Abstract
In the first part of this paper, we give a survey on convergence rates analysis of quasi-Newton methods in infinite Hilbert spaces for nonlinear equations. Then, in the second part we apply quasi-Newton methods in their Hilbert formulation to solve matrix equations. So, we prove, under natural assumptions, that quasi-Newton methods converge locally and superlinearly; the global convergence is also studied. For numerical calculations, we propose new formulations of these methods based on the matrix representation of the dyadic operator and the vectorization of matrices. Finally, we apply our results to algebraic Riccati equations.
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References
Abou-Kandil H., Freiling G., Ionescu V., Jank G.: Matrix Riccati Equations in Control and Systems Theory. Birkhauser Verlag, Berlin (2003)
Bartels R.T., Stewart G.W.: Solution of the matrix equation AX+BX=C. Comm. ACM 15, 820–826 (1972)
Benahmed, B.: Méthodes du second ordre et problèmes extrémaux. Thèse de doctorat d’état, Univesité d’Oran, Algérie (2007)
Benahmed, B., de Malafosse, B., Yassine, A.: Matrix transformation and quasi-Newton methods, IJMMS, 2007, Article ID25704, 17 doi:10.1155/2007/25705
Benahmed B., Fares A., de Malafosse B., Yassine A.: The quasi-Newton method and the infinite matrix theory applied to the continued fractions, Afr Diaspora J Math, Special Issue in Memory of Prof. Ibn Oumar Mahamat Salah 8(2), 1–16 (2009)
Broyden C.G., Dennis J.E. Jr., Moré J.J.: On the local and superlinear of convergence quasi-Newton methods. Math. Comput. 12, 223–246 (1973)
Broyden C.G.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593 (1965)
Broyden C.G.: Quasi-Newton methods and their applications to function minimization. Math. Comp. 21, 368–381 (1967)
Dennis J.E. Jr., Moré J.J.: A characterization of superlinear convergence and its applications to quasi-Newton methods. Math. Comput. 28, 549–560 (1974)
Gay D.M.: Some convergence properties of Broyden’s method. SIAM J. Numer. Anal. 16, 623–630 (1979)
Griewank A.: The local convergence of Broyden-like methods on lipschitzian problems in Hilbert spaces. SIAM J. Numer. Anal. 24, 684–705 (1987)
Gruver W.A., Sachs E.W.: Algorithmic Methods in Optimal Control. Pitman Publishing limited, London (1980)
Guo C.-H.: Newton’s method for discrete algebraic Riccati equation when the closed-loop matrix has eigenvalues on the unit circle. SIAM J. Matrix Anal. Appl. 20(2), 279–294 (1998)
Horwitz L.B., Sarachik P.E.: Davidon’s method in Hilbert space. SIAM J. Appl. Math. 16(4), 676–695 (1968)
Juang J.: Existence of algebraic matrix Riccati equations arising in transport theory. Linear Algebra Appl. 230, 89–100 (1999)
Juang J., Lin W.W.: Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices. SIAM J. Matrix Anal. Appl. 20, 228–243 (1999)
Kantorovich L.V.: On Newton’s method for functional equations. Dokl. Akad. Nauk SSSR 59, 1237–1240 (1948) (in Russian)
Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford University Press, (1995). xvii+480 pp. ISBN 0-19-853795-6
Kelley C.T., Sachs E.W.: A quasi-Newton method for elliptic boundary value problems. SIAM J. Numer. Anal. 24, 516–531 (1987)
Kelley C.T., Sachs E.W.: Quasi-Newton methods and unconstrained optimal control problems. SIAM J. Control Optim. 23, 1503–1516 (1987)
Kelley C.T., Sachs E.W.: A new proof of superlinear convergence for Broyden’s method in Hilbert space. SIAM J. Optim. 1(1), 146–150 (1991)
Laub A.J.: A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Control. AC-24, 913–921 (1979)
Lûtkepohl H.: Handbook of Matrices. John Wiley & Sons edition, New York (1996)
Man F.T.: The Davidon method of solution of the algebraic matrix Riccati equation. Int. J. Control 10(6), 713–719 (1969)
Martinez J.M.: Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math. 124, 97–121 (2000)
Ortega J.M., Rheinboldt W.C.: Iterative Solution of Nonlinear Variable Equations in Several Variables. Academic Press, San Diego (1970)
Sachs E.W.: Broyden’s method in Hilbert space. Math. Program. 35, 71–82 (1986)
Sherman J., Morrison W.J.: Ajustment of an inverse matrix corresponding to changes in the elements of a given comumn or a given row of the original matrix. Ann. Math. Stat. 20, 621 (1949)
Stoer J.: Two examples on the convergence of certain rank-2 minimization methods for quadratic functionals in Hilbert space. Linear Algebra Appl. 20, 217–222 (1979)
Tokumaru H., Adachi N., Goto K.: Davidon’s method for minimization problems in Hilbert space with application to control problems. SIAM J. Control 8(N°2), 163–178 (1970)
Wen-huan Y.: A quasi-Newton method in infinite-dimentional spaces and its application for solving a parabolic inverse problem. J. Comput. Math. 16(4), 305–318 (1998)
Wen-huan Y.: A quasi-Newton approach to identification of a parabolic system. J. Austral. Math. Soc. Ser. B 40, 1–22 (1998)
Yamamoto T.: Historical developments in convergence analysis for Newton’s and Newton-like methods. J. Comput. Appl. Math. 124, 1–23 (2000)
Zeidler, E.: Nonlinear Functional Analysis and its Applications, Part I, Fixed-Point Theorems. Springer-Verlag (1986)
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Benahmed, B., Mokhtar-Kharroubi, H., de Malafosse, B. et al. Quasi-Newton methods in infinite-dimensional spaces and application to matrix equations. J Glob Optim 49, 365–379 (2011). https://doi.org/10.1007/s10898-010-9564-2
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DOI: https://doi.org/10.1007/s10898-010-9564-2