A Class of Hybrid Projection Methods with Perturbations - Advanced Research on Computer Education, Simulation and Modeling - page 219

A Class of Hybrid Projection Methods with
Perturbations

Meixia Li

School of Mathematics and Information Science
Weifang University
Weifang, Shandong Province, China
[e-mail]

Abstract. In this paper, a new kind of method which are called hybrid
projection methods with perturbations are proposed and nonmonotone line
search technique is employed. At the same time, global convergence of these
methods is proved only in the case where the gradient function is uniformly
continuous on an open convex set containing the iteration sequence.

Keywords: Hybrid projection method; Perturbation; Global convergence.

1 Introduction

We consider the unconstrained optimization problem

{ }

min f ( x ) : x ∈ R

n

, (1.1)

where : is a

continuously differentiable function. There are many iterative schemes for solving
(1.1). Among them the line search method has the form

n
R denotes the n dimensional Euclidean space and f R

n

→ R

x = x + λ d , k = 0 2 , , (1.2)

K

k + 1 k k k

where

d is a descent direction of

f ( x ) at

x and

λ is a step size. Denote

x the k

k x the current iterate at the k th iteration. Generally, we

initial point and

denote xf by f , respectively.

And the set that consists of all the stationary points of problem (1.1) is denoted by
*

f ( x ) by

f ,

∇ f ( x ) by g ( x ) , ∇ f ( x ) by

g and

( * )

k k

{ }

Ω

*

= x ∈ R

n

g ( x ) = 0

d is generally required to

Ω , that is, . The search direction

satisfy

T
k d

g < 0 . (1.3)

k

There are many methods for solving (1.1), for example, gradient method, conjugate
gradient method, Newton method, quasi-Newton method, trust region method, et al
(see [1,7,8]). In line search methods, if the search direction

d is given at the k th