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An Improved Boosted DC Algorithm for Nonsmooth Functions with Applications in Image Recovery

ZeYu Li, Te Qi, TieYong Zeng

TL;DR

The authors address nonsmooth DC programs of the form $\phi(x)=g(x)-h(x)$ by introducing an improved boosted DC algorithm (IBDCA) that enforces monotone descent even when the first DC component is nonsmooth. Building on BDCA, IBDCA anchors line searches to the previous DCA iterate, switching to DCA steps when the extrapolated direction is non-descent, and preserves convergence properties under the Kurdyka–Łojasiewicz framework. They establish global convergence and rates for subanalytic, coercive objective functions, and demonstrate the method on image restoration problems involving Cauchy noise, where IBDCA yields faster convergence and better restoration quality than DCA, nmBDCA, and ADMM. The work provides both theoretical guarantees and practical evidence that IBDCA is an effective, efficient tool for large-scale nonsmooth DC problems with real-world applications in image processing. Overall, IBDCA broadens the applicability of boosted DC strategies and offers a robust approach for solving challenging nonsmooth optimization tasks.

Abstract

We propose a new approach to perform the boosted difference of convex functions algorithm (BDCA) on non-smooth and non-convex problems involving the difference of convex (DC) functions. The recently proposed BDCA uses an extrapolation step from the point computed by the classical DC algorithm (DCA) via a line search procedure in a descent direction to get an additional decrease of the objective function and accelerate the convergence of DCA. However, when the first function in DC decomposition is non-smooth, the direction computed by BDCA can be ascent and a monotone line search cannot be performed. In this work, we proposed a monotone improved boosted difference of convex functions algorithm (IBDCA) for certain types of non-smooth DC programs, namely those that can be formulated as the difference of a possibly non-smooth function and a smooth one. We show that any cluster point of the sequence generated by IBDCA is a critical point of the problem under consideration and that the corresponding objective value is monotonically decreasing and convergent. We also present the global convergence and the convergent rate under the Kurdyka-Lojasiewicz property. The applications of IBDCA in image recovery show the effectiveness of our proposed method. The corresponding numerical experiments demonstrate that our IBDCA outperforms DCA and other state-of-the-art DC methods in both computational time and number of iterations.

An Improved Boosted DC Algorithm for Nonsmooth Functions with Applications in Image Recovery

TL;DR

The authors address nonsmooth DC programs of the form by introducing an improved boosted DC algorithm (IBDCA) that enforces monotone descent even when the first DC component is nonsmooth. Building on BDCA, IBDCA anchors line searches to the previous DCA iterate, switching to DCA steps when the extrapolated direction is non-descent, and preserves convergence properties under the Kurdyka–Łojasiewicz framework. They establish global convergence and rates for subanalytic, coercive objective functions, and demonstrate the method on image restoration problems involving Cauchy noise, where IBDCA yields faster convergence and better restoration quality than DCA, nmBDCA, and ADMM. The work provides both theoretical guarantees and practical evidence that IBDCA is an effective, efficient tool for large-scale nonsmooth DC problems with real-world applications in image processing. Overall, IBDCA broadens the applicability of boosted DC strategies and offers a robust approach for solving challenging nonsmooth optimization tasks.

Abstract

We propose a new approach to perform the boosted difference of convex functions algorithm (BDCA) on non-smooth and non-convex problems involving the difference of convex (DC) functions. The recently proposed BDCA uses an extrapolation step from the point computed by the classical DC algorithm (DCA) via a line search procedure in a descent direction to get an additional decrease of the objective function and accelerate the convergence of DCA. However, when the first function in DC decomposition is non-smooth, the direction computed by BDCA can be ascent and a monotone line search cannot be performed. In this work, we proposed a monotone improved boosted difference of convex functions algorithm (IBDCA) for certain types of non-smooth DC programs, namely those that can be formulated as the difference of a possibly non-smooth function and a smooth one. We show that any cluster point of the sequence generated by IBDCA is a critical point of the problem under consideration and that the corresponding objective value is monotonically decreasing and convergent. We also present the global convergence and the convergent rate under the Kurdyka-Lojasiewicz property. The applications of IBDCA in image recovery show the effectiveness of our proposed method. The corresponding numerical experiments demonstrate that our IBDCA outperforms DCA and other state-of-the-art DC methods in both computational time and number of iterations.
Paper Structure (12 sections, 12 theorems, 76 equations, 11 figures, 1 table, 4 algorithms)

This paper contains 12 sections, 12 theorems, 76 equations, 11 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic conceptions and results in convex analysis
  4. Assumptions
  5. Monotone Boosted DC Algorithms
  6. Improved BDCA
  7. Basic Properties of IBDCA
  8. Global Convergence Properties Based on the Kurdyka–Łojasiewicz Inequality
  9. Application in Image Restoration
  10. Restoring degraded images corrupted by Cauchy noise
  11. Numerical Experiments
  12. Conclusion

Key Result

Proposition 1

Let $f: \mathbb{R}^{m} \rightarrow \mathbb{R} \cup\{+\infty\}$ be an extended real-valued function, then $f$ is convex if and only if $f$ is $\rho$-strongly convex ($\rho>0$) if and only if

Figures (11)

  • Figure 1: The iterative behavior of DCA, nmBDCA, BDCA and IBDCA in \ref{['eg3.5']}.
  • Figure 2: The iterative behavior of DCA, nmBDCA, and improved BDCA with the initial point $x^{0} = (2.2, 0.4)$ in \ref{['eg3.6']}.
  • Figure 3: Test images: cameraman, house, butterfly, man, birds, and squares. Among these images, cameraman, house, butterfly, birds, and squares are of the size $256 \times 256$. Man is of the size $512 \times 512$.
  • Figure 4: Visual comparisons on the selected restored image, which features Cauchy noise with $\gamma = 3$. Zoom in for better visualization.
  • Figure 5: Visual comparisons on the selected restored image, which features Cauchy noise with $\gamma = 3$. Zoom in for better visualization.
  • ...and 6 more figures

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1: Pallaschke2016AnEP
  • Definition 4
  • Proposition 2: Property of Clarke subdifferential, clarke1990optimization
  • Definition 5
  • Remark 1
  • Proposition 3: First-order necessary optimality condition, fukushima1981generalized
  • Proposition 4
  • ...and 21 more