A Family of Modified Spectral Projection Methods for Nonlinear Monotone Equations with Convex Constraint

Peixin Li,Lin Li,Zhensheng Yu,George S Dulikravich

doi:10.1155/2020/2018243

Peixin Li, Lin Li + Show 2 more

Open Access

https://doi.org/10.1155/2020/2018243

Copy DOI

Abstract

In this paper, we propose a family of modified spectral projection methods for nonlinear monotone equations with convex constraints, where the spectral parameter is mainly determined by a convex combination of the modified long Barzilai–Borwein stepsize and the modified short Barzilai–Borwein stepsize. We obtain a trial point by the spectral method and then get the iteration point by the projection technique. The algorithm can generate a bounded iterative sequence automatically, and we obtain the global convergence of the proposed method in the sense that every limit point is a solution of the nonlinear equation. The proposed method can be used to resolve the large-scale nonlinear monotone equations with convex constraints including smooth and nonsmooth equations. Numerical results for nonlinear equation problems and the ℓ 1 -norm regularization problem in compressive sensing demonstrate the efficiency and efficacy of our method.

Highlights

In this paper, we consider the nonlinear monotone equations with constraints as follows:F x∗􏼁 0, x∗ ∈ Ω, (1)where F: Ω ⟶ Rn is continuous and monotone and Ω⊆Rn is a nonempty closed convex set
In order to observe the performance of the new algorithm, we first discuss the selection of convex combination coefficients λk and make preparations for the numerical experiments
An adaptive truncated cyclic (ATC) scheme is proposed by Dai et al in [32], where λk arg min􏼌􏼌􏼌􏼌􏼌λα􏽥BkB1 +(1 − λ)α􏽥BkB2 − αk− 1􏼌􏼌􏼌􏼌􏼌, (50)

Summary

Introduction

We consider the nonlinear monotone equations with constraints as follows:. Dai et al [32] proposed a family of spectral gradient methods for unconstrained optimization, whose stepsize is determined by a convex combination of the long Barzilai–Borwein (BB) stepsize and the short BB stepsize, and the algorithm presents good numerical performance. Our algorithm can be seen as a family of modified spectral gradient projection methods for the nonlinear monotone equations with convex constraints, where the spectral parameter is mainly determined by a convex combination of the modified long BB stepsize and the modified short BB stepsize.

Algorithm

Convergence Analysis

Preliminary Numerical Results

Application in Compressive Sensing

Conclusion