ON VECTOR VARIATIONAL INEQUALITY AND

ON VECTOR VARIATIONAL INEQUALITY AND ITS APPLICATION TO VECTOR TRAFFIC EQUILIBRIA* X.Q. Yang y and C.J. Gohz Abstract.

We motivate the study of vector variational inequality by a practical trac ow problem, namely a generalization of the well-known Wardrop user equilibrium law. Both weak and strong form of the vector variational inequality will be discussed and their relationships to vector optimization problems will be established under various convexity assumptions. An exact analytical method for solving a special case of the weak vector variational inequality involving only ane functions is proposed, which also allows one to partially solve for a special case of vector trac equilibria.

Keywords. Vector variational inequality, vector optimization, trac equilibria, active set method.

* This work is supported by a grant from the Australian Research Council y Research fellow, Department of Mathematics, University of Western Australia, Nedlands, WA 6009 z Associate professor, Department of Mathematics, University of Western Australia, Nedlands, WA 6009 1

1. Introduction Hitherto, almost all the models for predicting urban trac equilibria are based on the well-known Wardrop's user equilibrium principle, which is based on a scalar cost (usually minimum time-delay) for determining whether a path should have positive ow. The resulting conditions for a ow to be in (scalar) Wardrop's equilibrium appear naturally in the form of a nonlinear complimentarity problem (see Magnanti (Ref. 1) or Florian (Ref. 2) for a comprehensive survey and summary of the trac equilibria problem). These conditions can be cast in an equivalent variational inequality (VI) formulation by choosing the closed positive octant as the underlying convex set. Under suitable assumptions (either separability or symmetry of the cost), these equilibrium conditions can also be cast into an equivalent optimization problem familiar to most operations research practitioners. Nevertheless it appears that equilibrium problems are still mostly solved in the more general formulation of variational inequality using a variety of techniques (see Narguney (Ref. 3) for a detailed discussion of various computational techniques). The assumption that a road user will choose a path only if it incurs minimum delay appears to be unnecessarily restrictive. In many situations, the minimum delay path may not be the cheapest path to travel on. In particular when toll is collected on freeways which are supposedly the minimum delay paths, there will always be other less direct paths which are cheaper to travel on because no toll is required. Other possibilities include the user's perception of the scenery of the path and other attractions. These criteria for choosing a path are often con icting and hence a generalization of the classical Wardrop's principle to account for multiple criteria, or vector cost, appears to be of both theoretical as well as practical interests. An equilibrium principle for vector-valued cost functions has previously been proposed by Chen and Yen (Ref. 4). This will be stated in rigorous term in the next section, but essentially it generalizes the Wardrop's principle by saying that the ow along a path is positive only if the (vector) cost for this path is not dominated by that of another path, i.e., the path incurs an Pareto optimal or ecient (vector) cost. Chen and Yen went on to establish several results, including existence, sucient and necessary conditions for vector equilibria under rather restrictive assumptions. It is the rst goal of this paper to establish some dierent sucient conditions leading naturally to vector variational inequality (VVI), both for a weak and strong form of vector equilibria to be de ned in section 2. Necessary conditions, however, remain elusive without making further restrictive assumptions. Section 3 of the paper is concerned with establishing the various relationships between 2

vector variational inequality (VVI) and vector optimization problems (VO), in both weak and strong forms (see Ref. 5 and 6). As in the case of scalar variational inequality, we show that, under a certain symmetric assumption on the cost, a solution to VVI is also the solution to a corresponding VO problem. Unfortunately the converse is not true. In a weak form, however, equivalence can be established under further convexity assumption. The last part of the paper describes an exact analytical technique for solving a special case of weak-VVI where all the underlying functions are ane and symmetric. This is equivalent to solving for the set of weak ecient solutions for a VO involving convex quadratic costs. Since there is no known technique for solving VVI to date, this appears to be the rst method for solving VVI, albeit only a special case. The solution to this special VVI is also a partial solution to the generalized trac equilibria. Before we describe the model rigorously, it is necessary to lay down some formal de nitions. De nition 1.1 Given ;  2 IRr , the following de nes the ordering relationships =< ; ; 6 ; ; 6> are de ned similarly. It is clear that 0 6  =) 0 6< . If r = 1, then both =  6  or  6<  implies    . It is sometimes useful to generalize these ordering relations by replacing IRr with some closed pointed cone. De nition 1.2 The vector variational inequality problem is de ned by: (Problem VVI)

