Convergence study on strictly contractive Peaceman–Rachford splitting method for nonseparable convex minimization models with quadratic coupling terms

Abstract

The alternating direction method of multipliers (ADMM) and Peaceman Rachford splitting method (PRSM) are two popular splitting algorithms for solving large-scale separable convex optimization problems. Though problems with nonseparable structure appear frequently in practice, researches on splitting methods for these problems remain to be scarce. Very recently, Chen et al. (Math Program 173(1–2):37–77, 2019) extended the 2-block ADMM to linearly constrained nonseparable models with quadratic coupling terms and established its convergence. However, theoretical researches about nonseparable PRSM or its variants are still lacking. To fill the gap, in this paper we focus on the strictly contractive PRSM (SC-PRSM) applied to 2-block linearly constrained convex minimization problems with quadratic coupling objective functions. Under mild conditions, we prove the convergence of our proposed SC-PRSM and establish its o(1/k) convergence rate. Moreover, we implement the SC-PRSM to solve a problem of calculating the Euclidian distance between two ellipsoids, and compare its performance with three ADMM type algorithms. The results show the nonseparable SC-PRSM outperforms the other three algorithms in terms of both the iteration numbers and CPU time.