An inexact golden ratio primal-dual algorithm with linesearch step for a saddle point problem

TL;DR

This paper introduces an inexact primal-dual algorithm with linesearch for saddle point problems, achieving faster convergence rates and efficiency improvements, especially under strong convexity conditions, demonstrated through numerical experiments.

Contribution

It proposes a novel inexact primal-dual algorithm with linesearch that allows larger stepsizes and achieves various convergence rates based on convexity assumptions.

Findings

Proves convergence and establishes an O(1/N) ergodic rate.

Achieves accelerated O(1/N^2) rate with strong convexity.

Demonstrates effectiveness on sparse recovery and image deblurring tasks.

Abstract

In this paper, we propose an inexact golden ratio primal-dual algorithm with linesearch step(IP-GRPDAL) for solving the saddle point problems, where two subproblems can be approximately solved by applying the notations of inexact extended proximal operators with matrix norm. Our proposed IP-GRPDAL method allows for larger stepsizes by replacing the extrapolation step with a convex combination step. Each iteration of the linesearch requires to update only the dual variable, and hence it is quite cheap. In addition, we prove convergence of the proposed algorithm and show an O(1/N) ergodic convergence rate for our algorithm, where N represents the number of iterations. When one of the component functions is strongly convex, the accelerated O(1/N2) convergence rate results are established by choosing adaptively some algorithmic parameters. Furthermore, when both component functions are strongy convex, the linear convergence rate results are achieved. Numerical simulation results on the sparse recovery and image deblurring problems illustrate the feasibility and efficiency of our inexact algorithms.