E L S E V I E R European Journal of Operational Research 106 (1998) 198-203

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

Fuzzy

T h e o r y a n d M e t h o d o l o g y

resolution on the infeasibility of variational inequality l

H s i a o - F a n W a n g *, H s u e h - L i n g L i a o

Department of lndustrial Engineering, National Tsing-Hua University, Hsinchu, Taiwan, ROC

Received 20 November 1996; accepted 1 July 1997

Abstract

In this study, we discuss variational inequality problems when there is no solution. By applying fuzzy set theory, we "soften" the original problem with fuzzy relation and denote it by VI(X,f). Then the solutions which satisfy the vari- ation inequality to some degree #, 0 ~< # ~< 1 are searched with minimum relaxation. According to the proposed existence theorem, an approach derived from the concepts of single objective mathematical programming problems for solving the VI(X,f) is proposed to find its fuzzy solution set. © 1998 Elsevier Science Inc. All rights reserved.

Keywords: Variational inequality; Generalized complementarity problems; Single objective programming problems; Fuzzy relation; Infeasibility

1. Introduction

A variational inequality problem [1] is to find x* c X such that the vector f (x*) must not make an obtuse angle with all feasible vectors emanating from x*. Thus, by denoting V I ( X , f ) , X is referred to the domain of the problem and f can be regard- ed as all the decision factors that affect the result. However, when the variational inequality problem is infeasible, there does not exist any solution of the problem, and we are interested in what is the min-

* Corresponding author. OR/1E, NCSU, Box 7913, Raleigh, NC 27695, USA, before July, 1998

1 This work was supported by National Science Council, Taiwan, Republic of China with the project number NSC84- 2213-E007.

imum degree of relaxation we should do so that the problem can exist at least one solution. In other words, we intend to answer a question of "how much tolerance one can bear so that the original problem is solvable". Since to determine such "tol- erance" level is not easy and uncertain, we apply the concept of fuzzy set theory to "soften" the orig- inal (crisp) variational inequality problem with a fuzzy relation. That is, we want to find x* c X such that the angles between the vector f(x*) and all fea- sible vectors emanating from x* are less than about 90 °. In other words, the solution x* must satisfy the variational inequality to some degree #, 0 ~</~ < 1. Hence, we focus on the resolution of the reformu- lated problem in the form of variational inequality problem with fuzzy relation, denoted b y VI (X , f ) . In this way the induced solutions of V I ( X , f ) can provide the information of " i f p value of relaxation

S0377-2217198/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PHS03 7 7-22 1 7 (98)003 1 2-3

H.-E Wang, H.-L. Liao I European Journal o f Operational Research 106 (1998) 198 203 199

is allowed on inequality relations, one will have the "desired" solutions x* that have ~vdegree of opti- mality". Thus, the solution set of VI(X,f ) is a fuz- zy solution set. Based on the equivalence between variational inequality and generalized comple- mentarity problem over a convex cone [2], the VI(X,f ) is reformulated as a generalized comple- mentarity problem with fuzzy relation, denoted by GCP(X, f ) where X is a convex cone. Then, bz_using fuzzy set theory [3,4] to defuzzify GCP(X, f ) , a fuzzy solution set can be obtained.

In Section 2, after defining VI(X,f) , we shall present an equivalent relation between VI(X,f) a n d a generalized complementarity problem GCP(X, f ) . Then, through a transformation, a sin- gle objective programming model is derived to fa- cilitate solution process. In Section 3, a numerical example is presented for illustration. Finally, in Section 4, summary and conclusions are drawn.

2. Problem formulation

A variational inequality problem with fuzzy re- lation is to find x* c X with some degree p, 0 ~< p ~< 1, such that

(X?I(X,f)) (f(x*), x - x*) ~ 0 for all x e X.

(1)

That is, a vector f ( x* ) corresponding to x* should satisfy that their inner product (-, .) is greater than or equal to zero to some degree /~ for all x e X, 0 <~/~ ~< 1. Hence to some degree ~, x* is a solution and denoted by (x*, #). Thus, the solution set is a fuzzy set. Suppose, the relaxed value is p , p >~ O, then by the definition of >m in fuzzy set theory we candraw the relation of >m as in Fig. 1, and re- write VI(Y,f ) Eq. (1) as

(VI(X,f) ' ) x* CX (f(x*), x -x* )>~ - p ,

p >/0 for all x C X (2)

with

0 ifp'~< - p ,

e_~ if 0 > 1)' > - p , # = p

1 if pl >~ 0.

/ 0

P

2>

<f(x*),x-x*>

Fig. 1. Membership function o f " >~ ""

where (f(x*), x - x * ) >~p' for all x E X. Because the variational inequality is equivalent

to the generalized complementarity problem (GCP) over a convex cone, let us define a GCP wit_h fuzzy relation below, and denote it by GCP X(~__f, ). Then, the relation between VI(X,f) and GCP(X, f ) is discussed.

(GCP(X,f)) x* ~ X, (3a)

(f(x*), x*)~ 0, (3b)

f ( x*) E X ±, where X j- (3c) = {z : (z, x)/> 0 for all x 6 X},

" ~ " denotes a fuzzy equivalence relation; and " ~ " presents a fuzzy belonging [4].

Then, we have the following results.

Theorem 2.1. VI(X,f) = G C P ( X , f ) , when X is a convex cone.

Proof. First, we show that VI(X,f) C GCP(X,f ) . Let x* with degree # be a solution of VI(X,f) (2). Suppose that either (i) (f(x*), x*) > ql, ql ~> 0 or (ii) (f(x*), x*) < q2, q2 < 0 is true. Since X is a convex cone, from Eq. (2) we have 0C(x*), x*) ~<p, p ~> 0 when x = 0, and (f(x*), x*) >~ - p / ( 2 - 1) when x = 2x* with 2 > 1. If (i) is

true, then p ~> ql and - p / ( 2 - 1) >/ql. This leads to a contradiction. If (ii) is true, then p ~< q2. There lies a contradiction. Therefore, q2 ~ (f(x*), x*) <~ql, ql >~ 0, q2 < 0, and we can say

200 H.-F Wang, H.-L. Liao / European Journal of Operational Research 106 (1998) 198-203

(f(x*), x* )~0 . Furthermore, from Eq. (2), we have 0C(x*), x)~> (f(x*), x * ) - p for all x E X, then (f(x*), X) >~q2 -- P, ( f ( x * ) , x) >~0 for all x E X, x ~X,_~.f(x*)~X ±. Hence x* is also a solution of GCP(X,f ) . Converse selY, suppose that x* with degree p is a solution of GCP(X, f ) (3), then (f(x*), x)>mO for all x E X Eq.(3c) and (f(x*), x* )~O Eq. (3b_). So, (f(x*), x - x * ) >~o, for all x E X. Hence GCP(X, f ) c VI(X,f) . There- fore V I ( X , f ) = GCP(X, f ) when X is a convex cone. []

Theorem 2.1 states an equivalence between a fuzzy GCP and a fuzzy variational inequality problem (VI) over a convex cone. Next, by the concepts of fuzzy set theory [3,4], we can defuzzify the fuzzy relation by the following process. Let X = { x : D x ~ O , D E ~ re×n, D=[di], i = 1 , 2 , . . . , m} and the vector d[ be normal to the row vectors di of D, i = 1 ,2 , . . . , m, and the value of tolerance be p,p > 0. Therefore, based on fuzzy set theory (shown in Theorem. 2.2), Eqs. (3a)-(3c) can be re- written into Eqs. (4a)-(4d) respectively. Then we can find that in model SOP the larger the value of 0 of an efficient solution, the larger is the re- quired relaxation of that solution. So, the best con- dition is 0 = 0 (0 ~< 0 ~< 1), and 1 - 0 represents a satisfactory degree which should be maximized. That is, if (x*, 0") is a feasible solution of SOP, then x* with degree 1 - 0* is a solution of G C P ( X , ~ in which p = 1 - 0'. Thus, given p > 0, GCP(X, f ) can be reformulated as a Single Objective Programming (SOP) model below.

(SOP) min 0 (4a)

s. t Dx >>. 0, (4b)

( - p + q)O <~ 0C(x), x) <<. qO, (4c)

(d[,f(x)) >~ ( - p + q)O,

i = 1 ,2 , . . . ,m , (4d)

- p + q < O , q > 0 , (4e)

0 ~< 0 ~< 1. (4f)

Then the relation between VI(X,f ) (2) and SOP is presented below.

Theorem 2.2. Let Xefr denote the set o f all efficient solutions o f an SOP. Then the point x* with

objective value O* belongs to Xeff i f and only if x* is a solution o f V I ( X , f ) to the degree o f l~ = 1 - 0".

Proof. "Only if". Suppose x* E Xeff and its objec- tive value is 0". Then from (4a), x*E X; from Eqs. (4c) and (4e), (f(x*), x * ) = 0 ; and from Eq. (4d), (f(x*), x)~>0 for a l l x E X . Sox* is a solution of GCP(X, f ) and (f(x*), x - x*) >1 -pO* for all x E X, then from the membership function defined in Eq. (2), x* satisfies VI(X,f) to the degree of (1 - 0').

"If". Suppose that x* is a solution of VI(X,f) to the degree of 1 - 0", then from Eq. (2), (f(x*), x -x~>>. -pO* for all x EX, and x* is a solution of GCP(X, f ) . Therefore, Eq. (3a) implies Eq. (4b); Eq. (3c) implies (f(x*), x)>m 0 for all x E X, f(x*) = D v v, v >~ O. That is, d[f(x*)>~O, i = 1,2 ..... m., where d[ is normal to di. Since from Eq. (3b), q'O*<<.(f(x*), x*) <<. qO*, where q' < 0, q > 0. Therefore (d~,f(x*)) >>-(-p+q)O* and - p + q < 0 imply Eqs.(4d)

and (4e). Without loss of generality, let q ' = - p + q in Eq. (4c). Then, x* with objective value 0* is an efficient solution of SOP. []

Thus, the efficient solution set of model SOP is a solution set of VI(X,f) with some relaxed value p. Each element of the set represents a solution val- ue with the degree of satisfaction to the original variational inequality problem. Therefore, if x* is with 0"= 0 in model SOP, then 0C(x*), x*)= 0 and (d~,f(x*)) >~ 0 for all i, that is (f(x*), x*) = 0 and f(x*) E X I. Then x* is a solu- tion of the crisp GCP(X, f ) and we have the fol- lowing result.

Corollary 2.1. The point x* with objective value O* = 0 is an efficient solution o f SOP if and only if x* is a solution o f crisp VI(X,f) .

In practice, we do not want to relax a problem too large to violate the original structure. There- fore we want to know the minimum value the p should be so that of the model SOP can exist at least one solution. In other words, we intend to de- termine the minimum degree of relaxation so that the corresponding VI is solvable. Therefore, in the

H.-F Wang, H.-L. Liao I European Journal o f Operational Research 106 (1998) 198-203 201

following, the model for determining the minimum relaxation of an infeasible VI will be derived.

First, we show the following corollary.

Corollary 2.2. Suppose VI(X,f) is infeasible and p* is an optimal solution of model SOP ~ below, then the corresponding VI (X, f ) has a feasible solution when p >~p*.

(SOP') min p (5)

s.t. Dx>/O,

( -p + q)O <~ (f(x), x) <~ qO,

(d;,f(x)) >~ ( -p + q)O, i = 1,2 . . . . ,m,

- p + q < 0 , q > 0 , p > 0 ,

0<0~<1.

Proof. Suppose VI(X,f) is infeasible, then there does not exist any feasible solution of SOP with p = 0. That is, p = 0 is not a feasible solution of SOP'. Suppose p* is an optimal solution of SOP'. Then p with a value greater than p* must be a feasible solution of SOP' and the model SOP with such p must be feasible. That is, the corresponding VI(X,f) is feasible by Theorem 2.2. Conversely, p with a value smaller than p* must be an infeasible solution of SOP' and the model SOP with such p must be infeasible, that is, the corresponding VI(X,f) is infeasible. []

Therefore, if a VI is infeasible, we can solve model SOP' to obtain the minimum relaxation p* of this infeasible VI. Hence, the solution set of the relaxed VI can be found by solving model

SOP. Based on Theorem 2.2, if the crisp VI(X,f) is infeasible and x* is a solution of the correspond- ing VI(X,f), then x* is a feasible solution of SOP and the relation between x* and its corresponding 0* and q* is shown in Table 1. Now, let us con- clude the above results in the following.

