# First-order primal-dual algorithm with correction

## TL;DR

This paper introduces a new primal-dual algorithm with correction and larger step sizes for convex optimization, improving efficiency and convergence rates through adaptive parameter selection and linesearch techniques.

## Contribution

The paper proposes a primal-dual algorithm with an expanded parameter range and correction mechanism, enabling larger step sizes and improved convergence for convex problems.

## Key findings

- Larger step sizes lead to faster convergence.
- The algorithm's parameter range is significantly expanded.
- Numerical experiments confirm improved efficiency.

## Abstract

This paper is devoted to the design of efficient primal-dual algorithm (PDA) for solving convex optimization problems with known saddle-point structure. We present a new PDA with larger acceptable range of parameters and correction, which result in larger step sizes. The step sizes are predicted by using a local information of the linear operator and corrected by linesearch to satisfy a very weak condition, even weaker than the boundedness of sequence generated. The convergence and ergodic convergence rate are established for general cases, and in case when one of the prox-functions is strongly convex. The numerical experiments illustrate the improvements in efficiency from the larger step sizes and acceptable range of parameters.