A continuity property of a parametric projection and an iterative process for solving linear variational inequalities

iterative process for solving linear variational inequalities

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Zhu, S. Q. (1988). A continuity property of a parametric projection and an iterative process for solving linear variational inequalities. (Memorandum COSOR; Vol. 8806). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum COSOR 88-06 A continuity property of a parametric projection and an iterative process for solving linear variational inequalities

by Zhu Siquan

Eindhoven, Netherlands February 1988

Abltract F(x,J,G) :=

p~(x)

(y) TariatioDAl Linear proces,; Iterative

-•

Projection inequality .

p~( )

(y) = Solution of min

/I

J - u

I/

a

x

u

t

K(x)

*

n*

Let the point-to-tet mapping K : RD - - ) RD be continuoUl (R consists of all

dOled conTex subset in Rn )

and

as

be the set of all symmetric pOlitin

definite

matrices of

lile n. Define

Department of mathematici

Xi

I

an Jiaotong uniTercity

China

A CONTINUITY PROPERTY OF A PARAMETRIC PROJECTION AND AN ITERATNE

PROCESS FOR SOLVING LINEAR VARIATIONAL INEQUALITTES

Siquan Zhu

Facultyofmathematics and computing science

Eindhoven university of technology

the Nethula.ndJ (current address)

Key words

Where

If

x " = xTGx x t Rn .

In

thil ptlper, it is lhown thtlt the operator F : RnIRnIGS

-

>

an defined. by

ia continuous at each point (X,y,G) t

RD.xRDxas.

Using this result, we can pIOn the convergence ofan iteratiTe procell for lolving linear va.natioD&! inequality problems •

-2-Inwdution

L....I- ..1.~ • • n nill nill

...w: pomt-to-tet mappmg K : R - - ) R be continuoUJ (R CODJirt, of allclOied

conTeI lublet inRn; the definition of K being continuoUJ will be given in section

2)

&Ild

as

be the let ofalllJID.Illetric positive definite matrices with ,isen. Define

P~(I)

(y):= Solution of min

II

y - u

II

_G u (

K(I)

Where" J:

II

=

ITI I ( Rn. Now we C8Il define &Il operator F :

R~RnIGS

- - ) Rn

F(I.y,G) :=

p~(J:)

(y) (1.1)

Inorder to study the variationlll ineqUiility problem, [Chen.1982J hu proved that F(I:,y,O) is a

continuou, mapping ofI and y when G :: I. In tID' paper, we mow that the operator F : Rn I

Rn I:

as - - )

Rn defined by (1.1) i, continuous at each point (I:,y,G) , RnIRn I

as.

U,ing

thi.relUlt, we can prove the convergence of 8Il iterative procell for ,olving a linear variational inequality problem. The iterative method (algorithm 3.2) designed in thi, paper i. a

generalisation of [Men.1977J', method, wIDch i, a very general iterative .cheme for solving the linea.r complementarity problem, which i. a special cue of the linear variational inequality problem. WitlUn our knowledge there are few method. .pecially tend to lolve the linear

T&Iiational inequalitie., although there are many iteratiTe methodl presented for .olTing the linea.r complementarity problem•. In fact, the method di,culled in thi, paper for lolving line&l' Tariational inequalitiel i, to reduce a line&l' Tariational inequality into a .eria .imple mb-linear Tariational inequalities wlUch C&ll be ewy .oIved. In .ection 2 we will prOTe the

continuity of the operator F(x, y, a). The iterative process will be pre.ented in .ection 3, 8Ild

the convergence theorem win be proved by using the remIt obta.ined in .ection 2.

2 The continuity of F

Let GS be the let ofall .ymmetric n by n po~tj,e definite matrices. For each G (GS, one can define 8Il inner product

v

U f C (2.2)

o

I = Pg (y) (I-y ,U-I)0

>

Pg

(I)

:= 101 min II

I -

y 11₀ ( 2.1 ) Y t C

Lemma 2.1 [Kin.1980] The following two statement. are equiTalent

< , )

:=

< , )

I II

/1:=

II

III

1

II I 11

₀

:= [<I,I)

0]2

Lemma 2.2 [Kin.1980] Foreach I,y f Rn

II

Pg

(I) -

Pg

(y) 11

0

~

II

I -

Y

/1

0

(2.3)

1)

2)

andanODn

then we have

Suppose C ( Rn

is

a dOled conva subset, andthe projectionPg : Rn- - ) Cis defined by

HO=I,~t

and

n n*

Let K : R - - ) R be continuoUi : K

i.

both upper .emicontinuou. and lower aemicontinuoUl. Recall that K is upper semicontmuous[Ber.1963] atI t

an,

iffor each open set

(2.5)

(2.!)

