Iterative methods for variational and complementarity problems

Mathematical Programming 24 (1982) 284- 313 North-Holland Publishing Company

ITERATIVE METHODS FOR VARIATIONAL AND COMPLEMENTARITY PROBLEMS*

J.S. P A N G * * and D. C H A N

Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.

Received 4 May 1981 Revised manuscript received 9 November 1981

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.

Key words: Variational Inequality, Complementarity. Iterative Methods, Convergence, Traffic Equilibria.

1. Introduction

Given a subse t K of R" and a mapp ing f f rom R" into itself, the var ia t ional

inequal i ty p rob lem VI(K, f ) is to find a vec to r x* E K such that

(y - x*)T f (x *) >- 0 for all y ~ K.

In the impor t an t special case where the set K is the nonne ga t i ve or thant R+, the

p rob lem VI(K, f ) is equ iva len t to the non l i nea r c o m p l e m e n t a r i t y problem: F ind a

vec tor x E R" so that [30]

x >- O, f ( x ) >- 0 and xTf(x) = 0, (1)

The var ia t ional inequa l i ty p rob lem has b e e n called var ious ly as the s ta t ionary

po in t p rob lem by Eaves [16] and the genera l ized equa t ion by R o b i n s o n [44]. In

this paper, we shall use the term var ia t ional inequal i ty .

The theory as well as the appl ica t ions of bo th the var ia t ional inequal i ty and

the non l inea r c o m p l e m e n t a r i t y p rob lems has been well d o c u m e n t e d in the

* This research was based on work supported by the National Science Foundation under grant ECS-7926320.

** Present address: School of Management and Administration, The University of Texas at Dallas, Richardson, TX 75080, U.S.A.

284

J.S. Pang and D. Chan/ Variational and complementarity problems 285

literature. Various extensions of these two problems have recently been introduced and studied by many authors including Saigal [46], Fang and Peterson [20] and Chan and Pang [6].

The present research was partly motivated by numerous recent works in which iterative methods are proposed for solving the variational inequality and the nonlinear complementarity problems arising from diverse applications. The papers by Eaves [16] and Josephy [28, 29] analyze methods of the Newton-type for solving the variational inequality problem with application to the PIES model [24]. The papers by Irwin [25, 26] and the dissertation of Ahn [3] deal with algorithms of the PIES-type. As pointed out by Ahn [3] and Thrasher [51], the original PIES algorithm [24] is really the well-known Jacobi iterative method for systems of linear and nonlinear equations [34] extended to treat the PIES-model. In a related paper [27], Irwin and Yang discuss a similar (iterative) method for solving a certain nonlinear complementarity problem arising from an economic spatial equilibrium model. Aashtiani, in his doctoral dissertation [1] proposes a Newton method for solving the traffic equilibrium problem formulated as a nonlinear complementarity problem. Except in a very special instance, convergence of the method has not been established. Incidentally, Asmuth [5] has formulated the same traffic equilibrium problem as a variational inequality problem and proposes the use of the Eaves-Saigal [17] fixed point algorithm for its solution. Along a slightly different line of approach, Dafermos [12, 13] and Fang [18, 19] have established the (global) convergence of an iterative method for solving the traffic equilibrium problem with fixed (i.e., inelastic) demand under certain strong monotonicity assumption on the cost function. Dafermos' result is more general and includes Fang's as a special case. Quite independently, Aganagic [2] has obtained exactly the same convergence result as Fang. In fact, their analyses are identical. Basically, the method of Dafermos, Fang and Aganagic is of the projection type. Such a method has been used to treat various extensions of the variational inequality and nonlinear complementarity problems referred to earlier, [20, 6].

In essence, all the methods discussed in the aforementioned references can be cast as special cases of the following scheme.

Algorithm. Given x k E K, let x k+l solve the variational inequality subproblem VI(K, fk) where /k is some mapping approximating the given f at the point x k.

Presumably, each fk should be such that the subproblem V I ( K , f k) is numerically easier to solve than the original problem VI(K,f). The actual computation of each x k+l is not of particular concern in the present paper, however.

We shall classify the method as a linear approximat ion me thod if fk is of the form

f k (x ) = f ( x k) + A(xk ) (x X k) (2)

286 J.S. Pang and D. Chan/ Variational and complementarity problems

where A ( x k) is an n by n matrix, or as a nonlinear approx imat ion method if fk is nonlinear. Included in the family of linear approximation methods are

(i) the Newton method in which the mapping f is differentiable and A ( x k) = V f ( x k) for each k, [16, 28];

(ii) the quasi-Newton methods in which A ( x k) is an approximation of the

gradient matrix Vf(xk) , [29]; (iii) the successive overrelaxation (SOR) methods in which A(x k) = L (x k) (or

U(xk)) + D(xk)/~o * where ~o* is the relaxation parameter such that 0 < ~o* < 2 and where D(xk) , L ( x k) and U ( x k) are respectively the diagonal, strictly lower and upper triangular parts of Vf(xk), [39, 34] (note that the matrix A ( x k) is

triangular in these SOR methods); (iv) the linearized Jacobi method in which A ( x k) = D(xk); and (v) the projection methods in which A ( x k) = G for each k where G is some

fixed symmetric positive definite matrix, [12, 13]. The reason why methods of the last category are termed projection methods is

due to the following geometrical interpretation of the iterates {xk}. Indeed, it is easy to show that if the set K is closed and convex - - an assumption which we shall maintain through the paper - -and if A ( x k) = G is symmetric positive definite, the vector x k+l solving the subproblem V I ( K , f k) is precisely the projection of the point x k - G-~f(x k) onto the set K where the projection is defined with respect to the G-norm, i.e.,

x k+l = PrK(x k - G l f(xk))

where for a given vector z, PrK(z) is the (unique) vector solving the program

minimize IIz- yllo, subject to y E K

and

Ilxll~ = (xTGx) '/2 (3)

is the G-norm of the vector x. The SOR methods (i.e., methods of the third category) originate from the solution of systems of linear equations [52] and have been discussed and applied extensively for solving many (large-scale) linear complementari ty problems, [7, 8, 10, 11, 32, 36, 40]. When o~* = 1, we obtain the linearized Gauss-Seidel method.

