Ill-Posed Nonlinear Optimization Problems and Uniform Accuracy Estimates of Regularization Methods. (23rd November 2020)

• Record Type: Journal Article
• Authors: Kokurin, M. Yu.
• Is Part Of: Numerical functional analysis and optimization. Volume 41:Number 15(2020)
• Page Start: 1887
• Page End: 1900
• Publication Date: 2020-11-23
• Subjects: Accuracy estimate -- convex set -- error level -- ill-posed optimization problem -- Minkowski functional -- regularizing operator -- strong convexity -- weakly lower semicontinuous functional; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2020.1845729
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: We investigate the ill-posed problem of minimizing weakly lower semicontinuous functionals on a convex closed set in a Hilbert space. The functionals to be minimized are available with errors. We prove that a necessary condition for the existence of a regularization procedure with a uniform accuracy estimate on the class of weakly lower semicontinuous functionals is the well-posedness of related optimization problems. We also study regularization methods that provide uniform accuracy estimates, linear with respect to the error level in cost functionals. Under appropriate assumptions on the feasible set, we establish a necessary condition for the existence of such regularizing procedures. In the case of unconstrained problems, the obtained condition reduces to the local strong convexity of functionals under minimization.