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A full splitting algorithm for structured difference-of-convex programs

Radu Ioan Bot, Rossen Nenov, Min Tao

TL;DR

This work develops a full-splitting framework for structured, nonconvex, nonsmooth DC programs by introducing an adaptive double-proximal DPFS algorithm that decouples linear operators from nonsmooth terms. The method yields subsequential convergence to approximate stationary points and global convergence under KL, with a concrete adaptive scheme that stabilizes step sizes and provides a rigorous convergence theory via a dedicated merit function. A comprehensive KL-based analysis clarifies when convergence rates are finite, linear, or sublinear, and the approach is validated on audio denoising tasks using a DC-regularizer in the time-frequency domain. The results show that DC regularization improves reconstruction quality and that line-search variants can further enhance convergence, highlighting the practical impact for large-scale, structured DC optimization problems.

Abstract

In this paper, we study a class of nonconvex and nonsmooth structured difference-of-convex (DC) programs, which contain in the convex part the sum of a nonsmooth linearly composed convex function and a differentiable function, and in the concave part another nonsmooth linearly composed convex function. Among the various areas in which such problems occur, we would like to mention in particular the recovery of sparse signals. We propose an adaptive double-proximal, full-splitting algorithm with a moving center approach in the final subproblem, which addresses the challenge of evaluating compositions by decoupling the linear operator from the nonsmooth component. We establish the subsequential convergence of the generated sequence of iterates to an approximate stationary point and prove its global convergence under the Kurdyka-Łojasiewicz property. We also discuss the tightness of the convergence results and provide insights into the rationale for seeking an approximate KKT point. This is illustrated by constructing a counterexample showing that the algorithm can diverge when seeking exact solutions. Finally, we present a practical version of the algorithm that incorporates a nonmonotone line search, which significantly improves the convergence performance.

A full splitting algorithm for structured difference-of-convex programs

TL;DR

This work develops a full-splitting framework for structured, nonconvex, nonsmooth DC programs by introducing an adaptive double-proximal DPFS algorithm that decouples linear operators from nonsmooth terms. The method yields subsequential convergence to approximate stationary points and global convergence under KL, with a concrete adaptive scheme that stabilizes step sizes and provides a rigorous convergence theory via a dedicated merit function. A comprehensive KL-based analysis clarifies when convergence rates are finite, linear, or sublinear, and the approach is validated on audio denoising tasks using a DC-regularizer in the time-frequency domain. The results show that DC regularization improves reconstruction quality and that line-search variants can further enhance convergence, highlighting the practical impact for large-scale, structured DC optimization problems.

Abstract

In this paper, we study a class of nonconvex and nonsmooth structured difference-of-convex (DC) programs, which contain in the convex part the sum of a nonsmooth linearly composed convex function and a differentiable function, and in the concave part another nonsmooth linearly composed convex function. Among the various areas in which such problems occur, we would like to mention in particular the recovery of sparse signals. We propose an adaptive double-proximal, full-splitting algorithm with a moving center approach in the final subproblem, which addresses the challenge of evaluating compositions by decoupling the linear operator from the nonsmooth component. We establish the subsequential convergence of the generated sequence of iterates to an approximate stationary point and prove its global convergence under the Kurdyka-Łojasiewicz property. We also discuss the tightness of the convergence results and provide insights into the rationale for seeking an approximate KKT point. This is illustrated by constructing a counterexample showing that the algorithm can diverge when seeking exact solutions. Finally, we present a practical version of the algorithm that incorporates a nonmonotone line search, which significantly improves the convergence performance.
Paper Structure (16 sections, 21 theorems, 153 equations, 2 figures, 4 tables, 2 algorithms)

This paper contains 16 sections, 21 theorems, 153 equations, 2 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic assumptions and stationary points of structured DC programs
  4. Double proximal full splitting algorithm
  5. Conceptual algorithmic framework
  6. DPFS
  7. Construction of dynamical step sizes to foster convergence
  8. Examining subsequential convergence to KKT points
  9. Adaptive DPFS
  10. Global convergence under the Kurdyka-Ł ojasiewicz property
  11. Convergence rates
  12. On the KL property of the merit function
  13. Application to audio signal processing
  14. Difference of Convex (DC) regularizer
  15. KL exponent of the merit function
  16. ...and 1 more sections

Key Result

Lemma 2.1

\newlabelepsappsol0 Let $g:{{\mathbb R}^n}\rightarrow{\overline{\mathbb R}}$ be a proper, convex and lower semicontinuous function and ${\mathbf{w}}\in{\text{\rm int}}({\text{\rm dom }}g)$. Let $\varepsilon>0$ and ${\@fontswitch{}{\mathcal{}} O}$ be a compact set such that $B({\mathbf{w}},\varepsi where ${\widehat{\varepsilon}}=2\kappa\varepsilon$. Furthermore, if $g$ is differentiable and $\nabl

Figures (2)

  • Figure 1: Convergence of the iterates ${\mathbf{x}}^k$ generated by Adaptive DPFS to their limit $\widebar{\mathbf{x}}$ expressed in terms of $\left \| {\mathbf{x}}^k - \widebar{\mathbf{x}} \right \|$.
  • Figure 2: Comparison of the ISNR achieved by Adaptive DFPS, Adaptive DFPS with line search, and the FBDC method.

Theorems & Definitions (43)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 3.2
  • Definition 3.3
  • Theorem 4.1
  • Proof 1
  • Theorem 4.2
  • Proof 2
  • Remark 4.3: Explicit step size regime
  • Example 4.4
  • ...and 33 more