Find x 2 C; s:t: F(x)(y ? x) 6 0; 8y 2 C;

where C is a closed convex subset of IRn and F : C ! IRrn is an C 1 matrix-valued function. This formulation of VVI is rst attributed to Giannessi (Ref. 7). De nition 1.3 The weak vector variational inequality problem is de ned by: (Problem WVVI)

Find x 2 C; s:t: F(x)(y ? x) 6< 0; 8y 2 C; 3

where C; F are as given in de nition 1.2. De nition 1.4 Given a C 1 function f : IRn ! IRr and C , a closed convex subset of IRn. A point x 2 C is said to be ecient if there exists no y 2 C such that f (y)  f (x). The set of ecient points is denoted by E(f ; C ). A point x 2 C is said to be weak ecient if there exists no y 2 C such that f (y) < f (x). The set of weak ecient points is denoted by WE(f ; C ). Clearly, E(f ; C )  WE(f ; C ): De nition 1.5 The vector (or multi-criteria) optimization problem is de ned by: (Problem VO) Find E(f ; C ) for V ? Min f (x); subject to x 2 C; where f : IRn ! IRr is C 1 and C is as given in de nition 1.2. De nition 1.6 The weak vector optimization problem is de ned by: (Problem WVO) Find WE(f ; C ) for Weak ? Min f (x); subject to x 2 C; where f and C are as given in de nition 1.5. De nition 1.7 A dierentiable vector function f : IRn ! IRr is said to be IRr+-convex if 8x; y 2 IRn, f (y) 2 f (x) + rf (x)(y ? x) + IRr+; or

f (y) => f (x) + rf (x)(y ? x); where rf (x) is the Jacobian of f , an r  n matrix. We say that f is strictly IRr+-convex if f (y) 2 f (x) + rf (x)(y ? x) + intIRr+; or

f (y) > f (x) + rf (x)(y ? x):

De nition 1.8 A dierentiable vector function f : IRn ! IRr is said to be IRr+-concave (resp, strictly IRr+-concave) if ?f is IRr+-convex (resp, strictly IRr+-convex). We often use the terms convex and strictly convex (resp., concave and strictly concave) to mean IRr+-convex and strictly IRr+-convex (resp., IRr+ -concave and strictly IRr+-concave) respectively. 4

2. Vector Wardrop's equilibrium principle. We shall introduce the necessary notation by summarizing the (scalar) trac equilibria problem. Our notation follows closely that of Ref. 1. Consider a transportation network G = (N ; A) where N denotes the set of nodes and A denotes the set of arcs. Let I be a set of origin-destination (O-D) pairs and Pi ; i 2 I denotes the set of available paths joining O-D pair i. For a given path k 2 Pi , let hk denote the trac ow on this path P and h = [hk ] 2 IRN ; N = i2I jPij. A (path) ow vector h induces a ow va on each arc a 2 A given by: XX va = ak hk i2I k2Pi

where
= [ak ] 2 IRjAjN is the arc path incidence matrix with ak = 1 if arc a belongs to path k and 0 otherwise. Let v = [va] 2 IRjAj be the vector of arc ow. Succinctly v =
h. We shall assume, P for the rest of this paper, that the demand of trac ow is xed for each O-D pair, i.e., k2P hk = di , where di is a given demand for each O-D pair i. In the scalar cost case, di can be easily generalized to be a function of the minimum cost (see Ref. 1), although it is not so clear at this stage how this can be done for vector cost without incurring complicated set-valued notations. For simplicity, we shall assume > 0 satisfying the demand is called a feasible ow. xed demand for the moment. A ow h = P > 0; k2P hk = di 8i 2 Ig. H is clearly a closed convex set. Let ta(v) Let H = fh j h = be a (scalar) cost on arc a (usually the delay) and is in general a function of all the arc