It is the purpose of this paper to develop a convergence theory for the general iterative scheme introduced above to solve the variational inequality and nonlinear complementari ty problems. Such a theory would provide (sufficient) conditions under which the sequence {x k} is well defined and converges, either locally or globally, to a solution of the original variational (or complementarity) problem. More importantly, the theory unifies and extends many existing convergence results, including those referred to above. We shall emphasize more on the family of linear approximation methods, although several convergence

J.s. Pang and D. Chan/ Variational and complementarity problems 287

results will also be obtained for the nonlinear methods. Such emphasis is partly justified because there are many efficient algorithms for solving linear variational and complementarity problems. In fact, if the set K is polyhedral (which is the case in many applications such as the aforementioned ones), then each subproblem VI(K,/k) where fk is given by (2) can be cast as a linear complementarity problem and thus can be solved by numerous algorithms including some of the more recent ones that are designed specifically for solving a linear variational inequality problem defined on a polyhedral set [14, 15, 37].

As mentioned in [40] for the case of the linear complementarity problem, there are four approaches that can be used to establish the desired convergence of the sequence of iterates generated by the general scheme. These are the symmetry approach, the norm-contraction approach, the vector-contraction approach and the monotone approach. The symmetry approach depends crucially on the assumption that the mapping f is the gradient of some numerical function g. Under such an assumption, a solution x* of the variational inequality problem VI(K, f) is then a stationary point of the optimization problem

minimize g(x),

subject to x ~ K

and the objective function g may be used to monitor the progress of the iterations. A necessary condition for such an 'integrability' assumption to hold is that the gradient matrix Vf(x ) should be symmetric. As this symmetry condition is generally not satisfied in such applications as those mentioned earlier, we shall in the rest of the paper, not further discuss the symmetry approach for convergence. The norm-contraction approach has been used by various people including Eaves [16], Josephy [28, 29], Irwin [25, 26], Ahn.[3], Thrasher [51] and Dafermos [12, 13]. The vector-contraction approach has been used by Ahn [4] and Pang [39]. The monotone approach has been used by Pang [38].

Motivated by much recent interest in the computation of traffic equilibria via the variational/complementarity approach, we shall study a special class of variational inequality problems of the form VI(K, f) with

\ h(z) ] + E + and K = K I × K 2 (4)

where g and h are mappings from R m and R ~ into themselves respectively, B and E are m x n and (n + l) x (n + l) matrices respectively, c and d are vectors in R" and R ~ respectively, and K1 and K2 are subsets of R" and R ~ respectively. See [1, 5] for the interpretation of the mappings g and h and matrices B and E in the context of the traffic equilibrium problem. (Incidentally, many other network equilibrium models also lead to variational inequality problems of the above type. Several relevant references are [21, 41, 36, 49].) This special class of variational problems is interesting, not only because of its many applications, but also because of its particular structure. For instance, one generally cannot


establish the convergence of the sequence of y-vectors , mainly because of the

presence of the matrix B. The convergence results obtained in the paper may be classified as local or

global. Typically, a local result would assert that if the initial iterate x ° is chosen in a sufficiently small neighborhood of a solution to the given variational problem (which is assumed to exist), then convergence of the iterates to the solution can be established under suitable conditions on the data. On the other hand, a global result generally does not require the iteration to start close to a solution.

The organization of the remainder of this paper is as follows. Section 2 discusses the norm-contract ion approach for convergence. It is divided into two subsections. In the first subsection, we establish a general local convergence result and derive f rom it several special cases. Included among them are the linearized Jacobi and Newton ' s methods. In Section 2.2, we establish a global convergence result and discuss how it can be used to prove the convergence of the project ion methods. The organization of Section 3 is similar to Section 2. It deals with the vector-contract ion approach and is also divided into two subsections, one for local results and the other for global. Local convergence of the SOR methods will be established here. Section 4 deals with the monotone approach. There, as a special case of the general convergence theory, we give a global convergence result for Newton ' s method. In Section 5, we establish the convergence of two nonlinear methods. Finally, in the sixth and last section, we

study the special variational problem (4).

2. The norm-contraction approach for convergence

2.1. Local results

To state our first convergence result, we recall some definitions. If G is a

symmetr ic positive definite matrix, then the G-norm of a vector x is given by (3) and the induced G-norm of a matrix B is

IIBII = max IIBxll . IIxLlc = 1

In defining IIBII~, it is unders tood that B is square and of the same order as G. If G is the identity matrix, then the G-norm becomes the standard Euclidean (i.e.,

2-) norm which we denote by I1" I1=, It is known that [33]

IIBII2 = p(B B)

where p denotes the spectral radius of a matrix. In particular, if B is symmetr ic ,

then IIBII = o(B). Note that

Ilxll --IIG'/=xll= where G it: denotes any nonsingular matrix such that G =(GI/2)TG1/Z. (For

instance, G ~/z may be the Cholesky factor of G.) The following fact will be used


later: if G is a symmetric positive definite matrix, then

IIG-'II2 = p(G- ' ) = l/ct

where a---minllyll2=~ yTGy. Given any square matrix B, we shall denote its symmetric part by/~, i.e.,/~ = ~(B + BT). The set of real square matrices of order n is denoted by R "×".

The following theorem is the main local convergence result for the family of linear approximation methods using the norm-contract ion approach. In stating the theorem, we assume that a solution to the variational problem exists. Conditions ensuring the existence of such a solution can be found in [30, 48].

Theorem 2.1. Let K be a nonempty closed and convex subset of R n. Let f :R" o R " and A:R" ~ R n×n be continuous. Suppose that x* solves the problem

VI(K, f). If there exist a positive definite matrix G and a scalar b < 1 such that A ( x * ) - G is positive semi-definite and for all x and y in some neighborhood N~

of X*,

IIG-l(.f(x) - f ( y ) - A(y)(x - Y))I[~ -< b l l x - y[[~, (5)

then provided that the initial vector x ° is chosen in a suitable neighborhood of x*,

the sequence {x k} generated by the linear approximation method is well defined and converges to the solution x*.

Remark. The matrix G is not assumed to be symmetric.

Proof of Theorem 2.1. First of all, note that the matrix A(x*) must be positive definite. Thus, since the mapping A is continuous, there exists a neighborhood N2 of x* such that A(y) is positive definite for each y in N2. By a classical result of Stampacchia [48], the variational problem VI(K, fY) where

fY(x) = f (y) + A(y)(x - y)

has a unique solution for y E N2. Since G is positive definite, so is its symmetric part G. Therefore , the G-norm in (5) is well defined. Let • > 0 be such that

0 < r = b/(1 - •) < 1. Then there exists a neighborhood N3 of x* such that for each y E N3

II(d-'/2)'r(A(x *) - A(y))d-'/=ll2 _< •.

With no loss of generality, we may assume that the neighborhoods Ni are defined with respect to the G-norm. Le t N = /") ~=~ Ni and let the iterate x ° be chosen in N. Then x ~ is well defined (and in fact unique). Since x 1 solves the subproblem VI(K, fo) and x* E K, we have

(x* - x ' ) r ( f (x °) + A(x°)(x 1 - x°)) >- O.

Similarly, since x* solves the original variational problem V I ( K , / ) and x l E K,


we have

(X 1 -- x * ) T f ( x *) >~ O.

Adding the last two inequalities and rearranging terms, we obtain

(X 1 -- x*)TA(xO)(X 1 - X*) <~ (X 1 -- x*)T(f(X *) -- f ( x °) -- A(x°)(x * - x°)).

Writing A(x °) = A(x*) + (A(x °) - A(x*)), we have

(X 1 -- x*)T A(xO)(x 1 -- X*) =

(X 1 -- x*)T A ( x * ) ( X 1 -- X*) + ( X 1 - - x * ) T ( A ( x O) -- A(x*))(x l - x * )

(X 1 - X c ¢ ) T G ( x l -- X g:)

-~- (X 1 - x*)T(GI/2)T(G-1/2)T(A(x O) -- A(x*))G-I/2d'/2(x ' - x* )

-> I t x ' - x*ll~(1 -II(d-'/z)T(A(x °) -- A(x*))d-'/=ll2) -> (1 - ~)llx 1_ x*ll~.

Furthermore,

(x ' - x*)T(/(X *) -- I ( X °) -- a ( x °)(x * - x °)) =

= (x ' - x*)T(d'/2)T(d-' /2)T(I(X * ) -- I(X °) -- A ( x ° ) ( x * - x ) )

--< I I x ' - x* l ld d - ' ( / ( x * ) - f ( x °) - A ( x ° ) ( x * - x°)lle

<- b l l x ' - x* l ldx ° - x*ll~.

Consequently, it follows that

I Ix ' - x*ll~ -< rll x ° - x*ll~.

By the definition of r, it follows that x 1 remains in the neighborhood N. Using the same argument, we may deduce that the whole sequence {x k} is well defined and satisfies

II x k + ' - x*ll~-< rll xk - x*ll~ for k -> 0.

By a standard contraction argument (see e.g. [34, p. 120]), the sequence {x k} converges to the solution x* as desired.

In what follows, we give several sufficient conditions for (5) (and thus Theorem 2.1) to hold.

Corollary 2.2. Let K, [, A and x* be as in Theorem 2.1. I f there exist a positive definite matrix G and a scalar b ' < a = minllzll2=l zTGz such that A ( x * ) - G is positive semi-definite and for all x and y in some neighborhood of x*,

IlI(x) - I ( y ) - A(y)(x - r)l12 -< b'llx - Yll~, (6)

then the conclusion of Theorem 2.1 holds.

Proof. It suffices to show that condition (6) implies (5) holds locally for some

constant b < 1. Note that [Ix - YlI~ -> ~llx - yllg and recall that IId-'llz = 1/~. Let x


and y be two v e c t o r s sa t i s fy ing (6). T h e n

I I d - ' ( / ( x ) - f ( y ) - A ( y ) ( x - Y) ) I I~ =

= ( f ( x ) - f ( y ) - A ( y ) ( x - y ) ) X ( ~ - l ( f ( x ) - f ( y ) - A ( y ) ( x - y))

-~ ( b ' l l x - y l 1 9 2 / a -< ( b ' l l x - y l l ~ ) 2 / ~ =

Thus wi th b = b'/a < 1, (5) fo l lows .

C o r o l l a r y 2.3. Let K, f, A and x* be as in Theorem 2.1. Let f be continuously differentiable in a neighborhood of x*. I f there exist a positive definite matrix G

and a scalar 8 < a = minllzll2-1 zTGz such that A ( x * ) - G is positive semi-definite and for all x in some neighborhood of x*,

IIA(x) - Vf(x)ll2 -< 8, (7)


Proof . By the m e a n va lue t h e o r e m , we h a v e

I (x) - f ( y ) - A(y ) (x - y) = (B(x, y) - A(y ) ) (x - y)

= (B(x, y) - Vf (y) ) (x - y)

+ (Vf(y) - A(y ) ) (x - y)

w h e r e B(x, y) is the ma t r i x w h o s e i th row is Vf~(x + t~(y - x)) for s o m e t~ ~ (0, 1).

By cond i t i on (7) and the con t i nuous d i f f e ren t i ab i l i t y on f, (6) fo l lows eas i ly . This

e s t a b l i s h e s the co ro l l a ry .

R e m a r k . C o n d i t i o n (7) o b v i o u s l y ho lds ( loca l ly at x*) if f is c o n t i n u o u s l y

d i f fe ren t i ab le and Vf(x*) = A(x*).

C o r o l l a r y 2.4. Let K, f, A and x* be as in Theorem 2.1. Suppose that f is continuously differentiable in a neighborhood of x* and that A(x*) is positive

definite. Let B(x*) = Vf(x*) - A(x*). I f

II(/(x*)-'/bTB (X *) A(x *)-"211= < 1, (8)


Proof . W e show tha t cond i t i on (5) wi th G = A ( x * ) ho lds loca l ly at x*. In fac t ,

f r om the p r o o f of C o r o l l a r y 2.3, we o b t a i n

IIG-~(f(x) - f ( y ) - A ( y ) ( x - Y))H~ =

= (x - y)'r[(B(x, y) - A(y))Td-~(B(x, y) - A(y))](x - y)

w h e r e B(x, y) is as g iven in C o r o l l a r y 2.3. W e m a y wr i t e

B(x, y) - A ( y ) = (B(x, y) - Vf(x*)) + (Vf(x*) - A(x*))

+ (A(x*) - A(y) ) .

292 J.S. Pang and D. Chan[ Variational and complementarity problems

By letting C ( x * , z ) = B ( x , y ) - V f ( x * ) where z = ( x , y ) and A ( x * , y ) = A ( x * ) - A(y), it follows that

I I d - ' ( / ( x ) - I ( y ) - A(y)(x - Y))II~ -- = (X -- y)T(OI/2)T(O-1/2)T[C(x*, Z) + B(X*) + A ( x * , y ) ] d -1/2

× ( O - ' / 2 ) T [ C ( x *, z) + B(x*) + A(x*, y ) ] d - ' / 2 d i / 2 ( x - y)

-< IIx - y U~U(d- ' /=)T(C(x *, z) + B(x*) + m(x*, y))G" -,/22112.

By the continuous differentiability of f and condition (8), we can easily deduce that (5) holds locally at x* for some constant b < 1. This establishes the corollary.

Remark. Condition (8) was motivated by Ahn's work [3] dealing with the PIES algorithm. It can be shown that if (8) holds, then the matrix Vf(x*)= A(x*) + B(x*) must be positive definite (see, e.g., [3, p. 33]).

The next result gives a sufficient condition for (8) to hold.

Proposition 2.5. Let A be a positive definite matrix and B a matrix of the same order as A with

IIBII~ < ,~ = min yTAy. HylI2 = '

Then, tI(A.-I/2)TBA-~/:II2 < 1.

Proof. The idea of the proof is very similar to that for Corollary 2.2. For completeness, we give a detailed proof. For x ~ 0, we have

- , / 2 2 _ 1 1 A_,/2 x -YzXY'"Y ' ' 2 - < ' ~ 1,, If2 a where y = . Ilxll~

Thus,

II(A-~NBA-"=x]I~ = xT(A-"=)TB~A- 'BA-"=X IIxlI~ IIxlI~

--< IIA-~II2 IIA-l/2x U ?z " tlxll~

Consequently, it follows that

I I (A- ' /=)TBA-I /211 : = m a x I[(A-I/2)TBA-I/2xH21HxH2 ~ HBll2/ot < 1.

Remark. The converse of Proposition 2.5 does not hold as the following example shows. Let

B = (0 2 ~), A = (~ ; ) and 41/2= (~ ~). Obviously, IIBH2 > 1 = minllyll2=, yTAy. But II(A 1/2)TBA '% = 2/3 < 1.


In what follows, we give two applications of the above results. The first application concerns Newton 's method in which A ( x ) = Vf(x) for all x and the second the linearized Jacobi method in which A(x) is the diagonal part of Vf(x).

Corollary 2.6. Let K, f and x* be as given in Theorem 2.1. Suppose that f is continuously differentiable with Vf(x*) being positive definite. Then there exists a neighborhood of x* such that if the initial iterate x ° is chosen there, the sequence {x k} generated by Newton 's method is well defined and converges to x*.

Moreover, if V f is Lipschitz continuous at x*, i.e., if there exists a neighborhood N of x* and a positive scalar 3' such that for all x and y in N,

I IW(x) - W(y)I I2 ~ 3"llx - yll=,

then {x k} converges quadratically to x*, i.e., there is a constant c such that

IIx ~÷1- x*l12~ cllx k - x*ll~ f o r all k.

ProoL The convergence is an immediate consequence of Corollary 2.3. To establish that the convergence is quadratic, we note that the proof of Theorem 2.1 gives

~(1 - ~)llx k + l - x*ll, ~ -< (x k + l - x * ) T q ( x *) - f ( x k) - Vf(x k)(x * - xk))

where a = minllrll2=i yTVf(x*)y > 0 and e > 0 is some small scalar less than 1. By the mean value theorem, we obtain

a(1 - e)[Ix k÷~- x*l12

-< s u p IlVf(x k + t (x* - x ~ ) ) - Vf(x~)lHIx ~ - x*ll= 0~<t_<l

-< 3'1ix ~ - x* l l~ .

From this last inequality, we may conclude that the convergence is quadratic.

Remark. The above convergence result for Newton ' s method has previously been obtained by Eaves [16] and Josephy [28]. In Eaves ' t reatment, Vf is not assumed positive definite but has a rather special form. In fact, f is given by

(c) f ( y , z ) = g(z)

where Vg is assumed positive definite. The quadratic convergence is established only for the z-variables. (See also the last section.) On the other hand, Josephy derived his convergence result using a rather involved theory based on Robin- son's earlier work [44]. He has also obtained some very detailed error estimates.


Corollary 2.7. Let K, f and x* be as given in Theorem 2.1. Suppose that f is continuously differentiable in a neighborhood of x* and that Vf(x*) has positive diagonals. Let Vf(x*) = D(x*) + B(x*) where D(x*) and B(x*) are the diagonal and off-diagonal parts of Vf(x*) respectively. If

IID(x*)-l/2B(x*)D(x*)-I/2112 < 1, (9)

then the conclusion of Theorem 2.1 holds for the linearized Jacobi method.

Proof. This is an immediate consequence of Corollary 2.4.

The linearized Jacobi method has the computational advantage in that each matrix A(x k) is diagonal. In the case of the nonlinear complementarity problem (1), this property is particularly useful because the solution of each subproblem then becomes trivial. Of course, the overall convergence of the method remains dependent on the quantity I1D(x*)-~/2B(x*)D(x*)-l/2112 and on the choice of the initial iterate.

In essence, we could state a similar convergence result for the SOR methods. However, we choose not to do this because we feel that the required conditions for the convergence of these methods (other than the linearized Jacobi) would be hard to satisfy in practice. (For one thing, we need to assume that the triangular matrix A(x*)= L(x*) (or U(x*))+ D(x*)/co* is positive definite.) Instead, more reasonable assumptions will be imposed when we study the vector-contraction approach.

To conclude this section, we give an easy-to-verify sufficient condition for (9) to hold.

Proposition 2.8. Let A be a row diagonally dominant and strictly column diagonally dominant matrix with positive diagonal entries, i.e.,

a,i-> Y. l a,jl for all i,

aij > ~ laij[ for all j.

Let A = D + B be the decomposition of A into its diagonal and off-diagonal parts. Then

IID-I/2BD-~/2112 < 1.

The same conclusion holds if A is column diagonally dominant and strictly row diagonally dominant with positive diagonal entries.

Proof. We only prove the conclusion under row and strict column diagonal


dominance. Let x be such that Ilxl12 = 1. Then we have

< E ]

Therefore ,

Ib'Jllxjl2 • dj

• dj

txf= 1.

(Cauchy-Schwartz in equality)

(row diagonal dominance)

(strict column diagonal dominance)

IID-1/2BD-'/2I]: = max liD 1/2BD-112xlI2 < 1. Ilxll2= 1

2.2. Global results

Basically, global convergence results can be obtained by imposing the global version of the conditions in Theorem 2.1 and in its Corollaries (2.1 to 2.4). In what follows, we give two global results based on Theorem 2.1 and Corollary 2.3 and show how they can be applied to the projection methods.

T h e o r e m 2.9. Let K be a nonempty closed and convex subset of R". Let f be continuous on K and A be bounded on compact subsets of K. I f there exists a positive definite matrix G with A(x ) - G positive semi-definite for all x in K and if there is a constant b < 1 such that condition (5) holds for all x and y in K, then the sequence of iterates {x k} generated by the linear approximation method

converges to a solution of the variational problem VI(K, f ) for any initial vector x ° ~ K .

R e m a r k . The existence of solution to the variational problem V I ( K , f ) is a consequence rather than an assumption of the theorem.

P r o o f o f T h e o r e m 2.2. Note that the matrix A(x) is positive definite for all x E K. Thus each x k+l is well defined. Since x k÷l solves VI(K, fk) and x k ~ K, we have

(X k -- xk+l)T( f (X k) -]- A ( x k ) ( X k+l - - xk)) >-- O.

Similarly, we have

( X k + l - - x k ) T ( f ( x k - l ) "~- A(xk -1 ) (xk -- x k - l ) ) :> 0 .


Adding the two inequalities and rearranging terms, we obtain

(X k+l -- xk)T A(xk)(xk+l-- X k) <__

<~ (X k -- xk+l)T{f(x k) - - f (x k-i) -- A(xk-1)(X k - x k-i)).

By using the same argument as in Theorem 2.1, we may easily deduce

II xk+*- xqlo -< bll x k - xk-lllo.

Consequently, by a standard contract ion result, it follows that the sequence {x k} converges to some vector x*. Since K is closed, x* E K. For each x ~ K, we have

(X -- x k + l ) T ( f ( x k) -~ A(xk)(X k+l -- Xk)) ~> 0 for all k.

Hence , passing the limit k ~ o~ and using the assumed propert ies on f and A, we conclude that x* solves the problem VI(K, f) . This establishes the theorem.

Corollary 2.10. Let K, f and A be given as in Theorem 2.9. Suppose that f is

differentiable. I f there exists a positive definite matrix G such that A ( x ) - G is

positive semi-definite for all x and if there exist scalars 3' and 8 such that

3' + 8 < a = minllsll2=l yTGy and for all x and y in K,

I IW(x)- W(y)II:-< 3' and I IW(x)- A(x)ll~-< 8,


Proof. By the mean value theorem, we have

Ill(x) - f ( y ) - A(y)(x - Y)II2 -<

_< [ s u p IIW(y + t ( x - y ) ) - Vf(y)ll2 + I lVf(y)- A(y)ll2JIIx - yll>

Hence, it follows that

Ilf(x) - f ( y ) - A(y)(x - y)ll2 -< (3' + 8)l[x - yll> Thus, the proof of Corollary 2.2 shows that condition (5) holds with b =

(3' + a)/a < 1.

Recall that the project ion methods are character ized by the fact that A ( x ) = G

for all x where G is some symmetr ic posit ive definite matrix. The following result is the specialization of Theorem 2.9 to such methods.

Corollary 2.11. Let K and f be as given in Theorem 2.9 and let G be a symmetric

positive definite matrix with Ix = p(G) and v = p(G-1). Suppose that f is Lipschi tz

continuous and strongly monotone with constants [3 and 3' respectively, i.e., .for

all x and y

Ilf(x)-ffy)ll2 -< [31Ix - yl12 a n d ( x - y)T(f(X) - f ( y ) ) -- vllx - yll,:.


If u2~2<2711~, then the conclusion of Theorem 2.9 holds for the projection method defined by the matrix G.

Proof. We show that under the given assumptions, condition (5) holds globally for some b < 1. Indeed, we have

IIG l ( f ( x ) - f ( y ) - G(x - y))ll~ =

= if(x) - f (y ) - G(x - y))TG-l(f(x) -- f (y) -- G(x - y))

= i f ( x ) - f ( y ) ) T G - ~ f f ( x ) - - f ( y ) ) - - 2 i f ( x ) - f(y))T(x -- y) + IIx - yl]~

-< ~ l l f ( x ) - f ( y ) l l ~ - 2v i i x - yll~ + IIx - y l ]~

- ~/3:l lx - y l l ~ - ¢ 2 v / ~ ) t l x - y l l~ + IIx - y l l~

_< (1 + ~ 2 / 3 2 - 2 v / ~ ) l l x - y l l~ .

Hence, if u2/32 < 27/ix, then b = 1 + u:fl 2 - 2,/[/z < 1 as desired.

Remark. It is known (see, e.g., [48]) that if f is strongly monotone, then the variational problem VI(K, f ) has a unique solution. Thus, under the conditions of Corollary 2.11, the sequence {x k} generated by the p.rojection method converges to the unique solution of the variational problem.

The condition v2/32< 2y[/x is generally not easy to be satisfied for an arbitrary symmetric positive definite matrix G. (For one thing, it imposes an upper bound on the condition number K ( G ) = p(G)p(G 1) of G; namely, K(G)<23,/[32v.) Nevertheless, given any symmetric positive definite matrix H, we may take G to be a positively scaled multiple )tH of H and choose the scalar )t so that the desired condition is met. The correct choice of )t is given in the next result.

Corollary 2.12. Let K and f be as given in Corollary 2.11 and let H be any symmetric positive definite matrix. I f )t > [32p(H)-~p(H-~)2/2% then the conclusion of Theorem 2.9 holds for the projection method defined by the matrix

G = )tH.

Proof. Indeed, with G =) tH, we have /x = p ( G ) = ) t o ( H ) and u = p(G-l) = )t-~p(H-l). If )t is chosen as such, it is easily seen that the condition v2/32 < 2V//~ holds.