Remarks. (1) By corollary 2.2.1, if x* is a solution of crisp VI~X,f), then x* must be a feasible solution of VI(X, f ) .

(2) By corollary 2.1, if x* with degree/~ ~ 1 is a solution of VI(X,f), then x* is not a solution of crisp VI(X, f ) .

(3) By corollary 2.2, if crisp VI(X,J) is infeasi- ble, then by solving SOP', the minimum value of relaxation can be determined, which guarantees the existence of at least one solution of VI. Con- versely, if crisp VI(X,f) is feasible, then the objec- tive value of the optimal solution of SOP' must be ze ro .

3. Examples

Now, we demonstrate the proposed method by an example of infeasible VI(X,f) below. First, a minimum relaxation value is found by solving SOP', and then, the relaxed VI is solved by solving SOP. Both models SOP and SOP' are nonlinear programming models. Adoption of software GINO [5] we can solve the problem with the com- putational complexity of O (n3).

Example 3 . 1 . L e t X = {x: - X l - x2 q- 3x3 7> 0,

- 2 x l -~ x2 - x3 ~ O, x : (X l ,X2 ,X3) E ~3} a n d

Table 1 The relation between x* and 0, q

x* = 0 f(x*)x* = 0 x on boundary f (x*) E X 1 f(x*)x* > 0 The angle of X > 90 ° 0 q

y . . . . . .

N N Y Y Y - ¢ 0 N N Y N Y - # 0 N N Y N N Y # 0 N N Y N N N # 0 N N N Y Y - ¢ 0 N N N N Y - # 0 N N N N N - # 0

7 b - p - p

o,-p ¢ - p # - p - p

# 0 , - p # - p

202 H.-F. Wang, H. -L. Liao I European Journal of Operational Research 106 (1998) 198-203

2X ~- 0 .2X~ - - 0 . 5 X 2 ~- 0.1x3 - 3]

f ( x ) = [ - 0 . 5 x , + x 2 + 0 . 1 x 3 + 0 . 5 ] ,

L 0 " 5 X l - - 0"2X2 q- 2X3 - - 0 . 5

it can be shown that this VI (X , f ) is infeasible [6]. Now, to find the minimum relaxation, we reformu- late the VI (X , f ) in the form of SOP' as follows:

[l13j D = - 2 1 -1 '

d i = ( - 1 - 1 3), d 2 = ( - 2 1 - 1 ) ,

d I = ( 5 2 - 8 ) , d ~ = ( - 2 4 9 - 5 ) ,

min p

s.t. - Xl - - X2 -~- 3X3 ~> 0, --2Xl + X2 -- X3 ~ 0,

0.2x 4 + .2x~ - 3Xl + 0.1x 4 + x i + 0.5x2 + 2x3 2

- - 0 .5X3 - - XlX2 ~- 0 . 6 X l X 3 - - 0 .2X2X3 - - qO <~ 0,

0.2x 4 + 2x~ - 3Xl + O.lx 4 + x22 + 0.5x2 + 2x 2

- - 0 .5X3 - - XlX2 -f- 0 . 6 X l X 3 - - 0 .2X2X3 - - qO

+pO >~ O,x~ + 5X 1 -{- 0 . 2 X ~ -]- l.lx2 -- 15.5x3

- 10 - qO +pO >1 O, -4.8x~ - 5 5 x 1 -[- 0.9x 3

+ 22x2 - 12.4x3 + 79 - qO +pO >>. 0,

- p + q < 0,q > 0,p > 0,0 < 0<~ 1.

