A primal-dual flow for affine constrained convex optimization

School of Mathematical Sciences, Peking University, Beijing 100871, China

^* Corresponding author: luohao@math.pku.edu.cn

Received: 10 March 2021
Accepted: 20 April 2022

Abstract

We introduce a novel primal-dual flow for affine constrained convex optimization problems. As a modification of the standard saddle-point system, our flow model is proved to possess the exponential decay property, in terms of a tailored Lyapunov function. Then two primal-dual methods are obtained from numerical discretizations of the continuous problem, and global nonergodic linear convergence rate is established via a discrete Lyapunov function. Instead of solving the subproblem of the primal variable, we apply the semi-smooth Newton iteration to the inner problem with respect to the multiplier, provided that there are some additional properties such as semi-smoothness and sparsity. Finally, numerical tests on the linearly constrained l₁-l₂ minimization and the tot al-variation based image denoising model have been provided.