A four-operator splitting algorithm for nonconvex and nonsmooth optimization

TL;DR

The paper tackles four-term nonconvex nonsmooth optimization by introducing a four-operator splitting algorithm that extends the Davis–Yin framework to handle $\Psi(x)= f(x)+g(x)+h(x)+p(x)$ with $f$ and $h$ smooth and proximable, $g$ proximable, and $p$ a continuous weakly convex perturbation. It establishes global subsequential convergence to stationary points and an $\mathcal{O}(1/T)$ rate for a first-order measure, under carefully derived stepsize conditions; the analysis yields larger allowable stepsizes than existing results for the classical DYS algorithm when $p=0$. The method specializes to known schemes in limiting cases and demonstrates practical speedups by leveraging the distinct structures of $f$ and $h$, with experiments showing faster convergence than three-term analogs and proximal DC variants. The work also compares favorably to recent approaches like Dao2024, offering broader applicability without reformulating the problem via Moreau envelopes, and highlights potential for adaptive stepsize strategies to further improve empirical performance. Overall, the proposed four-operator splitting framework broadens the applicability and efficiency of splitting methods for complex nonconvex nonsmooth optimization problems.

Abstract

In this work, we address a class of nonconvex nonsmooth optimization problems where the objective function is the sum of two smooth functions (one of which is proximable) and two nonsmooth functions (one proper, closed and proximable, and the other continuous and weakly concave). We introduce a new splitting algorithm that extends the Davis-Yin splitting (DYS) algorithm to handle such four-term nonconvex nonsmooth problems. We prove that with appropriately chosen stepsizes, our algorithm exhibits global subsequential convergence to stationary points with a stationarity measure converging at a global rate of $1/T$, where $T$ is the number of iterations. When specialized to the setting of the DYS algorithm, our results allow for larger stepsizes compared to existing bounds in the literature. Experimental results demonstrate the practical applicability and effectiveness of our proposed algorithm.

Figures (1)

• Figure 1: Comparison of stepsize upper bounds (denoted by $\bar{\alpha}$) for the DYS algorithm in this paper (with $\tau = 1$) and in BianZhang21.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Remark 2.2
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof