CONVERGENCE THEOREMS OF ITERATIVE METHODS FOR SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES

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193

CONVERGENCE THEOREMS OF ITERATIVE METHODS FOR SYSTEM OF NONCONVEX VARIATIONAL INEQUALITIES

Issara Inchan

Department of Mathematics and Computer, Uttaradit Rajabhat University, Uttaradit, Thailand Email: [email protected]

ABSTRACT

In this paper, we suggest and analyze some iterative methods for solving the system of nonconvex variational inequalities with two nonlinear mappings by using the projection technique. We show that the strong convergence of these iterative methods to the solutions of the system of nonconvex variational inequalities. Consequently, the results presented in this paper can be viewed as an improvement and refinement of some known results from the literature.

Keywords: Lipschitz continuous; strongly monotone mapping; Nonconvex; Uniformly prox regular.

2000 Mathematics Subject Classification: 46C05, 47D03, 47H09, 47H10, 47H20.

1. INTRODUCTION

The theory of variational inequalities is a branch of the mathematical sciences dealing with general equilibrium problems. It has a wide range of applications in economics, operations research, industry, physical, and engineering sciences. Many research papers have been written lately, both on the theory and applications of this field. Important connection with main areas of pure and applied science have been made, see for example [3, 6, 7] and the references cited therein.

Variational inequalities theory, which was introduce by Stampacchia [20], provides us with a simple, natural general and unified framework to study a wide class of problems arising in pure and applied science. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solutions to important classes of problems. On the other hand, it also enables us to develop highly efficient and powerful new numerical methods for solving, for example, obstacle, unilateral, free, moving, and complex equilibrium problems.

Using the projection operator, Noor [9] has established the equivalence between the general nonconvex variational inequalities and the fixed point problem. In this paper, we prove a new characterization of the projection operator for the prox-regular sets. Using this characterization, one can easily show that the nonconvex projection operator is Lipschitz continuous, which is a new result.

In 2010, N. Petrot [19], introduced some existence theorems and provide the conditions for existence solutions of the variational inequalities problems in nonconvex setting and prove the strongly monotonic assumption of the mapping may not need for the existence of solutions.

In this paper we show that the convergence of iterative methods requires two nonlinear mappings with Lipschitz continuous and strongly monotone mappings. Our main result represents an improvement and refinement of the known results.

2. PRELIMINARIES

Let

C

be a closed subset of a real Hilbert space

H

with inner product

, 

and norm

 

respectively. Let us recall the following well-known definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis.

Definition 2.1 Let

u  H

be a point not lying in

C

. A point

v  C

is called a closest point or a projection of

u

onto

C

if

d

_C

( u ) =  u  v 

when

d

_C is a usual distance. The set of all such closest points is denoted by

P

_C

(u )

; that is,

}.

= ) ( : {

= )

( u v C d u  u v 

P

_C



_C



(2.1)

Definition 2.2 Let

C

be a subset of

H

. The proximal normal cone to

C

at

x

is given by

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194

)}.

( 0;

>

: {

= )

( x z H x P x z

N

_C^P

   

_C

 

(2.2)

The following characterization of

N

_C^P

(x )

can be found in [4].

Lemma 2.3 Let

C

be a closed subset of a Hilbert space

H

. Then

. ,

, 0,

>

)

( x if and only if z y x y x

²

y C

N

z 

_C^P

           

(2.3)

Clark et al. [5] and Poliquin et al. [18] have introduced and atudied a new class of nonconvex sets, which are called uniformly prox-regular sets. This class or uniformly prox-regular sets has played an important part in many nonconvex applications such as optimization, dynamic systems, and differential inclusions.

Definition 2.4 For a given

r  (0, ]

, a subset

C

of

H

is said to be uniformly

r

-prox-regular with respect to

r

if, for all

x  C

and for all

0 =  z  N

_C^P

( x )

, one has

. 2 ,

, 1 x x

²

x C

x r z x

z      

  



^(2.4)

It is well known that a closed subset of a Hilbert space is convex if and only if it is proximally smooth of radius

r > 0

. Thus, in Definition 2.4, the case of

r = 

, the uniform

r

-prox-regularity

C

is equivalent to convexity of

C

. Then, it is clear that the class of uniformly prox-regular sets is sufficiently large to include the class

p

-convex sets,

C

1,1 submanifolds (possibly with boundary) of

H

, the images under a

C

^1,1 diffeomorphism of convex sets, and many other nonconvex sets; see [5, 18].

Let

C

be a closed subset of a real Hilbert space

H

with is uniformly

r

-prox-regular(nonconvex), set

}

<

) , ( : {

:= x H d x C r

C

_r



. For given nonlinear mappings

T

₁

, T

₂

: C

_r

 H

, we consider the problem of finding

x

^*

, y

^*

 C

_r such that

0 >

, 0,

,

^*

*

1



 T y  x  y x  x    x  C

_r



0,

>

, 0,

,

^*

*

2



 T x  y  x x  y    x  C

_r



(2.5)

which is called the system of nonconvex variational inequalities.

If

T

₁

= T

₂

= T

, then the problem (

SNVI ) is equivalent to finding

x

^*

, y

^*

 C

_r such that

0 >

, 0,

,

^*

*



 Ty  x  y x  x    x  C

_r



0. >

, 0,

,

^*

*



 Tx  y  x x  y    x  C

_r



(2.6)

It is worth mentioning that if

T

₁

= T

₂

= T

and

x

^*

= y

^*

= u

, then problem ( 2.5) is equivalent to finding

u  C

_r

such that

, 0,

, v u v C

_r

Tu     



^(2.7)

which is known as nonconvex variational inequalities introduced and studied by Bounkhel et. al.[1] and Noor [10, 11].

It is known that problem ( 2.7 ) is equivalent to finding

u  C

_r such that

), ( 0 Tu N

^P

u

Cr





(2.8)

which

N

^P

(u )

Cr denote the normal cone of

C

_r at

u

. The problem (2.8) is called the variational inclusion associated with nonconvex variational inequalities ( 2.7).

We now recall the well-known lemmas of the uniform prox-regular sets.

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195

Lemma 2.5 [19] Let

C

be a nonempty closed subset of

H

,

r  (0, ]

and set

}.

<

) , ( : {

=

; x H d x C r

C

_r



If

C

is uniform

r

-uniformly prox-regular, then the following hold:

(1) for all

x  C

_r

, P

_C

( x ) =  

,

(2) for all

s  (0, r ), P

_C is Lipschitz continuous with constant

s r t

_s

r

= 

on

C

_s, (3) the proximal normal cone is closed as a set-valued mapping.

Let

H

be a real Hilbert space. A mapping

T : H  H

is called

  strongly monotone

if there exists a constant

 > 0

such that

, , x y  x y 

²

Ty

Tx     

 

(2.9)

for all

x , y  H .

A mapping

T

is called

  Lipschitz

if there exists a constant

 > 0

such that

 ,



 Tx  Ty   x  y

(2.10)

for all

x , y  H .

Lemma 2.6 In a real Hilbert space H, there holds the inequality

1.

 x  y 

²

  x 

²

 2  y , x  y  x , y  H

and

 x  y 

²

=  x 

²

 2  x , y    y 

², 2.

 tx  (1  t ) y 

²

= t  x 

²

 (1  t )  y 

²

 t (1  t )  x  y 

²

,  t  [0,1]

.

3. MAIN RESULTS

In this section we first establish the equivalent between the system of nonconvex variational inequalities (2.5) with the projection technique.

Lemma 3.1 For given

x

^*

, y

^*

 C

_r is a solution of system of nonconvex variational inequalities ( 2.5), if and only if

], [

=

^* ₁ ^*

*

P y T y

x

_C

 

], [

=

^* ₂ ^*

*

P x T x

y

_C

 

(3.1)

where

P

_C is the projection of

H

onto the uniformly prox-regular set

C

_r.

Proof. Let

x

^*

, y

^*

 C

_r be a solution of (2.5), from (2.8) and for a constant

 > 0

^{, we have}

] [

) )(

(

= ) (

0 T

₁

y

^*

x

^*

y

^*

N x

^*

I N

^P

x

^*

y

^*

T

₁

y

^*

Cr P

Cr

 



      



if and only if

], [

= ] [

) (

=

¹ ^* ₁ ^* ^* ₁ ^*

*

I N y T y P y T y

x

^P _C

Cr

 

  



^

where we have used the well-known fact that

= ( 

^P

)

^¹

Cr

C

I N

P 

.

Similarly, we obtain

].

[

=

^* ₂ ^*

*

P x T x

y

_C

 

This prove our assertions.

Algorithm 3.2 For arbitrarily chosen initial points

x

₀

, y

₀

 C

_r, the sequence

{ x

_n

}

and

{ y

_n

}

in the following way:

0 >

], [

=

_C _n



₂ _n



n

P x T x

y 

0,

>

], [

) (1

=

₁

1



_n _n



_n _C _n



_n



n

x P y T y

x

_

  

(3.2)

(4)

196 where

{ 

_n

}

is a sequence in

[0,1]

.

Now, we suggest and analyze the following explicit projection method (3.2) for solving the system of nonconvex variational inequalities (2.5).

Theorem 3.3 Let

C

be a uniformly

r

-prox-regular closed subset of a Hilbert space

H

^{, and let}

H

C T

T

₁

,

₂

: 

be such that

T

₁ is a



₁-Lipschitz continuous and



₁-strongly monotone mapping,

T

₂ is a



₂

-Lipschitz continuous and



₂-strongly monotone mapping. If there exists constant

 ,  > 0

such that

1 2 1

2 1 2

2 2 1 2 2 1 2

1 2

1

( 1) , 1 <

|<

| 









    



_s _s

s s s

s

t t t

,

<

1 1) ,

|< (

|

2 2 2

2 2 2 2 2 2 2

2 2

2











    



_s _s

s s s

s

t t t

t t

t

^(3.3)

where

s r t

_s

r

= 

for some

s  (0, r )

. If the sequence of positive real number



_n

 [0,1]

with



_n^₌₀



_n

= 0

, then the sequences

{ x

_n

}

and

{ y

_n

}

obtained from Algorithm 3.2 converge to a solution of the system of nonconvex variational inequalities (2.5).

Proof. Let

x

^*

, y

^*

 C

_r be a solution of (2.5) and from Lemma 2.6, we have



 x

_n_₁

 x

^*

= (1  

_n

) x

_n

 

_n

P

_C

[ y

_n

  T

₁

y

_n

]  x

^*



 (1 )( ) ( [ ] [ ])

=  

_n

x

_n

 x

^*

 

_n

P

_C

y

_n

  T

₁

y

_n

 P

_C

y

^*

  T

₁

y

^*



 [ ] [ ]

)

(1  

_n

x

_n

 x

^*

 

_n

P

_C

y

_n

  T

₁

y

_n

 P

_C

y

^*

  T

₁

y

^*





 ( ) ( )

)

(1  

_n

x

_n

 x

^*

 

_n

t

_s

y

_n

  T

₁

y

_n

 y

^*

  T

₁

y

^*



. ) (

) (

) (1

=  

_n

 x

_n

 x

^*

  

_n

t

_s

 y

_n

 y

^*

  T

₁

y

_n

 T

₁

y

^*



(3.4) Since

T

₁ are both



₁-Lipschitz continuous and



₁-strongly monotone mapping and from Lemma

lem2.1 , we consider

2

* 1 1 2

* 1 1

* 2

* 1 1

*

) ( ) = 2 ,

(     

 y

_n

 y   T y

_n

 T y y

_n

 y    y

_n

 y T y

_n

 T y    T y

_n

 T y

2

* 2

1 2 2

* 1 2

*

 2    

 y

_n

 y  y

_n

 y  y

_n

 y

   

. )

2 (1

=  

₁

 

²



₁²

 y

_n

 y

^*



²

It follows that

. 2

1 ) (

)

(

^* ₁ ₁ ^*



₁ ² ₁²



^*



 y

_n

 y   T y

_n

 T y       y

_n

 y

(3.5)

Replace (eq3.5) into (eq3.4), we have

. 2

1 )

(1

^* ₁ ² ₁² ^*

*

1

    

 x

_n_

 x   

_n

x

_n

 x  

_n

t

_s

     y

_n

 y

(3.6)

On the other hand, we can compute that



 y

_n

 y

^*

= P

_C

[ x

_n

  T

₂

x

_n

]  y

^*



 [ ] [ ]

= P

_C

x

_n

  T

₂

x

_n

 P

_C

x

^*

  T

₂

x

^*



 ( x T

₂

x ) ( x

^*

T

₂

x

^*

) t

_s _n

 

_n

  



. ) (

) (

= t

_s

 x

_n

 x

^*

  T

₂

x

_n

 T

₂

x

^*



(3.7)

Similarly, from

T

₂ are both



₂-Lipschitz continuous and



₂-strongly monotone mapping, we have

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197

2

* 2 2 2

* 2 2

* 2

* 2 2

*

) ( ) = 2 ,

(     

 x

_n

 x   T x

_n

 T x x

_n

 x    x

_n

 x T x

_n

 T x    T x

_n

 T x

2

* 2 2 2 2

* 2 2

*

 2    

 x

_n

 x  x

_n

 x  x

_n

 x

   

. )

