Solution of Ill-Posed Nonconvex Optimization Problems with Accuracy Proportional to the Error in Input Data
- 7 Downloads
The ill-posed problem of minimizing an approximately specified smooth nonconvex functional on a convex closed subset of a Hilbert space is considered. For the class of problems characterized by a feasible set with a nonempty interior and a smooth boundary, regularizing procedures are constructed that ensure an accuracy estimate proportional or close to the error in the input data. The procedures are generated by the classical Tikhonov scheme and a gradient projection technique. A necessary condition for the existence of procedures regularizing the class of optimization problems with a uniform accuracy estimate in the class is established.
Keywords:ill-posed optimization problem error Hilbert space convex closed set Minkowski functional Tikhonov’s scheme gradient projection method accuracy estimate
This work was supported by the Russian Foundation for Basic Research, project no. 16-01-00039a.
- 1.F. P. Vasil’ev, Methods for Solving Extremal Problems (Nauka, Moscow, 1981) [in Russian].Google Scholar
- 2.A. N. Tikhonov, A. S. Leonov, and A. G. Yagola, Nonlinear Ill-Posed Problems (Fizmatlit, Moscow, 1995; CRC, London, 1997).Google Scholar
- 3.A. B. Bakushinskii and M. Yu. Kokurin, Iterative Methods for Solving Ill-Posed Operator Equations with Smooth Operators (Editorial URSS, Moscow, 2002) [in Russian].Google Scholar
- 11.M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreyko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximate Solution of Operator Equations (Nauka, Moscow, 1969; Springer-Verlag, Berlin, 1972).Google Scholar
- 12.V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis: A University Course (RKhD, Moscow, 2011) [in Russian].Google Scholar
- 13.M. M. Vainberg, Variational Method and the Method of Monotone Operators in the Theory of Nonlinear Equations (Nauka, Moscow, 1972) [in Russian].Google Scholar
- 15.A. B. Bakushinskii and A. V. Goncharskii, Iterative Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1989) [in Russian].Google Scholar