ows, and t(v) = [ta(v)]. If ta is only a function of va, we say that the cost is separable. If @ta =@va = @ta =@va, we say that the cost is symmetric. Note that a separable cost is a special case of symmetric cost. The cost along a path k is assumed to be the sum of all the arc cost along this path, thus i

i

0

0

k (h) =

X

a2A

ak ta(v):

Let  = [k ] 2 IRN . Succinctly  (h) =
>t(v): Given a ow vector h, the minimum cost for an O-D pair i is de ned by ui = kmin  (h): 2P k i

Wardrop's user principle is a behavioral principle which asserts that, at equilibrium, users only choose minimum cost paths to travel on, i.e., a ow h 2 H is said to be in Wardrop equilibrium if (Ref. 1 or 2)

:Wardrop equilibrium principle: 8i 2 I ; 8k 2 Pi; k (h) > ui =) hk = 0: 5

Amongst other possible ways of stating the conditions for Wardrop equilibrium, the most popular one seems to be in the form of a (scalar) variational inequality (VI): (Problem VI) Find h 2 H; s:t:  (h)>(h ? h)  0; 8h 2 H: Existence and Uniqueness for the trac equilibria can be established under easily satis able conditions such as continuity and strict monotonicity of the function  . As pointed out in the introduction, the assumption that users choose their path based on a single criterion may be unreasonably restrictive. In reality, users may choose their path based on several criteria, for example, time delay and (monetary) cost, amongst others. In general, these costs are often con icting. We generalize the scalar costs ta(h) and k (h) to vector costs ta (h) and k (h) 2 IRr . Let T(h) be an r  N matrix with columns given by k (h). Chen and Yen (Ref. 4) proposed the following vector equilibrium principle, which asserts that, at equilibrium, users only choose Pareto Optimal or ecient paths to travel on.  Vector equilibrium principle: A ow h 2 H is said to be in vector equilibrium if: 8i 2 I ; 8k; k 2 Pi; k (h) ? k (h)  0 =) hk = 0: Here we de ne an ecient cost in the same way as in de nition 1.4. A weaker form of the vector equilibrium which turns out to be rather important is de ned as follows.  Weak vector equilibrium principle: A ow h 2 H is said to be in weak vector equilibrium if: 8i 2 I ; 8k; k 2 Pi; k (h) ? k (h) > 0 =) hk = 0: The latter is called weak since a ow h satisfying k (h) ? k (h) > 0 and hk > 0 (i.e., not weak vector equilibrium) also satis es k (h) ? k (h)  0 and hk > 0 (i.e., not vector equilibrium). We may now establish a sucient condition for a ow h to be in vector equilibrium. Propostion 2.1. h 2 H is in vector equilibrium if h solves the VVI: Find h s:t: T(h)(h ? h) 6 0; 8h 2 H: (2:1)

Proof: The proof here is similar to that of Chen and Yen (Ref. 4), although they lead to dierent conclusion. Let h satisfy (2.1), Choose h to be such that 8 < hj if j = 6 k or k; h j = : 0 if j = k; hk + hk if j = k. 6

P P j 2Pi hj = j 2Pi h j = di . Now XX T(h)(h ? h) = (h j ? hj )j (h) i2I j 2Pi = (h k ? hk )k (h) + (h k ? hk )k (h) = hk (k (h) ? k (h)) 6 0:

(2:2)

k (h) ? k (h)  0;

(2:3)

Clearly h 2 H since 8i 2 I ;

If

then (2.2) and (2.3) together imply that hk = 0. A similar sucient condition can be established for weak vector equilibrium: Propostion 2.2. h 2 H is in weak vector equilibrium if h solves the WVVI: Find h s:t: T(h)(h ? h) 6< 0; 8h 2 H:

