Existence and Asymptotic Behavior of Second-Order Difference Equation with Tikhonov Regularization. (10th December 2018)

• Record Type: Journal Article
• Authors: Balhag, Aicha; Chbani, Zaki; Riahi, Hassan
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 16(2018)
• Page Start: 1727
• Page End: 1741
• Publication Date: 2018-12-10
• Subjects: Maximal monotone operator -- second-order difference equation -- strong convergence -- Tikhonov regularization; 39A12 -- 39A10 -- 47H05; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1492937
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: We study existence and asymptotic behavior of a bounded solution of the following Tikhonov regularized second-order difference equation (D S x, ε ):u n + 1 − 2 u n + u n − 1 ∈ c n A u n + ε n c n u n ; for n ≥ 1, with ; u 0 = x . where, A is a multivalued maximal monotone operator defined on a Hilbert space andc n, ε n are positive real parameters. We first prove under conditionA − 1 ( 0 ) ≠ ∅, existence of unique bounded solution of (D S x, ε ). For asymptotic behavior, we use a suitable assumption on cn andε n to prove strong convergence of un to the element of minimal norm ofA − 1 ( 0 ) . Some applications are thereafter discussed with respect to minimization and saddle-point problems. Specially, we study the rate of convergence of optimal values in convex minimization and convex-concave problems. We end the paper by concluding remarks and noticing some research perspectives.