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Primal-Dual algorithms for Abstract convex functions with respect to quadratic functions

Ewa Bednarczuk, The Hung Tran

TL;DR

This work develops primal-dual methods for saddle-point problems with abstract $\Phi$-convexity relative to a quadratic function class, extending Chambolle–Pock type algorithms to the $\Phi_{lsc}^{\mathbb{R}}$ setting via a $\Phi_{lsc}^{\mathbb{R}}$-proximal operator. It presents both full and reduced (L=I) variants, deriving convergence results under mild assumptions (e.g., $\tau\sigma\|L\|^2<1$, $1+2\sigma a_n>\sqrt{\sigma\tau}\|L\|$) and providing ergodic rates for weakly convex objectives. The paper also introduces subdifferential-based conjugate updates for the dual and analyzes their convergence to KKT points. Numerical experiments on toy 1D problems and a binary-tomography application demonstrate the practical effectiveness and robustness of the proposed methods across convex and nonconvex regimes.

Abstract

We consider the saddle point problem where the objective functions are abstract convex with respect to the class of quadratic functions. We propose primal-dual algorithms using the corresponding abstract proximal operator and investigate the convergence under certain restrictions. We test our algorithms by several numerical examples.

Primal-Dual algorithms for Abstract convex functions with respect to quadratic functions

TL;DR

This work develops primal-dual methods for saddle-point problems with abstract -convexity relative to a quadratic function class, extending Chambolle–Pock type algorithms to the setting via a -proximal operator. It presents both full and reduced (L=I) variants, deriving convergence results under mild assumptions (e.g., , ) and providing ergodic rates for weakly convex objectives. The paper also introduces subdifferential-based conjugate updates for the dual and analyzes their convergence to KKT points. Numerical experiments on toy 1D problems and a binary-tomography application demonstrate the practical effectiveness and robustness of the proposed methods across convex and nonconvex regimes.

Abstract

We consider the saddle point problem where the objective functions are abstract convex with respect to the class of quadratic functions. We propose primal-dual algorithms using the corresponding abstract proximal operator and investigate the convergence under certain restrictions. We test our algorithms by several numerical examples.
Paper Structure (18 sections, 11 theorems, 118 equations, 3 figures, 5 algorithms)

This paper contains 18 sections, 11 theorems, 118 equations, 3 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. -Convex Functions
  4. Proximal Operator
  5. Primal-Dual Algorithms
  6. Fully - Setting
  7. Applying convex subdifferentials to the conjugate
  8. Using --Subdifferentials for conjugate function
  9. Numerical Examples
  10. Toy Examples
  11. Application
  12. Setting:
  13. Numerical tests
  14. Funding
  15. Solving minimization problem on the class of elementary functions
  16. ...and 3 more sections

Key Result

Proposition 2.6

Let $f:X\to (-\infty,+\infty]$ be proper, $x\in \mathrm{dom}\, f$ and $\phi\in \Phi_{lsc}^\mathbb{R}$, we have

Figures (3)

  • Figure 1: Performance of Primal-dual algorithms for Example \ref{['ex1']}.
  • Figure 2: $\Phi_{lsc}^\mathbb{R}$-Primal Dual algorithms for Example \ref{['ex2: nonconvex example']}.
  • Figure 3: Comparisons between different solutions. From left to right: original image, least squares solution, thresholded least squares solution, solution obtained with the dual method proposed in kadu2019convex, solution obtained with our model.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2: $\Phi_{lsc}^\mathbb{R}$-conjugate
  • Example 2.3
  • Remark 2.4
  • Definition 2.5: $\Phi_{lsc}^\mathbb{R}$-subdifferentials
  • Proposition 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 30 more