Let

F :

RDxRnIGS - - ) Rn Which. is defined by 4

-Define

Ncontaining f(I), there emts a neighborhood Uof I such thatfeU)

=

U f(J) ( N, and Kis

ytU

lower aemicontinuoUi [Ber.1963] at I t Rn, if for each. open let N meeting f(I~ there is a neighborhood U of I such. that

f(y)

meeh N for all y t U. IT K is upper ( or lower ) semicontinuous at each point in Rn, we say

K

is upper ( or lower) IemicontinuOUI in Rn.

the following lemma is provedin[Chan.1982].

...

Lemma 2.3 Suppose that K :Rn - - ) Rn is continuoUM, thenf : RnIRn- - ) Rn defined by (2.!) iscontinuo~.

Denote

Inthis section, we aregoing to prove

Theorem 2.! Let K : Rn ) Rn

*

be continuous.Then the operator F : RDxRDxos - - ) Rn defined by

(2.5)

is continuous at each point (I,y,O) t RnIR~GS.

(2.6)

( k - ) . )

(k--)1»)

and

Henceit

is

IUfficient to prove From lemma 2.3. onekIWWS that

By

(2.2)

and

and

Settingy = v

_k

in

(2.7)

andy

=

Uk in

(2.8)

gives

) 0 I.e.

6

-'then

hence

1

~mrn

(Gk)

II

uk- Yk

ll

<

lI

u_{k - Ykllk}

1

<

~

;"U

_{(G k)} min

(II

Y-YO

II

+

II

YO -Yk

II)

ytK(I

k)

Now it is aufficient to prove that {( min

II

Y- YO

II )}

is bounded. Let

ytK(I

_k)

By the continuity of

K(I),

there eIi~{'\} &uch that

(3.1) 8

-*

Ij;

<

I - I j f(.a. .>

>

:> 0

hence { (i'k} I i bounded. Thiil competes the proof.

J.e.

Let f : Rn - - - ) Rllb~ a mapping and C be a closed convex IUbset in Rn, the variational inequality problem ca.u. be itatt:'d at; followti

It!

Problem VIP(f,ill FiItd A { C lu(;h that

H C = Rn, theJl }itublem VIP(f.C) is equivalent to find a aolution of the equa.tion f(I) = O.

Given alliatr!I A t RllIll a.ud .:1vectoL b t Rn. When f(I) = AI - b, we say Problem VIP(f,C) is aline& ·va:riCitjolLd.llliequaljtj pfoblt:'lli and denoted by LVIP(A.b.C).

iii

It ill well knOW-ll [FiiIlg.l982; Kjn~1980;etc.] thatI satidies (3.1)ifand only if

where G t. GS. II\:J.LL~, jf fit; (U.iI1.UIUUllt> 6.Ild C is bounded, then by Brouwer's fixed point

theorem [KjI1.1980. d(], littJ11x.iIlVJP(fjC) alwil}i; has a solution. Iff is strictly monotone, then VIP(f,C) h&i i:it illutit IJJlc [;ulutiull i:W.d jf f iti titrongl} llionotone. VIP(fC) has a unique

lolutio.n.[Kill,1930,dt.j

III

Find I t Rn luch that

+

Problem LCP~

III iii III

AI - b t R:

<

I AI - b

>

= 0 (3.-t) IfAt GS, then we have an interesting fact

... A ( -1 )

I = Pc A b (3.3)

i.e. the lolution of LVIP(A,b,C) iI the projection of the IOlution of the equation AI=b under

the norm

II II

_{A on C. Using (3.2) one can check (3.3).}

If C = R~, then the problem LVIP(A,b,C) il equivalent to the linear complementary problem i.e.

