A Modified Spectral PRP Conjugate Gradient Projection Method for Solving Large-Scale Monotone Equations and Its Application in Compressed Sensing

Jie Guo,Zhong Wan

doi:10.1155/2019/5261830

Abstract

In this paper, we develop an algorithm to solve nonlinear system of monotone equations, which is a combination of a modified spectral PRP (Polak-Ribière-Polyak) conjugate gradient method and a projection method. The search direction in this algorithm is proved to be sufficiently descent for any line search rule. A line search strategy in the literature is modified such that a better step length is more easily obtained without the difficulty of choosing an appropriate weight in the original one. Global convergence of the algorithm is proved under mild assumptions. Numerical tests and preliminary application in recovering sparse signals indicate that the developed algorithm outperforms the state-of-the-art similar algorithms available in the literature, especially for solving large-scale problems and singular ones.

Highlights

In many fields of sciences and engineering, solution of a nonlinear system of equations is a fundamental problem
Since it is critical to choose an appropriate step length to improve the performance of the iterate scheme (3), as well as determination of search directions, we present an inexact line search rule to determine αk in (3)
Our aim is to show that the line search rule (17) termisnuaptepsofsine itthelayt,wfoitrhsaopmoesiittievreatseteipndleenxgetshsαukc.hBayscko∗n,trcaodnidcittiioonn, (17) does not hold

Summary

Introduction

In many fields of sciences and engineering, solution of a nonlinear system of equations is a fundamental problem. In [1, 2], both a Nash economic equilibrium problem and a signal processing problem were formulated into a nonlinear system of equations. In the past five decades, numerous algorithms and some software packages in virtue of those powerful algorithms have been developed for solving the nonlinear system of equations. In practice, no any algorithm can efficiently solve all the systems of equations arising from sciences and engineering. It is significant to develop a specific algorithm to solve the problems with different analytic and structural features [17, 18]

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Methods

Conclusion