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Oct.

2007

CHINESE

JOURNAL

ENGINEERING

MATHEMATICS

No.

Article

ID:

1005-3085 (2007) 05-0788-07

New

Nonmonotonic

Self-adaptive

ust

Region

AIgorithm

with

Line

Search*

YANG

Yang

,2,

SUN

Wen-yu

(1- School of Mathematics and Physics Science

时

10U

Institute of Technology, Xuzhou 221008;

2- School of Mathematics and Computer Science

, Nanjing Normal University, Nanjing 210097)

Abstract:

We propose a new

trust

region algorithm which

the

trust

region radius

updated

variable rate. Moreover

the

new algorithm performs a backtracking line search from the

ed point instead of resolving

the

trust

region subproblem. A nonmonotonic criterion

also used

speed up

the

convergence.

establish

the

global convergence of

the

new

algorithm. Numerical results are also presented.

Keywords:

unconstrained optimization;

trust

region; line search; nonmonotonic; global convergence

Classification:

AMS(2000) 90C30

CLC

number:

022

Document

code:

Introduction

consider

the

following

unconstrained

optimization

problem

JEi:Lf(z)

、‘

1·''

咱

''''-‘、、

where

f(x)

is twice

continuously

differentiable.

'fr

ust

region

method

is a

very

effective

and

robust

technique

for solving

problem

(1)

find

the

global

optimal

solution.

the

kth

step

calculates

trial

step

solving

the

following

trust

region

subproblem

Lκ

嘈

t--q

，"

十

儿

<一

、}/

GHH

飞

岛川川川

A4EU

队

(2)

(3)

where

儿

f(Xk)

and

f(Xk)

are

the

function

value

and

the

gradient

vector

the

current

approximation

iterate

respectively

n x n

symmetric

matrix

which

may

the

exact

Hessian

H(Xk)

the

quasi-Newton

approximation

and

ßk

> 0 is

the

trust

region

radius.

this

paper

the

notation

11.11

denotes

the

Euclidean

norm

Rn.

After

obtaining

the

trust

region

method

computes

the

ratio

between

the

actual

reduction

and

the

predicted

reduction

the

objective

function

check

dk is

acceptable.

Then

the

trust

region

radius

ßk

also

updated

according

the

value

, since

reflect

the

extent

which

the

quadratic

Received

date:

2005-11-25.

Biography:

Yang Yang

(Born

1981)

Female

, Master,

Teachi

吨

Ass

istant. Research

field:

Nonlinear Programming.

*Foundation

item: The National Natural

Science

Foundation

China

(10231060)

, the Special Research

nd of Doctoral Program of Higher Education

China

(20040319003)

and the Graduates' Creative Project

Jiangsu Province, China

NO.

Yang

Sun

Wenyu:

Nonmonotonic Self-adaptive

τrust

Region

Algorithm with

Line

789

model

如

(d)

approximates

the

objective

function f(Xk +

d).

Hei[lJ

proposed

a self-adaptive

trust

region

algorithm

which

ð.

k is

updated

R-function

a variable

rate.

Since

the

line search

strategy

and

the

trust

region

method

have

their

own

merits

, Nocedal

and

Yuan[2J suggested a

combination

the

trust

region

and

line

techniques.

Zhu[3J

also

proposed

method

combine

trust

region

and

line

techniques.

onmonotonic

line search

technique

for

unconstrained

optimization

is first

proposed

Grippo

[4J.

Due

its

high efficiency,

many

authors

generalized

the

nonmonotonic

tech-

nique

trust

region

methods

and

proposed

nonmonotone

trust

region

methods

, see

叫-

this

paper

, we will modify

and

improve

the

self-adaptive

trust

region

algorithm

proposed

adopting

the

above

ideas:

backtracki

吨

line

and

nonmo

∞

tonic

technique.

the

section we discuss

our

new

algorithm.

section 3,

the

global convergence

property

the

new

algorithm

is established.

section 4,

the

results

numerical

experiments

are

reported.

Our

Algorithm

First

all, we

introduce

the

definition

the

R-function.

Definition

Any

one-dimensional

function

(t)

defined in R =

(一∞，+∞)

with

the

parameter

ηε(0

，

1) is

R-function

and

only if

satisfies:

R,., (t) is non-decreasing in

(一∞，+∞);

limt-+

一∞鸟

(t)

函，

whereβ1

巳

，

is a

small

constant;

R，.，

(t)

三

一节，

for all t

ηwhereγ1ε(0

，

一

β1)

is a

constant;

札

η)=1+

节，

whereγ2

E (0,

+∞)

is a

constant;

的

1imt

→+∞

R，.，

(t)

岛，

where

ß2

巳

+币，+∞)

is a

constant

the

definition we

can

easily

get

the

following

important

properties

R-functions:

Theorem

R-function

札

(where

ηε(0

，

1)) satisfies

0<β1

三

R，.，

(t)

三

一

γ1

< 1,

ε(

一矶时，

1<1+γ2

三

Rη

(t)

三

β2<

十∞

，

tE[

，+∞)

solve

the

subproblem

(2)-(3)

obtain

the

trial

step

satisfying

<<P

(0)

一

<<P

k(d

)

主

rllgkll

叫ð.

，

llgkII/IIBkll}

and

d[gk

三

-rllgkll

min{

ð.

k, IIgkll/llBkll}

(4)

(5)

(6)

(7)

Relaxing

the

accepting

condition

, we use

the

nonmonotonic

trust

region technique.

Let

fl(k) = f(Xl(k)) =

^}l~aX"

，

f(Xk-j)

。三

三

m(k)

where

m(O)

= 0

and

m(k)

min{m(k

-1)

+ 1,

and

M is a

nonnegative

integer.

Then

the

actual

reduction

f(x)

is defined

Aredk = fl(k) - f(Xk + dk)

and

the

predictive

reduction

f(x)

is Predk =

<<P

k(O)

一

<<P

k(dk).

Now

we give a

description

our

new

algorithm.

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