Corollary 2.12 is basically the same result as the one obtained by Dafermos [12] in the context of the traffic equilibrium problem. The only difference is that instead of assuming the Lipschitz continuity condition on f, she assumes the slightly stronger condition of differentiability and works with the quantity maxxEr p[Vf(x)TH-IVf(x)]. We also point out that the result obtained by Fang [18, 19] and Aganagic [2] is a special case of Corollary 2.12 where H is the identity matrix.


3. The vector-contraction approach for convergence

3.1. Local results

One of the necessary conditions implied by the assumptions in the local convergence theorems of Section 2.1 is that the matrix A(x*) must be positive definite. As we have pointed out, such is a rather strong requirement in case A(x*) is triangular (as in the SOR methods). The vector-contract ion approach offers an alternative way to study the convergence of the linear approximation methods which does not require the positive definiteness of A(x*). However , the approach is applicable only to the nonlinear complementari ty problem and not to the more general variational inequality problem. So, throughout this section, we restrict our discussion to the nonlinear complementari ty problem (1).

To state the convergence results, we recall some matrix definitions. Let M be a real square matrix. Then M is Z (P respectively) if its off-diagonal entries are nonpositive (if its principal minors are all positive); M is K if it is both P and Z. Among the numerous characterizations of a K-matrix, the following is particularly useful. Let M be a Z-matrix, then M is K if and only if its inverse M -1 exists and is nonnegative [22]. The matrix M is H if its comparison matrix /~ defined by

~,j__{ IM,,I i f i = j , -IM,~I if i ~ j

is K. See [42, 35] for various characterizations of an H-matr ix . In particular, it is known that M is H if and only if there exists a positive diagonal matrix D such that MD is strictly row diagonally dominant and that an H-matr ix with positive diagonal entries must be P.

Note that in the case of the nonlinear complementari ty problem (1), each subproblem solved in the linear approximation method is a linear complementarity problem defined by the matrix A(xk). Hence if A(x k) is an H-matr ix with positive diagonals, then by a classical result in linear complementari ty theory [47], x k÷l exists and is unique.

Finally, recall that a vector norm I1" II is monotonic [33] if Ixl-<IYl implies Ilxll-<llyll where for each vector z, Izl is the vector whose components are the absolute values of the corresponding ones of z. We say that a matrix norm is monotonic if it is induced by a monotonic vector norm.

Theorem 3.1. Let f : R n ~ R n and A : R n o R n×" be continuous and let x* solve the

nonlinear complementarity problem (1). Suppose that A(x*) is H and has positive diagonals. If there exists a matrix C such that for all x and y in some neighborhood NI of x*,

I f ( x ) - / ( y ) - A(y)(x - y)[-< C[x - Yl (10)

and that II (x*)-lcII < 1 for some monotonic norm I1" II, then the sequence of

J.S. Pang and D. Chanl Variational and complementarity problems 299

iterates {x k} generated by the linear approximation method is well defined and converges to the solution x* for all x ° chosen in a sufficiently small neighborhood

of x*.

Proof. Since A(x*) is an H-mat r ix with positive diagonals and A is continuous, there exists a neighborhood N2 of x* such that for each y E N2, A(y) is an H-mat r ix with positive diagonals. Moreover , there exists an e > 0 such that . 4 ( x * ) - e E where E is the matrix of ones, remains a K-matr ix and H(A(x*)- eE)-IcII < 1. For such an e, there is a neighborhood N3 of x* such that for all x in N3,

[5 , (x*)- .4(x)l-< eE. (11)

Let N = A~=I Ni and let the initial iterate be chosen in N. Then x ~ is well defined and satisfies

x ~ >- O, f (x °) + A(x°)(x 1- x °) >- O, (X 1)T(f(X O) "]- A(x°)(x 1 - x°)) = O.

Since x* solves the problem (1), we have

x*>-O, f(x*)>-O and (x*)Tf(x * )=0 .

We evaluate Ix*-x~]. Consider an index i for which

Ix* - x ' l , = ( x * - x ' ) , .

If f(x*)i = O, then we have

(f(x °) + A(x°)(x 1 - x °) - f ( x * ) ) i >- O.

Since A(x°)ii = A(x°)ii and Ix* - x~li = (x* - xl)i we obtain

( A , ( x ° ) l x * - xlj), = A ( x ° ) , , ( x * - x l ) , - ~ , I A ( x ° ) ~ j ( x * - x ' ) j l

<_ ( A ( x ° ) ( x * - x l ) ) ,

<- - I f ( x * ) - I ( x °) - A ( x ° ) ( x * - x° ) ] , <- ( C I x * - x°l) , .

Thus by writing ~,(x °) = ~/,(x*)+ (/~(x °) - A ( x * ) ) and by using (11), we obtain

[ ( X ( x * ) - ~ E ) l x * - x ' l ] , -< [ C I x * - x°l] , .

On the other hand, if x* = O, then Ix*-x'l,--0 and the above inequality also holds because the left hand term is ---O. Similarly, we can show that the same inequality must hold for those indices i for which Ix*-x'l,--(x'-x*),. Con- sequently, we deduce

( ~ ( x * ) - , E ) l x * - x' l - C I x * - x°l.

Since (fi,(x*) - eE) has a nonnegative inverse, we obtain

Ix* - x ' l - < ( . ~ ( x * ) - ~ E ) - ' C I x * - x°l .


Since the norm I1" II is monotonic , it follows that

IIx* - xl l l -< I I ( , i (x*) - E)- CIIIIx * - x°ll.

From this inequality, we may conclude easily that the entire sequence {x k} is well defined and converges to the solution x*.

Remark. We require that the norm (instead of the spectral radius) of the matrix A(x*)-IC to be less than 1 in order for the whole sequence {x k} to contract inside

a suitably chosen neighborhood of the solution x*. (See also Theorem 3.6.)

Corollary 3.2. Let [, A and x* be as given in Theorem 3.1. Suppose that A(x*) is an H-matrix with positive diagonal entries and that f is continuously differentiable in a neighborhood of x*. If

II~,(x*)-'lB(x*)lll< 1 where B(x*) = V / ( x * ) - A(x*) (12)

for some monotonic norm I1" II, then the conclusion of Theorem 3.1 holds.