(2:4)

Proof: The proof follows exactly as in proposition 2.1 but with 6 in (2.2) replaced by 6 and  in (2.3) replaced by >.
0; 8h 2 H:

(2:5)

Note that (2.5) is a much stronger condition than (2.1) or (2.4). Unfortunately this necessary condition requires the ecient frontier to be a singleton, i.e., an utopia point, which seldom occurs in practice. If it does, then there is no need to examine vector cost anyway. Without this assumption, however, necessary conditions seem to be dicult to obtain, due to diculty in summing inequalities of the form (2.1) or (2.4). Since (2.1) (resp., (2.4)) is sucient but not necessary, the solution of which is only a partial solution to the vector equilibrium (resp., weak vector equilibrium) problem. Hitherto, no known methods are available to solve for either VVI (2.1) or WVVI (2.4). In the 7

next section, we shall establish some relations between VVI and WVVI with the vector optimization problems, VO and WVO. Under further assumptions, we are able to solve a special class of WVVI analytically, which in turn yields (partial) analytical solution to the weak vector equilibrium problem. In Ref. 2, it was shown that there exists a solution to the the scalar trac equilibrium problem if the cost is positive and continuous. In the case of vector trac equilibrium, a sucient condition for the existence of solution to the WVVI problem (2.4), and hence the existence of solution to the vector trac equilibrium, is that T is continuous and copositive, the later meaning T(h)h > 0 8h 2 IRN . A proof of this existence result can be found in Ref. 8.

3. Relationships between VVI, WVVI, VO and WVO It is well-known that a special case of VI, namely when the (scalar) cost is separable or has symmetric gradient, then it has an equivalent optimization problem. This result has a direct extension in the case of VVI.

De nition 3.1. A (cost) matrix function F : IRn ! Rrn is said to be conservative if @Fki(x) = @Fkj (x) ; 8 i; j = 1;


; n; 8 k = 1;


; r:

@xj @xi Thus a cost matrix is conservative if the Jacobian of each row of the matrix is symmetric. This nomenclature is a generalization of conservative eld in physics. In many trac

ow problems, the cost matrix is often separable, in the sense that Fki(x) = Fki(xi ) 8k = 1; 2;


; r. Clearly a separable cost matrix is also conservative. The following is a wellknown extension of Green's theorem to functions in IRn:

Proposition 3.1 If F(x) is conservative, then there exists f : IRn ! IRr , with f (x) = (f1 (x);


; fr (x))> , denoted by fk (x) = such that

Ix

Fk (y)dy; k = 1;


; r;

F(x) = rf (x):

Proof: See theorem 4.1.6 of Ortega and Rheinboldt (Ref. 9). 8

(3:1)

f is sometimes referred to as the potential of F. The following establishes when a solution of VVI is also a solution of VO,

Proposition 3.2 Assume that (3.1) holds. If f is IRn+-convex and x solves VVI, then x

is an ecient solution of VO. Proof. Suppose that x is not an ecient solution of VO. Then there is an y 2 C such that f (y)  f (x). Thus f (y) ? f (x) 2 ?IRr+ n f0g: Since f is IRn+-convex, we have

rf (x)(y ? x) 2 f (y) ? f (x) ? IRr+  ?IRr+ n f0g ? IRr+ = ?IRr+ n f0g: Thus rf (x)(y ? x)  0, a contradiction. The following shows that an ecient solution of VO may not be a solution of VVI, i.e., VVI is not necessary for VO.

Example 3.1. Consider the problem V ? min f (x); subject to x 2 [?1; 0]; where f (x) = (x; x2 )>. It is clear that every x 2 [?1; 0] is an ecient solution. Let x = 0. Then for y = ?1   rf (x)(y ? x) = ?01  0: Thus x = 0 is not a solution of VVI.

However for the weak vector variational inequality (WVVI), the following result holds.