2 (1

=  

₂

 

²



₂²

 x

_n

 x

^*



²

It follows that

. 2

1 ) (

)

(

^* ₂ ₂ ^*



₂ ² ₂²



^*



 x

_n

 x   T x

_n

 T x       x

_n

 x

(3.8)

Replace (eq3.8) into (eq3.7), we have

. 2

1

₂ ² ₂² ^*

*

  

 y

_n

 y  t

_s

     x

_n

 x

(3.9)

Moreover, from (eq3.6 ) and (eq3.9) we put



₁

= t

_s

1  2 

₁

 

²



₁²

, 

₂

= t

_s

1  2 

₂

 

²



₂² , it follows that



 x

_n_₁

 x

^*

 (1  

_n

) x

_n

 x

^*

 

_n



₁



₂

x

_n

 x

^*



^*

2

1

) )

(1 (1

=     

_n

x

_n

 x . )

) (1

(1

₁ ₂ ₀ ^*

0

=



 x x

i n

i



  ^ ^ ^

^(3.10)

Since



^_n₌₀



_n

= 

and (eq3.3), we obtain

0. = ) ) (1

lim (1

1 2

0

=

i n

n i







 



 (3.11)

It follows from (eq3.11) and (eq3.10), we have

0. lim  x

_n

x

^*

 =

n





 (3.12)

From (eq3.9) and (eq3.12), we have

0. lim  y

_n

y

^*

 =

n





 (3.13)

Which is

x

^*

, y

^*

 C

_r satisfying the system of nonconvex variational inequalities (2.5). This completes the proof.

Corollary 3.4 Let

C

be a uniformly

r

H

, and let

T : C  H

be such that

T

are both





-strongly monotone mapping. If there exists constant

0 >

, 



such that

,

<

1 1) ,

< ( , 1) <

(

₂

2 2

2 2 2 2 2 2 2

2 2 2 2 2 2

2











 

 







 _ ^ ^ _ ^ ^ _

s s s

s s

s

t t t

t

^(3.14)

where

s r t

_s

r

= 

for some

s  (0, r )



_n

 [0,1]

with



_n^₌₀



_n

= 0

, then the sequences

{ x

_n

}

and

{ y

_n

}

generated by for arbitrarily chosen initial points

x

₀

, y

₀

 C

_r

0 >

], [

=

_C _n



_n



n

P x Tx

y 

0,

>

], [

) (1

1

= 

_n _n



_n _C _n



_n



n

x P y Ty

x

_

  

(3.15)

converge to a solution of the system of nonconvex variational inequalities (2.6).

Proof. From Theorem 3.3, if

T

₁

= T

₂

= T

we have a result.

C

be a uniformly

r

H

, and let

T : C  H

be such that

T

are both





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198

0 >

, 



such that

,

<

1 1) ,

< ( , 1) <

(

₂

2 2

2 2 2 2 2 2 2

2 2 2 2 2 2

2











 

 







 _ ^ ^ _ ^ ^ _

s s s

s s

s

t t t

t

^(3.16)

where

s r t

_s

r

= 

for some

s  (0, r )

. Then the sequences

{ x

_n

}

and

{ y

_n

}

generated by for arbitrarily chosen initial points

x

₀

, y

₀

 C

_r

0 >

], [

=

_C _n



_n



n

P x Tx

y 

0,

>

], [

1

=

_C _n



_n



n

P y Ty

x

_



(3.17)

converge to a solution of the system of nonconvex variational inequalities (2.6).

T

₁

= T

₂

= T

and



_n

= 1

for any

n  1

, we have a result.

4. APPLICATIONS

In this section, we can applied Theorem 3.3 to the system of general nonconvex variational inequalities, for given nonlinear mappings

T , g : C

_r

 H

, we consider the problem of finding

x

^*

, y

^*

 C

_r such that

0 >

, 0,

) ( ), ( ) ( )

(

^* ^* ^* ^*

1



 T g y  g x  g y x  g x    x  C

_r



0,

>

, 0,

) ( ), ( ) ( )

(

^* ^* ^* ^*

2



 T g x  g y  g x x  g y    x  C

_r



^(4.1)

which is called the system of general nonconvex variational inequalities.

If

T

₁

= T

₂

= T

, then the problem (4.1) is equivalent to finding

x

^*

, y

^*

 C

_r such that

0 >

, 0,

) ( ), ( ) ( )

(

^* ^* ^* ^*



 Tg y  g x  g y x  g x    x  C

_r



0. >

, 0,

) ( ), ( ) ( )

(

^* ^* ^* ^*



 Tg x  g y  g x x  g y    x  C

_r



(4.2)

Now, similarly of the proof of Lemma 3.1, we have the result.

