Two spectral gradient projection methods for constrained equations and their linear convergence rate

Jing Liu,Yongrui Duan

doi:10.1186/s13660-014-0525-z

Abstract

Due to its simplicity and numerical efficiency for unconstrained optimization problems, the spectral gradient method has received more and more attention in recent years. In this paper, two spectral gradient projection methods for constrained equations are proposed, which are combinations of the well-known spectral gradient method and the hyperplane projection method. The new methods are not only derivative-free, but also completely matrix-free, and consequently they can be applied to solve large-scale constrained equations. Under the condition that the underlying mapping of the constrained equations is Lipschitz continuous or strongly monotone, we establish the global convergence of the new methods. Compared with the existing gradient methods for solving such problems, the new methods possess a linear convergence rate under some error bound conditions. Furthermore, a relax factor γ is attached in the update step to accelerate convergence. Preliminary numerical results show that they are efficient and promising in practice.

Highlights

In this paper, we consider the problems of finding a solution of the following constrained equations, denoted by CES(F, C),F x∗ = subject to x∗ ∈ C, ( )where F : C → Rn is a given continuous nonlinear mapping and C is a nonempty closed convex set of Rn
In [ ], Yu et al proposed a spectral gradient projection method for solving monotone CES(F, C), which can be applied to nonsmooth constrained equation, and works quite well even for large-scale CES(F, C)
In this paper, motivated by the projection methods in [, ] and the spectral gradient method in [ ], we propose two spectral gradient projection methods for solving nonsmooth constrained equations, which can be viewed as combinations of the wellknown spectral gradient method and the famous hyperplane projection method, and they possess a linear convergence rate under some error bound conditions

Summary

Introduction

In [ ], Yu et al proposed a spectral gradient projection method for solving monotone CES(F, C), which can be applied to nonsmooth constrained equation, and works quite well even for large-scale CES(F, C). Can we design a spectral/conjugate gradient projection method with a linear convergence rate for CES(F, C)? For the system of constrained nonlinear equations, we shall establish the locally R-linear convergence of the spectral gradient method in this paper.

Results

Conclusion