Proposition 3.3 Assume that (3.1) holds. If x is a weak ecient solution of WVO, then x is a solution of WVVI. If f is IRn+-convex, then the converse is true. Proof. See Proposition 4.1 of Chen and Yang (Ref. 5). Thus if f is IRn+-convex and if (3.1) holds, then solving WVO is equivalent to solving WVVI. Obviously, a solution of VVI is also a solution of WVVI, but not the converse; and a solution to VO is also a solution to WVO, but not the converse. Nevertheless, under a further assumption on f , the converse is also true. 9

Proposition 3.4 Assume that (3.1) holds and f is strictly IRn+-convex. If x solves WVO, then x also solves VO. Proof: Suppose that x solves WVO but not VO, then there exists y 2 C such that f (y)  f (x). By the strict convexity of f , f (y) ? f (x) > rf (x)(y ? x): By Proposition 3.3, we have

0  f (y) ? f (x) > rf (x)(y ? x) 6< 0; implying that

0 > rf (x)(y ? x) 6< 0;

which is a contradiction. We summarize the nding so far with the following diagram:

f convex; F = rf VVI as in def :1:2 ?! # f convex; F = rf ?!? WVVI as in def :1:3 F = rf

VO as in def :1:4

#" f strictly convex WVO as in def :1:5

Unfortunately, we have not been able to establish the converse result between WVVI and VVI, i.e., the question of, under what conditions is a solution to WVVI also a solution to VVI, is yet to be answered. The following result, however, is of some theoretical interest.

Proposition 3.5 Assume that (3.1) holds and f is strictly IRn+-concave. If x solves WVO, then x also solves VVI. Proof: Suppose that x solves WVO but not VVI, then there exists y 2 C such that rf (x)(y ? x)  0: By the strict concavity assumption,

f (y) ? f (x) < rf (x)(y ? x)  0; 10

i.e.,

f (y) < f (x);

which contradicts that x solves WVO.

Corollary 3.6 If (3.1) holds, f is strictly IRn+-concave, and x solves VO, then x also solves VVI.

Proof: If x solves VO, it also solves WVO unconditionally. The result thus follows from proposition 3.5.

Unfortunately a function cannot be convex and strictly concave simultaneously, hence no equivalent relationship can be established between VVI and VO.

4. An active set method for solving ane weak variational inequality

In Goh and Yang (Ref. 10), an active set method for solving a special case of VO where each component of the vector cost is a convex quadratic function has been reported. The set of ecient solutions are obtained in analytical form. This method will now be extended to obtain the set of weak ecient solutions for the following weak vector optimization problem called weak multicriteria convex quadratic program: >  (Problem WMCQP ) Weak ? Min 21 x>Q1 x + p>1 x;


; 21 x>Qr x + p>r x ; subject to ci(x) = a>i x ? bi = 0; i 2 E = f1; 2;


; mg; (4:1) ci(x) = a>i x ? bi  0; i 2 I = fm + 1;


; m + lg; (4:2)

where Q1 ;


; Qr 2 IRnn; a1;


; am+l; p1;


; pr 2 IRn. Let C = fx j x satis es (4.1) amd (4.2) g be the feasible set, which is assumed to be non-empty. C is clearly a closed convex set. Two basic assumptions are needed:

Assumption 4.1. The set of vectors fai 2 IRn : i 2 E [ I g are linearly independent. P ?1 Assumption 4.2. The matrices Q1 ; Q2;


; Qr are such that Q() = ir=1 i Qi + (1 ? Pr?1   i=1 i )Qr is symmetric positive de nite 8 2  , where  is de ned by: r?1 X  1 r ? 1 > i  1;  6= 1g n f0g:  = f = ( ;


;  ) j i  0; i=1

11

Note that  is a proper subset of a simplex  in IRr?1 space:  = f = ( ;


; r?1 )> 1

j i  0;

r?1 X i=1

i  1g:

We shall now relate WMCQP to a special case of WVVI, which can in turn be solved be solved by a variant of the active set method previously described in Goh and Yang (Ref. 10). Consider the following ane weak vector variational inequality problem: (Problem AWVVI ) Find x 2 C s:t: F(x)(y ? x) 6< 0 8 y 2 C; where C is the same convex set of WMCQP, and the kth row of F is given by an ane vector function: Fk (x) = x>Qk + p>k .

