# Golden ratio primal-dual algorithm with linesearch

arXiv: https://arxiv.org/abs/2105.07108 · HTML: https://mlsanity.com/paper/2105.07108/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2021.

## Built on by (most cited first, 2 of 2)

- [Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient](https://mlsanity.com/paper/2301.04431/) (2023)
- [Auto-conditioned primal-dual hybrid gradient method and alternating direction method of multipliers](https://mlsanity.com/paper/2410.01979/) (2024)

## Abstract (arXiv)

Golden ratio primal-dual algorithm (GRPDA) is a new variant of the classical Arrow-Hurwicz method for solving structured convex optimization problem, in which the objective function consists of the sum of two closed proper convex functions, one of which involves a composition with a linear transform. In this paper, we propose a linesearch strategy for GRPDA, which not only does not require the spectral norm of the linear transform but also allows adaptive and potentially much larger stepsizes. Within each linesearch step, only the dual variable needs to be updated, and it is thus quite cheap and does not require any extra matrix-vector multiplications for many special yet important applications, e.g., regularized least squares problem. Global convergence and cal O(1/N) ergodic convergence rate results measured by the primal-dual gap function are established, where N denotes the iteration counter. When one of the component functions is strongly convex, faster cal O(1/N2) ergodic convergence rate results are established by adaptively choosing some algorithmic parameters. Moreover, when both component functions are strongly convex, nonergodic linear converge results are established. Numerical experiments on matrix game and LASSO problems illustrate the effectiveness of the proposed linesearch strategy.