Abstract
We examine the convergence rate of approximations generated by Tikhonov’s scheme as applied to ill-posed constrained optimization problems with general smooth functionals on a convex closed subset of a Hilbert space. Assuming that the solution satisfies a source condition involving the second derivative of the cost functional and depending on the form of constraints, we establish the convergence rate of the Tikhonov approximations in the cases of exact and approximately specified functionals.
Similar content being viewed by others
References
F. P. Vasil’ev, Methods for Solving Extremal Problems (Nauka, Moscow, 1981) [in Russian].
A. N. Tikhonov, A. S. Leonov, and A. G. Yagola, Nonlinear Ill-Posed Problems (Nauka, Moscow, 1995; CRC, London, 1997).
M. Yu. Kokurin, “Convexity of the Tikhonov functional and iteratively regularized methods for solving irregular nonlinear operator equations,” Comput. Math. Math. Phys. 50 (4), 620–632 (2010).
A. B. Bakushinskii and A. V. Goncharskii, Iterative Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1989) [in Russian].
H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, Dordrecht, 2000).
V. A. Morozov, “Estimates of the accuracy of the regularization of nonlinear unstable problems,” Comput. Math. Math. Phys. 35 (9), 1139–145 (1995).
S. Fitzpatrick and R. R. Phelps, “Differentiability of the metric projection in Hilbert space,” Trans. Am. Math. Soc. 270, 483–501 (1982).
R. B. Holmes, “Smoothness of certain metric projections on Hilbert space,” Trans. Am. Math. Soc. 184, 87–100 (1973).
M. Yu. Kokurin, “Sourcewise representability conditions and power estimates of convergence rate in Tikhonov’s scheme for solving ill-posed extremal problems,” Russ. Math. 58 (7), 61–70 (2014).
M. Yu. Kokurin, “Reduction of variational inequalities with irregular operators on a ball to regular operator equations,” Russ. Math. 57 (4), 26–34 (2013).
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control (Fizmatlit, Moscow, 2005) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © M.Yu. Kokurin, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 7, pp. 1103–1112.
Rights and permissions
About this article
Cite this article
Kokurin, M.Y. Convergence rate estimates for Tikhonov’s scheme as applied to ill-posed nonconvex optimization problems. Comput. Math. and Math. Phys. 57, 1101–1110 (2017). https://doi.org/10.1134/S0965542517070090
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542517070090