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Primal-dual algorithm for weakly convex functions under sharpness conditions

Ewa Bednarczuk, The Hung Tran, Monika Syga

TL;DR

This work introduces a modified duality gap function, which is a lower bound of the standard duality gap function, and identifies the area around the set of saddle points where the convergence of the primal-dual algorithm is obtained.

Abstract

We investigate the convergence of the primal-dual algorithm for composite optimization problems when the objective functions are weakly convex. We introduce a modified duality gap function, which is a lower bound of the standard duality gap function. Under the sharpness condition of this new function, we identify the area around the set of saddle points where we obtain the convergence of the primal-dual algorithm. We give numerical examples and applications in image denoising and deblurring to demonstrate our results.

Primal-dual algorithm for weakly convex functions under sharpness conditions

TL;DR

This work introduces a modified duality gap function, which is a lower bound of the standard duality gap function, and identifies the area around the set of saddle points where the convergence of the primal-dual algorithm is obtained.

Abstract

We investigate the convergence of the primal-dual algorithm for composite optimization problems when the objective functions are weakly convex. We introduce a modified duality gap function, which is a lower bound of the standard duality gap function. Under the sharpness condition of this new function, we identify the area around the set of saddle points where we obtain the convergence of the primal-dual algorithm. We give numerical examples and applications in image denoising and deblurring to demonstrate our results.

Paper Structure

This paper contains 14 sections, 15 theorems, 138 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Primal-Dual relationship of Composite Minimization Problem with weakly convex function
  4. Sharpness
  5. Primal-Dual algorithm with Dual Update
  6. Primal-Dual algorithm with primal update first
  7. Solution to the Primal Problem
  8. Numerical Examples
  9. Inf-sharpness
  10. Setting:
  11. $\ell_1$-regularization
  12. Setting:
  13. Image Denoising with Total Variation
  14. Setting:

Key Result

Proposition 1

Let $X$ be a Hilbert space, and $F$ is proper lsc $\rho$-weakly convex on $X$ with $\rho\geq 0$. Then $\text{\normalfont dom}\, \partial_{\rho} F \subset \text{\normalfont dom}\, F,$ and for every $\varepsilon> 0$ we have the equality $\text{\normalfont dom}\, \partial_{\rho}^{\,\varepsilon} h = \te

Figures (12)

  • Figure 1: Lagrange function from Example \ref{['exm: inf-sharp 2nd case']}
  • Figure 2: Example \ref{['exm: inf-sharp 2nd case']}; From left to right: contour plots of $\mathcal{H}_\mathcal{L}(x,y) - \mu \sqrt{x^2+y^2}$ at $\mu = 1, 0.5, 0.1$.
  • Figure 3: Example \ref{['ex: Lagrange inf-sharp 3']}; From left to right: contour plots of $\mathcal{H}_\mathcal{L}(x,y) - \mu \text{dist} ((x,y),(0,0))$ at $\mu = 0.5, 0.9, 1$.
  • Figure 4: Function $\mathcal{K}(x,y)$ from Example \ref{['ex: lagrange inf-sharp 4']}.
  • Figure 5: Example \ref{['ex: lagrange inf-sharp 4']}; From left to right: contour plots of $\mathcal{H}_\mathcal{K}(x,y) - \mu \text{dist} ((x,y),(0,0))$ at $\mu = 0.5, 0.9, 1$.
  • ...and 7 more figures

Theorems & Definitions (29)

  • Definition 1: $\rho$-weak convexity
  • Example 1
  • Definition 2
  • Definition 3: Global proximal ${\varepsilon\text{-subdifferential}}$
  • Proposition 1: bednarczuk2023calculus
  • Theorem 1
  • Lemma 1
  • Definition 4: Sharpness
  • Definition 5: Inf-Sharpness
  • Example 2
  • ...and 19 more