New inertial proximal gradient methods for unconstrained convex optimization problems

Peichao Duan,Qinxiong Bu,Yiqun Zhang

doi:10.1186/s13660-020-02522-6

Peichao Duan, Qinxiong Bu + Show 1 more

Open Access

https://doi.org/10.1186/s13660-020-02522-6

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Abstract

The proximal gradient method is a highly powerful tool for solving the composite convex optimization problem. In this paper, firstly, we propose inexact inertial acceleration methods based on the viscosity approximation and proximal scaled gradient algorithm to accelerate the convergence of the algorithm. Under reasonable parameters, we prove that our algorithms strongly converge to some solution of the problem, which is the unique solution of a variational inequality problem. Secondly, we propose an inexact alternated inertial proximal point algorithm. Under suitable conditions, the weak convergence theorem is proved. Finally, numerical results illustrate the performances of our algorithms and present a comparison with related algorithms. Our results improve and extend the corresponding results reported by many authors recently.

Highlights

Let H be a real Hilbert space with the inner product ·, · and the induced norm ·, and let C be a nonempty closed convex subset of H
We will deal with the unconstrained convex optimization problem of the following type: min f (x) + g(x), (1.1)
For instance, [23], where the author introduced the properties and iterative methods for the lasso as a special case of (1.1); due to the involvement of the l1 norm, which promotes sparsity, we can get a good result on solving the corresponding problem

Summary

Introduction

Let H be a real Hilbert space with the inner product ·, · and the induced norm · , and let C be a nonempty closed convex subset of H. We will deal with the unconstrained convex optimization problem of the following type: min f (x) + g(x),. Many researchers have already proposed some algorithms to solve problem (1.1) and have discussed a lot of weak and strong convergence results, such as [1, 6, 12, 23, 25], just to name a few. Lots of important optimization problems can be cast in this form. For instance, [23], where the author introduced the properties and iterative methods for the lasso as a special case of (1.1); due to the involvement of the l1 norm, which promotes sparsity, we can get a good result on solving the corresponding problem

Duan et al Journal of Inequalities and Applications

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