An Indefinite-Proximal-Based Strictly Contractive Peaceman-Rachford Splitting Method (Journal of Computational Mathematics, 2023)
The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints. However, itsconvergence was not guaranteed without extra requirements. He et al. (SIAM J. Optim.24: 1011-1040, 2014) proved the convergence of a strictly contractive Peaceman-Rachfordsplitting method by employing a suitable underdetermined relaxation factor. In this paper,we further extend the so-called strictly contractive Peaceman-Rachford splitting methodby using two different relaxation factors. Besides, motivated by the recent advances on theADMM type method with indefinite proximal terms, we employ the indefinite proximalterm in the strictly contractive Peaceman-Rachford splitting method. We show that theproposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method isconvergent 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 theproposed method.
Keywords: Indefinite proximal; Strictly contractive; Peaceman-Rachford splitting method; Convex minimization; Convergence rate.