Proposition 4.1 AWVVI is equivalent to WMCQP. Proof. Let fk (x) = 21 x>Qk x + p>k x. Clearly Fk (x) = rfk (x), or F(x) = rf (x), hence

condition (3.1) holds. Furthermore each fk is convex, i.e., each Qk is positive semi-de nite by assumption 4.2. The equivalence thus follows from proposition 3.3. The active set method of Ref. 9 solves the strong version of WMCQP analytically, i.e., the (partial) set of ecient solutions are obtained in parametrized form. Although the extension to this method for solving weak WMCQP is only of minor dierence, for completeness, we shall nevertheless summarize the key ideas of the method and present a brief outline of the algorithm here. Essentially, the main idea is based on the fact that each weak ecient solution of WMCQP corresponds to a solution of the scalarized problem of WMCQP. Consider for a given  2 : Problem P1() : subject to

min 21 x>Q()x + p>()x; ci(x) = a>i x ? bi = 0; i 2 E; ci(x) = a>i x ? bi  0; i 2 I;

P ?1 i P ?1 i P ?1 i P ?1 i where Q() = ir=1  Qi + (1 ? ir=1  )Qr and p() = ir=1  pi + (1 ? ir=1  )pr :

Proposition 4.2 A point x 2 C is a weak ecient solution of WMCQP if and only if there exists  2  such that x is a solution of P1 (). Proof: See theorem 3.9 and theorem 4.15 of Ref. 10. 12

At optimality and for a given , some of the inequality constraints in I will be active. If we know a priori what the optimal active constraints are, then P1() reduces to a quadratic program with only equality constraints, which can be easily solved by solving for a set of linear equation parametrically in  (see section 14.1 of Ref. 12). If P1 () has already been solved for a given , then as  varies, the optimal active set will eventually change. The optimal active set of a neighboring optimal solution can be quite easily found from the previous active set by solving for the root of some rational equations, or by nding the boundary at which the optimal active set changes. These can be determined parametrically as a function of . By doing this repeatedly, we are able to obtain the parametrized solution for the whole of  and hence obtain the complete set of ecient solutions to MCQP.

De nition 4.1 The restricted scalarized equality program of MCQP associated with an active set Ak [ E; Ak  I is de ned to be Problem P2 (; Ak ) min 21 x>Q()x + p()>x; subject to : ci(x) = a>i x ? bi = 0; i 2 Ak [ E: We denote the optimal solution to P2(; Ak ) as x(; Ak ), with i (; Ak ) to be the corresponding multiplier to each of the active constraints in Ak [ E which are given by (pp. 425 of Ref. 12):  (; Ak ) = (AQ()?1 A> )?1 [AQ()?1 p() + b]; x(; Ak ) = ?Q()?1 [I ? A>(AQ()?1 A>)?1 AQ()?1 ]p() + Q()?1 A>(AQ()?1 A>)?1b;

(4:1) (4:2)

where A is a matrix composed by the putting a>i ; i 2 Ak [ E as its row. From (4.1) and (4.2), (; Ak ), x(; Ak ) are rational vector functions of . i (; Ak ) is a component of  (; Ak ), so is a rational function of . ci (x(; Ak )) is a linear function of x(; Ak ), hence it must also be a rational function of . Suppose that for some xed 0 , the corresponding optimal active set is Ak . We de ne -domain L(Ak ) as the set of all  2  such that Ak remains as the optimal active set for all  arsing from this set: De nition 4.2 The -domain L(Ak ) is an (open) subset of  de ned by

