Optimal Leveraging of Smoothness and Strong Convexity for Peaceman--Rachford Splitting

TL;DR

This work tackles the problem of minimizing $f(x)+g(x)$ for two smooth strongly convex components by applying a novel variant of the Peaceman–Rachford splitting that redistributes strong convexity and smoothness via a quadratic augmentation. By introducing parameters $\delta$, $\eta$, and $\tau$, the authors construct an equivalent problem whose PRS iteration converges linearly with a rate $r(\tau,\eta,\delta)$, and they derive an explicit optimal rate $r^*$ that improves on existing bounds. The rate is shown to be independent of the redistribution parameter in a certain sense, and optimal parameter choices $\delta^*$, $\eta^*(\delta)$, $\tau^*(\delta)$ yield the best linear convergence; the results recover known bounds in limiting cases. Numerical experiments in academic scenarios and image-processing tasks (restoration and CT) demonstrate practical improvements in convergence speed and computational efficiency with the proposed PRS lev method.

Abstract

In this paper, we introduce a simple methodology to leverage strong convexity and smoothness in order to obtain an optimal linear convergence rate for the Peaceman--Rachford splitting (PRS) scheme applied to optimization problems involving two smooth strongly convex functions. The approach consists of adding and subtracting suitable quadratic terms from one function to the other so as to redistribute strong convexity in the primal formulation and smoothness in the dual formulation. This yields an equivalent modified optimization problem in which each term has adjustable levels of strong convexity and smoothness. In this setting, the Peaceman--Rachford splitting method converges linearly to the solution of the modified problem with a convergence rate that can be optimized with respect to the introduced parameters. Upon returning to the original formulation, this procedure gives rise to a modified variant of PRS. The optimal linear rate established in this work is strictly better than the best rates previously available in the general setting. The practical performance of the method is illustrated through an academic example and applications in image processing.

Figures (8)

• Figure 1: Theoretical bounds and numerical errors for random instances in different dimensions.
• Figure 2: Normalized error vs Iteration number.
• Figure 3: Original Images
• Figure 4: $\sigma =0.5$, $\rho = 0.11$
• Figure 5: $\sigma =0.6$, $\rho = 0.013$

Theorems & Definitions (10)

• Theorem 2.1
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Theorem 3.4
• proof
• Remark 3.5
• proof