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A proximal splitting algorithm for generalized DC programming with applications in signal recovery

Tan Nhat Pham, Minh N. Dao, Nima Amjady, Rakibuzzaman Shah

TL;DR

This work tackles a generalized DC optimization problem of the form $F(x)=f(x)+h(x)-g(x)$, where $f$ has a Lipschitz continuous gradient, $h$ is lower semicontinuous, and $g$ is convex. It introduces the backward-Douglas–Rachford (BDR) algorithm, a proximal-splitting method that uses Fenchel conjugate-based evaluations for the concave part and three-operator splitting to handle the remaining terms. The authors prove subsequential convergence to a critical point and, under the KL property, global convergence of the full sequence with convergence rates, without requiring differentiability of the concave component. Numerical experiments on synthetic and real-data signal recovery tasks show competitive performance and notably faster convergence compared to leading splitting methods, highlighting potential applications in rapid data imputation for time-series in power systems. Overall, the approach extends DC programming techniques to a broader class of nonconvex problems with weak convexity assumptions and delivers practical, scalable performance improvements.

Abstract

The difference-of-convex (DC) program is an important model in nonconvex optimization due to its structure, which encompasses a wide range of practical applications. In this paper, we aim to tackle a generalized class of DC programs, where the objective function is formed by summing a possibly nonsmooth nonconvex function and a differentiable nonconvex function with Lipschitz continuous gradient, and then subtracting a nonsmooth continuous convex function. We develop a proximal splitting algorithm that utilizes proximal evaluation for the concave part and Douglas--Rachford splitting for the remaining components. The algorithm guarantees subsequential convergence to a {\color{black}critical} point of the problem model. Under the widely used Kurdyka--Łojasiewicz property, we establish global convergence of the full sequence of iterates and derive convergence rates for both the iterates and the objective function values, without assuming the concave part is differentiable. The performance of the proposed algorithm is tested on signal recovery problems with a nonconvex regularization term and exhibits competitive results compared to notable algorithms in the literature on both synthetic data and real-world data.

A proximal splitting algorithm for generalized DC programming with applications in signal recovery

TL;DR

This work tackles a generalized DC optimization problem of the form , where has a Lipschitz continuous gradient, is lower semicontinuous, and is convex. It introduces the backward-Douglas–Rachford (BDR) algorithm, a proximal-splitting method that uses Fenchel conjugate-based evaluations for the concave part and three-operator splitting to handle the remaining terms. The authors prove subsequential convergence to a critical point and, under the KL property, global convergence of the full sequence with convergence rates, without requiring differentiability of the concave component. Numerical experiments on synthetic and real-data signal recovery tasks show competitive performance and notably faster convergence compared to leading splitting methods, highlighting potential applications in rapid data imputation for time-series in power systems. Overall, the approach extends DC programming techniques to a broader class of nonconvex problems with weak convexity assumptions and delivers practical, scalable performance improvements.

Abstract

The difference-of-convex (DC) program is an important model in nonconvex optimization due to its structure, which encompasses a wide range of practical applications. In this paper, we aim to tackle a generalized class of DC programs, where the objective function is formed by summing a possibly nonsmooth nonconvex function and a differentiable nonconvex function with Lipschitz continuous gradient, and then subtracting a nonsmooth continuous convex function. We develop a proximal splitting algorithm that utilizes proximal evaluation for the concave part and Douglas--Rachford splitting for the remaining components. The algorithm guarantees subsequential convergence to a {\color{black}critical} point of the problem model. Under the widely used Kurdyka--Łojasiewicz property, we establish global convergence of the full sequence of iterates and derive convergence rates for both the iterates and the objective function values, without assuming the concave part is differentiable. The performance of the proposed algorithm is tested on signal recovery problems with a nonconvex regularization term and exhibits competitive results compared to notable algorithms in the literature on both synthetic data and real-world data.
Paper Structure (8 sections, 4 theorems, 21 equations, 5 figures, 10 tables, 1 algorithm)

This paper contains 8 sections, 4 theorems, 21 equations, 5 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed algorithm and convergence analysis
  4. Numerical results
  5. Sparse signal reconstruction on synthetic data and real power system voltage data
  6. Reconstruction of signals from incomplete and noisy signals
  7. An analytical example of \ref{['eq:P']} with nonconvex $h$
  8. Conclusion

Key Result

Proposition 2.1

Let $f\colon \mathbb{R}^d\to (-\infty, +\infty]$ be proper and let $x,v \in \mathbb{R}^d$. Then the following hold:

Figures (5)

  • Figure 1: Convergence of the BDR with different $\nu$ values.
  • Figure 2: Electric voltage signal and its sparsity (for illustration purpose, only the first 2000 samples of the signal are shown here).
  • Figure 3: An illustration of the case study.
  • Figure 4: Load data and its sparsity.
  • Figure 5: Contour lines of $\ell_{1-2}$ vs $\ell_{1,\mathcal{T}}-\ell_2$.

Theorems & Definitions (5)

  • Proposition 2.1
  • Remark 3.3
  • Lemma 3.4
  • Theorem 3.5: Subsequential convergence
  • Theorem 3.6: Full sequential convergence