We present the definition and study the dependence on the initial approximation of the asymptotic rate of convergence of a two-layer symmetrizable iterative method of the variational type. The explicit expression is obtained for the substantial (with respect to the Lebesgue measure) range of its values. Its domain of continuity is described.
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R. Fletcher, “A limited memory steepest descent method,” Math. Program. A, 135, 413–436 (2012).
R. de Asmundis, D. di Serafino, F. Riccio, and G. Toraldo, “On spectral properties of steepest descent methods,” in: Optim. Methods and Software for Inverse Problems (2012), pp. 1–20.
K. van den Doel and U. Ascher, “The chaotic nature of faster gradient descent methods,” Univ. British Columbia, Canada, 1–27 (2011).
Y. Narushima, T. Wakamatsu, and H Yabe., “Extended Barzilai–Borwein method for unconstrained minimization problems,” Pacif. J. Optim., 6, No. 3, 591–613 (2010).
M. Andretta, E. G. Birgin, and J. M. Martinez, “Partial spectral projected gradient method with active-set strategy for linearly constrained optimization,” Numer. Algorithms, 53, 23–52 (2010).
J. Barzilai and J. M. Borwein, “Two-point step size gradient methods,” IMA J. Numer. Anal., 8, 141–148 (1988).
A. A. Samarskii and E. S. Nikolaev, Methods for the Solution of Finite-Difference Equations [in Russian], Nauka, Moscow (1978).
H. Akaike, “On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method,” Ann. Inst. Statist. Math. Tokyo, 11, 1–16 (1959).
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 12, pp. 1622–1635, December, 2013.
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Zhuk, P.F., Musina, A.A. Asymptotic Rate of Convergence of a Two-Layer Iterative Method of the Variational Type. Ukr Math J 65, 1793–1808 (2014). https://doi.org/10.1007/s11253-014-0898-7
- Lebesgue Measure
- Initial Approximation
- Steep Descent
- Variational Type
- Iterative Sequence