

Ansari QUddin M(2024)Modified proximal point algorithms for variational inclusion problems on Hadamard manifoldsSoft Computing - A Fusion of Foundations, Methodologies and Applications10.1007/s00500-023-09562-228:9-10(6595-6606)Online publication date: 1-May-2024
https://dl.acm.org/doi/10.1007/s00500-023-09562-2
Zhao ZZeng QXu YQian YYao T(2022)A projection algorithm for pseudomonotone vector fields with convex constraints on Hadamard manifoldsNumerical Algorithms10.1007/s11075-022-01464-y93:3(1209-1223)Online publication date: 2-Dec-2022
https://dl.acm.org/doi/10.1007/s11075-022-01464-y
Yao TZhao ZBai ZJin X(2021)A Riemannian derivative-free Polak–Ribiére–Polyak method for tangent vector fieldNumerical Algorithms10.1007/s11075-020-00891-z86:1(325-355)Online publication date: 1-Jan-2021
https://dl.acm.org/doi/10.1007/s11075-020-00891-z
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Convergence analysis of inexact proximal point algorithms on Hadamard manifolds
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cover image Journal of Global Optimization

Journal of Global Optimization Volume 61, Issue 3

March 2015

205 pages

ISSN:0925-5001

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Copyright © Copyright © 2015 Springer Science+Business Media New York.

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Kluwer Academic Publishers

United States

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Published: 01 March 2015

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Cited By

Ansari QUddin M(2024)Modified proximal point algorithms for variational inclusion problems on Hadamard manifoldsSoft Computing - A Fusion of Foundations, Methodologies and Applications10.1007/s00500-023-09562-228:9-10(6595-6606)Online publication date: 1-May-2024
https://dl.acm.org/doi/10.1007/s00500-023-09562-2
Zhao ZZeng QXu YQian YYao T(2022)A projection algorithm for pseudomonotone vector fields with convex constraints on Hadamard manifoldsNumerical Algorithms10.1007/s11075-022-01464-y93:3(1209-1223)Online publication date: 2-Dec-2022
https://dl.acm.org/doi/10.1007/s11075-022-01464-y
Yao TZhao ZBai ZJin X(2021)A Riemannian derivative-free Polak–Ribiére–Polyak method for tangent vector fieldNumerical Algorithms10.1007/s11075-020-00891-z86:1(325-355)Online publication date: 1-Jan-2021
https://dl.acm.org/doi/10.1007/s11075-020-00891-z
Batista EBento GFerreira O(2016)Enlargement of Monotone Vector Fields and an Inexact Proximal Point Method for Variational Inequalities in Hadamard ManifoldsJournal of Optimization Theory and Applications10.1007/s10957-016-0982-2170:3(916-931)Online publication date: 1-Sep-2016
https://dl.acm.org/doi/10.1007/s10957-016-0982-2

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