Doubly relaxed forward-Douglas–Rachford splitting for the sum of two nonconvex and a DC function

In this paper, we consider a class of structured nonconvex nonsmooth optimization problems whose objective function is the sum of three nonconvex functions, one of which is expressed in a difference-of-convex (DC) form. This problem class covers several important structures in the literature includi...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 104; no. 2; p. 35
Main Authors: Dao, Minh N., Pham, Tan Nhat, Tung, Phan Thanh
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2025
Springer Nature B.V
Subjects:
Algorithms
Computational Mathematics and Numerical Analysis
Convergence
Convex analysis
Hilbert space
Machine learning
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Optimization
Performance evaluation
Sparsity
Splitting
Synthetic data
Theoretical
65K05
three-operator splitting
global convergence
Composite optimization
Douglas–Rachford splitting
90C26
49M27
nonconvex optimization
DC program
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:In this paper, we consider a class of structured nonconvex nonsmooth optimization problems whose objective function is the sum of three nonconvex functions, one of which is expressed in a difference-of-convex (DC) form. This problem class covers several important structures in the literature including the sum of three functions and the general DC program. We propose a splitting algorithm and prove the subsequential convergence to a stationary point of the problem. The full sequential convergence, along with convergence rates for both the iterates and objective function values, is then established without requiring differentiability of the concave part. Our analysis not only extends but also unifies and improves recent convergence analyses in nonconvex settings. We benchmark our proposed algorithm with notable algorithms in the literature to show its competitiveness on a low rank matrix completion problem and a simultaneously sparse and low-rank matrix estimation problem. Our algorithm exhibits very competitive results compared to notable algorithms in the literature, on both synthetic data and public dataset.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-025-02950-w