In thil lection, we are only interested in the numerical. method for lolving LVIP(A,b,C), especially, in designing an iteration method which suits any kinds of closed convu sublet Cin

Rn. There are many iterative methodi for solving LCP(A,b1 in [Pang.1982], one can lee a

compact cuney. Motivated by leveral. earlier works, [Man.1971] proposed a fairly general. iterative algorithm. for lolving LCP(A,b1 which is

Algorithm 3.1[Ma.n.1977] LetI

O~ 0

where

O ( l ( l i ",>0

{Dk} aDd {~} are bounded sequences of matricel in RDIn, with each

1\

being a po.itive diagonal matrix .atilfying

(3.6)

(3.7.2) for lome 0-

>

O.

Up to our knowledge, there a.re few methodl for solving LVIP(A,b,C1 when C i••upposed

to be an arbitrary dOled convex IUbset.Inthis lection, we a.re going to fill partof this gap and

to generalise algorithm 3.1 for solving LVIP(A,b,C).

Algorithm 3.2 Let I Ot C 2) ifD= 0-1 (0-

>

0), then where

-10-o

< '"

~ "'k

<

1 (3.7.1) {D

k} and {El } axe bounded sequence. of matrices in .nxn, with each Dk t

as

matriI

atilfying

PI?.

_Dk

?.

0-1

for lome 0- and

P

>

0 •

Rema.rk 1) In algorithm 3.2, we do not u.e

w,

becaUie, in fact, the

w

in algorithm 3.1 can be put into

1\.

2) Hone choo.e. D

k to be diagonal and C

=

R~, then algorithm 3.1 and algorithm 3.2 are identical. One can prove this Itatement u.ing the first part of the following lemma

(3.8)

yT_C

.\1

1D_k

+

E

k -

+

A)y

~

..,

It

yll 2 Vk Vy ( R

n

Thin tach

&Ccumulation point of { Ik } genera.ted by algorithm 3.2, is a ,olution of LVIP(

A,b,C ).

Proof Let

and {I

1 } are generatedby dgorithm 3.2, then

.One can directly use the definition to check this lemma.

Theorem 3.1 We posit

1)

A is &ymme1;ric;

2) (3.7.1) and (3.7.2) hold;

12

-~ 0

(3.10)

Sup~e that thereenlts a subsequence {IkJin { I

k}such that J

.

I_k.

>

I

J

BecaUie

{g(I

_{k) }}i8 monotone decrea..ing and

one get.

From

(3.10),

we know that

II

(I

_k

+

_{1 -Ii)}

II

>()

. ConJequentiy"wehaTe ,i ]

s;ince

i

I _k

.+

_{1- - )} I J

-14-wea1Johave

Yk-

>

%

1

Since {

I\}

and { E_k}arebounded sequences, without lOll generality we can aaume that

1\.

>

_{D ; Ek.}

>

E

J 1

ByaBlumption, D i& stillinas. Accordingto (3.5)

ow uling theorem 2.4, we get

.. D · 1 ·

% = Pc

[I -

D-

(AI -

b

)1

.

Y (3.2), one knows that I is a solution of LVIP(A,b,C). This completes the proof.

Coronary 3.4 We pout the assumptions oftheorem 3.4.IfA is positive then

•

%k

>

I

•

. _hereI is the unique solutionofLVIP(A,b,C).

Proof Firat, we know that

%.

=

P~

(A-

1

b)

_ the unique IOlution of LVIP(A,b,C). According to theorem 3.4, each accumulation point of

{~k} isa IOlution of LVIP(A,b,C). Therefore it is sufficient to prove that

{Ik}

is bounded.

VI

The .implest way toUle algorithm 3.2i. to put

:gk ::

0 i

r}::

~

I for .ome

~

>

0

one bowl that {

II}

iJ bounded. Thi. completes the proof.

in this cue, according to lemma 3.3, each .tep in algorithm 3.2 i. to compute

Yl = Pc [Ik -

~

-1 ( AIl - b ) ]

Remark Thefollowing two problem. are worthto be conaidered.

I) the con:9'ergence of algorithm 3.2under a weaker condition then Abeing positive; 2) [Ahn.1981] proved the convergence of algorithm 3.1 for the cue that A i. not IJmmetric.the lame problem can be posed for algorithm 3.2.

Acknowledgement

lid like to thank Dr. C. Praagman for hi. careful reading of the original manUlCfipt and

many mggeltiou. I am still grateful to Prof. You Zhaoyoug for hi. guidance on this subject

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