Proof. As in Corollary 2.4 we obtain

[ (x) - f (y ) - A(y)(x - y)

= [(B(x, y) - V/(x*)) + (V/(x*) - A(x*)) + (A(x*) - A(y))](x - y)

where B(x, y) is as given in Corollary 2.3. Hence if x and y are sufficiently close to x*, we have

If(x) - f ( y ) - A(y ) (x - Y)I -< [eE + IB(x*)l]lx - yl

where E is the matrix of ones and e > 0 is such that [li,(x*)-'(~E + In(x*)l)ll < 1. Thus, condition (10) holds with C = EE + IB(x*)l. This establishes the corollary.

Remark. I f (12) holds, then o(5,(x*)-'lB(x*)l)< 1. By [39, Corollary 3.5], this implies that the matrix Vf (x*)=A(x*)+B(x*) is itself H with posit ive diagonals.

An immediate consequence of Corollary 3.2 is the following convergence result for Newton ' s method.

Corollary 3.3. Let f and x* be as given in Theorem 3.1. Suppose that [ is continuously differentiable in a neighborhood of x* and that Vf(x*) is an H-matrix with positive diagonals. Then the conclusion of Theorem 3.1 holds for Newton's method.

In the remainder of this subsection, we investigate the application of Corollary 3.2 to the SOR methods. Recall that in these methods A(x) = L(x) (or U(x))+ D(x)/to*. In what follows, we consider only the fo rmer case as the t rea tment for


the la t ter is similar. W e no te that in these S O R me thods , the solut ion of each

s u b p r o b l e m is tr ivial b e c a u s e A(x) is t r iangular .

Lemma 3.4. Let A be a strictly row diagonally dominant matrix with positive diagonals. Let A = D + L + U be the decomposition of A into its diagonal, strictly lower and strictly upper triangular parts respectively. Then for each to ~ (0, to*) where to* = 2 mini aii/~,j lai~l, it holds that

II(D - tolLI)-~(l(1 - to )D + toUI)ll~ < 1

where I1" I[~ is the standard ~-norm for vectors and matrices.

R e m a r k . to* E (1, 2].

Proof of L e m m a 3.4. L e t x be such tha t Ilxll~ = 1. I t suffices to show tha t Ilyll~ < 1 whe re

y = ( D - to lLI)- ' ( l l - t o l d + tolUl)x.

We show b y induct ion tha t lyil < 1 for each i. Since

(D - tolLl)y -- (11 - tolD + tol Ul )x (13)

we have

lyil ~ (11 - tola,,Ix,I + to E la,sllx, I)/la,,I

(11 - tolall + to ~ lalsl)/la,I.

I f to -- 1, then

lyd ~ ((1 - to)all + to ~ la~jl)la, < 1 j~l

because al l > Ej~I lalsl. On the o ther hand if co ~ (1, to*), then

lyll ~ ((to - 1)al~ + to ~ lalsl)/aH < 1 ]41

by the cho ice of co*. N o w suppose tha t max~_<i_<k_~ lYll < 1. T h e n f r o m (13), we obta in

ly~l ~ (11 - tola~klxkl + to E lakJllxil + to E laksllYJl)/a~ j>k j<k

-~(11 - tolakk ÷ to ~ lak~l)lak~. j~k

B y the s ame a r g u m e n t as fo r the case k = 1, we deduce tha t [Yk[ < 1, comple t i ng the induct ion and the p roof .

302 zs. Pang and D. Chan/ Variational and complementarity problems

Corollary 3.5. Let f and x* be as given in Corollary 3.3. Then there exists a to* E (1, 2] such that for all to ~ (0, to*), the conclusion of Theorem 3.1 holds for the SOR method with A(x ) = L(x) + D(x)ltg.

Proof. Let B(x) = U(x) + ( 1 - 1/to)D(x). Since f (x*) is an H-matr ix , there is a positive diagonal matrix ~ such that V/(x*)O is strictly row diagonally dominant (by a remark made earlier). Hence by Lemma 3.4, there exists a to* E (1, 2] such that for all to E (0, to*),

11o-'5,(x*)-llB(x*)l ll < 1.

The norm I1" [I defined by

Ilxll = Ila- xll

is certainly monotonic. It is easily seen that the induced matrix norm is given by

IICII = Ilo-'c ll .

Consequently, for this monotonic norm, (12) holds. The desired conclusion now follows from Corollary 3.2.

3.2. Global results

In what follows, we derive two global convergence results based on the vector-contract ion idea.

Theorem 3.6. Let f : R n ~ R n and A : R ~ R ~×n be continuous. Suppose that A(x )

has positive diagonals for all x E R~. I f there exists a matrix C such that (10) holds for all vectors x and y in R~, and if there exists a K-matr ix G such that A(x) >- G for all x ~ R~_ and p(G-IC) < 1, then for all initial iterate x ° in R~+, the sequence {x k} generated by the linear approximation method is well defined and converges to some solution of the nonlinear complementarity problem (1).

Proof. By a result in [22], the matrix A(x) is H for all x ~ R~. Thus each x k is well defined. Using the same argument as in the proof of Theorem 3.1, we may easily deduce that

I x k + l - xkl _< G - ' C I x _ x -ll.

By a contraction argument, it follows that the sequence {x ~} converges to some vector x*. Since f and A are continuous, we conclude easily that x* solves the nonlinear complementari ty problem (1).

Parallel to Corollary 2.6, we establish:

Corollary 3.7. Let A:R" ~ R "×" be continuous and f :R n->R" differentiable. Suppose that A(x ) has positive diagonals for all x ~R"+. I f there exists a

J.S. Pang and D. Chart/Variational and complementarity problems 303

K-matr i x G such that A ( x ) >- G for all x in R~+ and if there exist matrices F and

with p (G- I (F + E)) < 1 and f o r all x and y in RT-,

I V f ( x ) - Vf(y) I <- F and ] V f ( x ) - A(x)l -< ~,

then the conclusion o f Theorem 3.6 holds.

Proof. We have, for all x and y in R~,

If(x) - f ( y ) - A ( y ) ( x - Y)I <- fiB( x, Y) - Vf(Y)I + IVf(Y) - A(Y)I]Ix - Y]

(r + )Ix - yl

where B(x, y) is the matrix defined in the proof of Corollary 2.3. Thus, condition (I0) holds with C = F + ~ and Theorem 3.6 applies.

