Accelerated primal dual fixed point algorithm

TL;DR

This paper introduces an Accelerated Primal-Dual Fixed-Point method that uses Nesterov acceleration to efficiently solve composite optimization problems, improving convergence rates and demonstrating effectiveness in practical applications.

Contribution

It presents a novel accelerated primal-dual fixed-point algorithm with decoupled iterations, extending Nesterov's acceleration to non-identity linear operators and achieving faster convergence.

Findings

Improved convergence rate from O(1/k) to O(1/k^2)

Validated efficiency through numerical experiments on logistic regression and CT reconstruction

Demonstrated theoretical and practical advantages of the proposed method

Abstract

This work proposes an Accelerated Primal-Dual Fixed-Point (APDFP) method that employs Nesterov type acceleration to solve composite problems of the form min f(x) + g(Bx), where g is nonsmooth and B is a linear operator. The APDFP features fully decoupled iterations and can be regarded as a generalization of Nesterov's accelerated gradient in the setting where B can be a non-identity matrix. Theoretically, we improve the convergence rate of the partial primal-dual gap with respect to the Lipschitz constant of the gradient of f from O(1/k) to O(1/k^2). Numerical experiments on graph-guided logistic regression and CT image reconstruction are conducted to validate the correctness and demonstrate the efficiency of the proposed method.