# A Globally Linearly Convergent Method for Pointwise Quadratically Supportable Convex-Concave Saddle Point Problems

## TL;DR

This paper introduces a new convergence analysis for the PAPC algorithm applied to convex-concave saddle point problems, demonstrating linear convergence under a relaxed supportability condition.

## Contribution

The paper extends the PAPC algorithm's theoretical guarantees by establishing linear convergence using pointwise quadratic supportability, a relaxation of strong convexity.

## Key findings

- Primal sequence converges R-linearly to an optimal solution.
- Primal-dual sequence converges globally Q-linearly.
- Effective application demonstrated in image denoising and deconvolution tasks.

## Abstract

We study the \emph{Proximal Alternating Predictor-Corrector} (PAPC) algorithm introduced recently by Drori, Sabach and Teboulle to solve nonsmooth structured convex-concave saddle point problems consisting of the sum of a smooth convex function, a finite collection of nonsmooth convex functions and bilinear terms. We introduce the notion of pointwise quadratic supportability, which is a relaxation of a standard strong convexity assumption and allows us to show that the primal sequence is R-linearly convergent to an optimal solution and the primal-dual sequence is globally Q-linearly convergent. We illustrate the proposed method on total variation denoising problems and on locally adaptive estimation in signal/image deconvolution and denoising with multiresolution statistical constraints.