Lemma 4.1 For given

x

^*

, y

^*

 C

_r is a solution of system of nonconvex variational inequalities (4.1), if and only if

)], ( )

( [

= )

( x

^*

P g y

^*

T

₁

g y

^*

g

_C

 

)], ( )

( [

= )

( y

^*

P g x

^*

T

₂

g x

^*

g

_C

 

(4.3)

where

P

_C is the projection of

H

onto the uniformly prox-regular set

C

_r.

Theorem 4.2 Let

C

be a uniformly

r

H

, and let

T

₁

, T

₂

: C  H

be such that

T

₁ is a



₁-Lipschitz continuous and



₁-strongly monotone mapping,

T

₂ is a



₂-Lipschitz continuous and



₂-strongly monotone mapping. If there exists constant

 ,  > 0

such that

1 2 1

2 1 2

2 2 1 2 2 1 2

1 2

1

( 1) , 1 <

|<

| 









    



_s _s

s s s

s

t t t

,

<

1 1) ,

|< (

|

2 2 2

2 2 2 2 2 2 2

2 2

2











    



_s _s

s s s

s

t t t

t t

t

^(4.4)

where

s r t

_s

r

= 

for some

s  (0, r )



_n

 [0,1]

with



_n^₌₀



_n

= 0

, then the sequence

{ x

_n

}

and

{ y

_n

}

is generated by for

x

₀

, y

₀

 C

_r,

(7)

199

0 >

)], ( )

( [

= )

( y

_n

P

_C

g x

_n

 T

₂

g x

_n



g 

0,

>

)], ( )

( [ )

( ) (1

= )

( x

_n₁



_n

g x

_n



_n

P

_C

g y

_n

 T

₁

g y

_n



g

_

  

(4.5)

strongly converge to a solution of the system of nonconvex variational inequalities (4.1).

Proof. Similar the proof in Theorem 3.3, let

x

^*

, y

^*

 C

_r be a solution of (4.1) and from Lemma 4.1, we can compute that

. ) ( ) ( ) ) (1 (1 )

( )

(

₁ ₂ ₀ ^*

0

=

*

1

  

 g x g x

_i

g x g x

n

i

n^

     ^ ^ ^ 

^(4.6)

where



₁

= t

_s

1  2 

₁

 

²



₁²

, 

₂

= t

_s

1  2 

₂

 

²



₂² . From



^_n₌₀



_n

= 

and conditions (4.4), we obtain

0. = ) ) (1

lim (1

1 2

0

=

i n

n i







 



 (4.7)

It follows from (4.6) and (4.7), we have

0. = ) ( )

lim  g ( x

_n

g x

^*



n





 (4.8)

And we can compute that

, ) ( ) ( )

( )

(

^*



₂



^*



 g y

_n

 g y   g x

_n

 g x

(4.9)

it follows that

0. = ) ( )

lim  g ( y

_n

g y

^*



n





 (4.10)

From

g

is injective mapping, we have

lim

n__

 x

_n

 x

^*

 = 0

and

lim

n__

 y

_n

 y

^*

 = 0

satisfying the system of general nonconvex variational inequalities (4.1). This complete the proof.

C

be a uniformly

r

H

, and let

T : C  H

be such that

T

are both





0 >

, 



such that

,

<

1 1) ,

< ( , 1) <

(

₂

2 2

2 2 2 2 2 2 2

2 2 2 2 2 2

2











 

 







 _ ^ ^ _ ^ ^ _

s s s

s s

s

t t t

t

^(4.11)

where

s r t

_s

r

= 

for some

s  (0, r )

. Then the sequence

{ x

_n

}

and

{ y

_n

}

is generated by for

x

₀

, y

₀

 C

_r,

0 >

)], ( )

( [

= )

( y

_n

P

_C

g x

_n

 Tg x

_n



g 

0,

>

)], ( )

( [

= )

( x

_n₁

P

_C

g y

_n

 Tg y

_n



g

_



(4.12)

strongly converge to a solution of the system of nonconvex variational inequalities (4.2).

T

₁

= T

₂

= T

and



_n

= 1

for any

n  0

, we have a result.

Acknowledgements. The author would like to thank The National Research Council of Thailand and Uttaradit Rajabhat University for financial support. Moreover, we would like to thank Prof. Dr. Somyot Plubiteng for providing valuable suggestions.

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