A Two-Step Inertial Primal-Dual Algorithm for Minimizing the Sum of Three Functions

Meng Wen,Yuchao Tang,Zhiwei Xing,Jigen Peng

doi:10.1109/access.2019.2951578

Abstract

In this paper, we introduce a two-step inertial primal-dual algorithm (TSIPD) for solving the minimizations of the sum a smooth function with Lipschitzian gradient and two non-smooth convex functions with linear operators. This is a complete splitting approach, in the sense that non-smooth functions are treated separately by their proximity operators. In order to prove the convergence of the TSIPD, we transform the problem into a fixed point equation with good performance, and prove the convergence of the algorithm base on the fixed point theory. This work brings together and significantly extends several classical splitting schemes, like the primal-dual method (PD3O) proposed by Yan, and the recent three-operator splitting scheme proposed by Davis and Yin. The validity of the proposed method is demonstrated on an image denoising problem. Numerical results show that our iterative algorithm (TSIPD) has better performance than the original one (PD3O).

Highlights

Minimizing the sum of finite functions has always been the core of mathematical optimization research
We provide a two-step inertial primal-dual algorithm for solving Problem (1)
Since the total variation term x TV can be expressed by a combination of convex function φ and linear transformation matrix B, i.e., x TV = φ(Bx)

Summary

INTRODUCTION

Minimizing the sum of finite functions has always been the core of mathematical optimization research. Condat [6] introduced a primal-dual splitting method to solve Problem (2), and proved the convergence of the algorithm. Li and Zhang [8] introduced a general primal-dual splitting algorithm to solve Problem (2), and proved the convergence of algorithms which included Condat and Vu’s algorithm as a special case. They proved the iteration schemes have O(1/k) convergence rate in the ergodic sense and the sense of partial primal-dual gap.

PRELIMINARIES

PROOFS OF CONVERGENCE

CONNECTIONS TO OTHER ALGORITHMS

APPLICATIONS

NUMERICAL EXPERIMENTS

CONCLUSION