TY - JOUR
T1 - An Indefinite-Proximal-Based Strictly Contractive Peaceman-Rachford Splitting Method
AU - Gu , Yan
AU - Jiang , Bo
AU - Han , Deren
JO - Journal of Computational Mathematics
VL - 6
SP - 1017
EP - 1040
PY - 2023
DA - 2023/11
SN - 41
DO - http://doi.org/10.4208/jcm.2112-m2020-0023
UR - https://global-sci.org/intro/article_detail/jcm/22102.html
KW - Indefinite proximal, Strictly contractive, Peaceman-Rachford splitting method, Convex minimization, Convergence rate.
AB - <p style="text-align: justify;">The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints. However, its
convergence was not guaranteed without extra requirements. He et al. (SIAM J. Optim.
24: 1011-1040, 2014) proved the convergence of a strictly contractive Peaceman-Rachford
splitting method by employing a suitable underdetermined relaxation factor. In this paper,
we further extend the so-called strictly contractive Peaceman-Rachford splitting method
by using two different relaxation factors. Besides, motivated by the recent advances on the
ADMM type method with indefinite proximal terms, we employ the indefinite proximal
term in the strictly contractive Peaceman-Rachford splitting method. We show that the
proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is
convergent and also prove the $o(1/t)$ convergence rate in the nonergodic sense. The numerical tests on the $l_1$ regularized least square problem demonstrate the efficiency of the
proposed method.</p>