TY - JOUR
T1 - The Glowinski-Le Tallec Splitting Method Revisited
T2 - A General Convergence And Convergence Rate Analysis
AU - Ma, Yaonan
AU - LIAO, Lizhi
N1 - Funding Information:
2020 Mathematics Subject Classification. Primary: 90C25; Secondary: 49M27. Key words and phrases. Splitting method, the θ-scheme, convergence, convergence rate, inexactness criterion. The work of L.-Z. Liao was supported in part by grants from Hong Kong Baptist University (FRG) and General Research Fund (GRF) of Hong Kong.
PY - 2021/7
Y1 - 2021/7
N2 - In this paper, we focus on a splitting method called the θ-scheme proposed by Glowinski and Le Tallec in [17, 20, 27]. First, we present an elaborative convergence analysis in a Hilbert space and propose a general convergent inexact θ-scheme. Second, for unconstrained problems, we prove the convergence of the θ-scheme and show a sublinear convergence rate in terms of objective value. Furthermore, a practical inexact θ-scheme is derived to solve l2-loss based problems and its convergence is proved. Third, for constrained problems, even though the convergence of the θ-scheme is available in the literature, yet its sublinear convergence rate is unknown until we provide one via a variational reformulation of the solution set. Besides, in order to relax the condition imposed on the θ-scheme, we propose a new variant and show its convergence. Finally, some preliminary numerical experiments demonstrate the efficiency of the θ-scheme and our proposed methods.
AB - In this paper, we focus on a splitting method called the θ-scheme proposed by Glowinski and Le Tallec in [17, 20, 27]. First, we present an elaborative convergence analysis in a Hilbert space and propose a general convergent inexact θ-scheme. Second, for unconstrained problems, we prove the convergence of the θ-scheme and show a sublinear convergence rate in terms of objective value. Furthermore, a practical inexact θ-scheme is derived to solve l2-loss based problems and its convergence is proved. Third, for constrained problems, even though the convergence of the θ-scheme is available in the literature, yet its sublinear convergence rate is unknown until we provide one via a variational reformulation of the solution set. Besides, in order to relax the condition imposed on the θ-scheme, we propose a new variant and show its convergence. Finally, some preliminary numerical experiments demonstrate the efficiency of the θ-scheme and our proposed methods.
KW - convergence
KW - convergence rate
KW - inexactness criterion
KW - Splitting method
KW - the θ-scheme
UR - http://www.scopus.com/inward/record.url?scp=85104052173&partnerID=8YFLogxK
U2 - 10.3934/jimo.2020040
DO - 10.3934/jimo.2020040
M3 - Journal article
AN - SCOPUS:85104052173
SN - 1547-5816
VL - 17
SP - 1681
EP - 1711
JO - Journal of Industrial and Management Optimization
JF - Journal of Industrial and Management Optimization
IS - 4
ER -