L(Ak ) = f 2 int j ci (x(; Ak )) < 0; i 2 I n Ak and i (; Ak ) < 0; i 2 Akg; 13

where x(; Ak ) and i (; Ak ) are the solutions to P2(; Ak ). The boundary of an domain L(Ak ) is then the union of some segments (note: not all) of the rational equations ci(x(; Ak )) = 0; i 2 I n Ak or i (; Ak ) = 0; i 2 Ak , plus possibly part of the boundary of . The idea of the active set method for WMCQP is to carve up the set  into a number of disjoint (except for the common boundaries) -domains L(Ak ); k = 1; 2;


; K whose union gives . In each -domain we have a dierent parametrization of the solution in the form of (4.1). We may now outline the active set method for solving WMCQP or AWVVI.

Algorithm 4.1 Step 0 Set k 1: Pick any  2 : Solve P1 () to nd the initial optimal active set A1. Solve P2(; A1 ) to obtain i (; A1 ); i 2 A1 and x(; A1 ) analytically as functions of . Let the set of unlabeled active set be fA1 g. Step 1 Solve the following equations of (rational) functions for  2  n fL(A1 ) [ ::: [ L(Ak?1 )g: ci(x(; Ak )) = 0; 8i 2 I n Ak ; i (; Ak ) = 0; 8i 2 Ak ;

(4:3) (4:4)

to nd the boundary segments for -domain L(Ak ). (Note that across a boundary segment, there is a change of active set.) Once this is done, label Ak permanently and delete it from

, i.e., n Ak . Step 2 Check on the boundary of (Ak ) for any unlabeled adjacent active set. If there is any (could be more than 1), include all of them in the set of unlabeled active sets. Go to Step 3. Step 3 If the set of unlabeled active set is empty, go to step 4. Otherwise, set k k +1, pick any unlabeled active set Ak from , solve P2(; Ak ), and return to Step 1. Step 4 Solve P1() for each of 0 = 0; 1> = (1; 0; 0;


; 0); 2> = (0; 1; 0;


; 0);


; r> = (0; 0; 0;


; 1): Denote the solution (which may be non-unique) to each of these as Z (i ); i = 0; 1; 2;


; r. Stop. Note that in Step 1, not necessarily all of the ci = 0; i 2 I nAk or i = 0; i 2 Ak constitute a boundary segment for L(Ak ), and some of the boundary segments are given by part of 14

the boundary of , i.e. []i = 0 or 1> = 1. At the termination of Algorithm 4.1, the set of ecient solution x() is then the union of all the x(; Ak ); k = 1; 2;


; K ; and Z (i ); i = 0; 1; 2;


; r. It is parameterized by r ? 1 parameters, hence it is a hyper-surface in IRn . This set of ecient solution also solves AWVVI by proposition 4.1.

5. An illustrative example Consider the following WVVI: 





y1 ? x1 6< 0; 8x 2 C; Find x 2 C; s:t: x01 2x0 y2 ? x 2 2 where C = fx j x1 + x2  1g. The equivalent WMCQP is Weak ? min (x21 ; 2x22 )> s:t: x1 + x2  1: The AWVVI and WMCQP here have 











F(x) = x01 x02 ; Q1 = 10 00 ; Q2 = 00 02 ; which satisfy assumptions 1 and 2. Application of algorithm 4.1 to the problem WMCQP yields the following solution to the problem AWVVI: 8 < f(t 0)> j t  1g if  = 0, x = : ( 2(12+?) ; 2+ )> if  2 (0; 1), >

f(0 t) j t  1g if  = 1.

6. Conclusion Hitherto vector variational inequality has mainly been studied in the theoretical framework. We show that it can be used to model a practical operations research problem in the form of vector trac equilibria. Other vector economics equilibria models can also be studied in the same vein. A number of results relating vector variational inequality and vector optimization problems have also been obtained. We also show how a special case of vector trac equilibria in the form of a weak vector variational inequality de ned by ane functions can be solved analytically. The solution of a general nonlinear vector variational 15

inequality without the symmetry assumption is still an outstanding problem. Research in this direction is currently underway.

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