(6) By using GINO, we obtain the optimal solu-

tion of p * = 8.014319 and 0 " = 1. That is, we can relax VI by 8.014319 to have a feasible solu- tion. In terms of VI(X, f ) , this is also equivalent to state that if the relaxed value is greater than 8.014319, VI (X , f ) is feasible. Since the degree of a solution which satisfies the variational in- equality is 1 - 0", a solution with smaller 0* is more preferable. Therefore, we define an upper bound of 0 which is smaller than 1 and apply model SOP' to obtain a relaxed value p* with more preferable degree 0". The results are shown in Table 2. From Table 2, we found that if we want to obtain a solution with a satisfactory level which is greater than 0.5, then the VI must be re- laxed to, at least, 16.028729.

So, let us relax VI by 16 in order to find its so- lution set. By utilizing model VI(X, f ) , the model is reformulated in the form of SOP as below:

T a b l e 2

T h e m i n i m u m re l axa t i on wi th respec t to 0 in the example

U p p e r b o u n d o f 0 p

1 8 .014319

0.5 16.028729

0.4 20.035793

0.3 26.714403

0.2 40 .071654

0.1 80.143158

min 0

s.t. - x l - x 2 + 3 x 3 ~ > 0 , - 2 x l + x z - x 3 ~ > 0 ,

0.2x 4 + 2x, - 3x, + O.lx 4 + xl + 05x2 +

- - 0 . 5 X 3 - - XIX 2 ~- 0 . 6 X l X 3 - - 0 . 2X2X 3 - - qO <<. O,

0.2x 4 + 2x 2 - 3xl + O.lx 4 + x~ + 0.5x2 + 2x 2

- - 0 .5X3 - - XIX2 q- 0 .6XlX3 - - 0 .2X2X3 - - qO

+ 160 ~> 0,x~ + 5xl + 0.2x~ + 1.1x2 - 15.5x3

- 10 - qO + 160 ~> 0, -4.8x~ - 55xl + 0.9x 3

+ 22x2 - 12.4x3 + 79 - qO + 160 >/0,

0 < q < 16, 0 < 0 ~ < 1 .

(7) Then, we use GINO to solve the above system.

Table 3 shows some solutions of the VI (X, f ) with relaxed value 16 when the lower bound of 0 in- creases incrementally from 0.1.

4. Summary and conclusion

This study proposed a method to solve infeasi- ble variational inequality problems by the concept of fuzzy relation. The purpose of utilizing fuzzy re- lation is to "relax" the original problem to certain level and then the solutions which satisfy the vari- ational inequality to certain degree can be found. An equivalent nonlinear programming model is designed to find the fuzzy solution set in which the computational complexity is polynomial. Be- sides, with the consideration of solvability and property reservation, the "minimum" relaxed val- ue is investigated and solved by a proposed model SOP'. Theoretical evidences are illustrated by a nu- merical example.

H.-F Wang, H.-L. Liao / European Journal of Operational Research 106 (1998) 198 203

Table 3 The feasible solutions of the example

203

Lower bound of 0 O* x* q*

0.1 0.500906 0.2 0.500906 0.3 0.500906 0.4 0.500906 0.5 0.500903 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

(-0.286021, -1.001075, -0.429032) (-0.286021, -1.001075, -0.429032) (-0.286021, -1.001075, -0.429032) (-0.286021, -1.001075,-0.429032) (-0.286021, -1.001075, -0.429032) (-0.285939,-1.000788, -0.428909) (-0.088088,-0.308308,-0.132132) (0.115659, 0.404806, 0.173488) (0.216640, 0.781082, 0.332574)

3.811410 3.811410 3.811410 3.811410 3.809352 1.436676 1.436676 1.436676 1.436676

References

[1] P.T. Harker, J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathemat- ical Programming 48 (1990) 161-220.

[2] S. Karamardian, Generalized complementarity problem, Journal of Optimization theory and Applications 8 (1971) 161-167.

[3] G.J. Klir, T.A. Folger, Fuzzy sets, uncertainty, and infor- mation, Prentice-Hall, New Jersey, 1988.

[4] H.J. Zimmermann, Fuzzy set theory and its applications, 3rd ed., Kluwer/Nijhoff Pub., Boston, 1988.

[5] J. Liebman et al., Modeling and optimization with GINO, The Scientific Press Publisher, 1988.

[6] H.F. Wang, H.L. Liao, Resolution of the variational inequality problems, Journal of Optimization Theory and Application (submitted).