4. The monotone approach for convergence

In the last two sections, we have derived many convergence results for the linear approximat ion methods. Here , we establish a few more such results by using a somewhat different kind of argument. We first recall some definitions. Let f : R" ~ R" be convex. Then an n by n matrix A is a subgradient of f at the point x if

f ( y ) >- f ( x ) + A ( y - x ) for all vectors y in RL (14)

The set of subgradients of f at x is denoted by Of(x). Let f : R" ~ R ~. Then f is called a Z-funct ion [43] if for each x E R ", the scalar-valued function

g t i ( t ) = f t ( x + t e l ) for i ~ j

where e ~ is the ith unit vector , is nonincreasing. If f ( x ) = A x + b for some n × n matrix A, then the function f is Z if and only if the matrix A is so. More generally, we have:

Lemma 4.1. Le t f : R" ~ R" be convex. Then f is a Z - f u n c t i o n if and only if each

subgradient matrix A E af(x) is Z f o r all x E R".

Proof. Sufficiency. Let x E R", i ~ j and t and s be two scalars with t -> s. Then by the subgradient inequality (14), we obtain

f t (x + se i) - f t ( x + te i) >- (s - t )Aje i >- 0

where A t is the j th row of a matrix A E Of (x+te l ) . Consequent ly , f is a Z-function.

Necessi ty . Let A E af(x). For each i~ j, choose y = x + e i. Then

0 >_ f j (x + e i) - f t (x) >- Atei = Ajl.

Thus A is a Z-matr ix.

304 J.s. Pang and D. Chan[ Variational and complementarity problems

The convergence results obtained below are global in nature but applies only to the nonlinear complementari ty problem (1). We recall that a vector x is feasible to the problem (1) if x -> 0 and f ( x ) >- O.

Theorem 4.2. Le t f : R n ~ R n be convex and A : R n ~ R n×" be such that A ( x ) U

Of(x) f o r all x. Suppose that there exists a K - m a t r i x X such that A ( x ) X is Z f o r

all x feas ib le to (1). Then if (1) is feas ib le and x ° is chosen to be feasible, there exists a well defined sequence {x k} where each x k÷~ solves the linear c o m -

p lementar i ty subproblem

x >-- O, f k ( x ) = f ( x k) + A (xk ) ( x -- X k) --> O, xTfk(x) = 0 (15)

such that {x k} converges to some solut ion o f the nonl inear complemen tar i t y

problem (1).

Proof. The assumption implies that each A ( x ) is hidden Z [9, 31]. Thus, provided that the linear complementari ty subproblem (15) is feasible, it has a solution x k÷~

which satisfies

X-lxk+l~ X - I x for all x feasible to (15). (16)

Consequently, to show that the sequence {x k} is well defined, it suffices to show that (15) is feasible. By means of an inductive hypothesis, we may assume that x k is feasible to the original nonlinear complementari ty problem (1). Then obviously, the same x k is feasible to (15). To complete the induction, we must show that x k+l is feasible to (1). By the fact that A ( x k) E Of(xk), we obtain

f (x k+l) -> f (x k ) + A(xk ) ( x k+l - x k ) --> 0

where the second inequality holds because x k÷l solves (15). It remains to show that the sequence {x k} converges to some solution of the nonlinear complementarity problem (1). By (16), we have

X - i x k+l ~ X - i x k.

Since X is a K-matrix, X - i x k >-0 for each k. Consequently, the sequence {X-ix k} is nonincreasing and bounded below, therefore converges. Hence, so does the sequence {xk}. Le t x* be the limit of {xk}. By [45, Theorem 24.7], the sequence {A(xk)} is bounded. Consequently, by passing the limit k-~ ~ in

x k+l --> O, f ( x k) + A (xk ) ( x k+l -- X k) ~ O,

(xk+l)T(f(x k) "q- A (xk ) ( x k+l - Xk)) = 0

we conclude that x* solves the problem (1). This completes the proof of the theorem.

Remark. It should be pointed out that the sequence of vectors {x k} generated in Theorem 4.2 can be obtained by linear programming. In fact, x k÷l is the unique

J.S. Pang and D. Chan[ Variational and complementarity problems 305

solution to the linear program

minimize eTX-1x,

subject to x ->0 and fk(x)>-O

for any positive vector e, [31, 9].

Combining Lemma 4.1 and Theorem 4.2, we obtain immediately:

Corollary 4.3. Let f : R n ~ R ~ be a convex Z- funct ion and let A:R"--*R "×" be such that A(x ) E Of(x) for all x. Then the conclusion of Theorem 4.2 holds.

Proof. This follows directly from Theorem 4.2 by noting that with X chosen as the identity matrix, A ( x ) X is Z by Lemma 4.1.

Remark. The convexi ty assumption in Corollary 4.3 is needed for the convergence of the iterative method. It is known (see, e.g., [50]) that if f is a Z-function, then the nonlinear cgmplementari ty problem (1) has a solution if it is feasible.

Specializing Theorem 4.2 to a differentiable function, we obtain the following global convergence result for Newton 's method.

Corollary 4.4. Let f : R " ~ R n be convex and differentiable. I f there exists a K-matr ix X such that V f ( x ) X is Z for all x feasible to problem (1), then the

conclusion of Theorem 4.2 holds for Newton ' s method. In particular, the conclusion must hold if f is a Z-function.

5. Convergence of some nonlinear methods

l'n the last three sections, we have investigated the convergence of many linear approximation methods for solving the variat ional/complementari ty problem. In what follows, we establish the convergence of two nonlinear methods. The first is the (nonlinear) Jacobi method applied to the variational inequality problem and the second is a method defined under a Z-funct ion splitting and is applicable only to the nonlinear complementari ty problem. The reasons why we choose not to consider other nonlinear methods are

(1) on a practical basis, we feel that the linear approximation methods should be more useful because there are efficient methods for solving the linear subproblems; and

(2) computationally, little is gained by solving nonlinear subproblems because in many cases, these subproblems are solved by a linearization method. The Jacobi method is interesting because the subproblems although still nonlinear, are separable (convex) programs to which many well studied methods are

306 • J.S. Pang and D. Chan/ Variational and complementarity problems

applicable. More importantly, the (Jacobi) method has been applied very suc- cessfully for solving the PIES model [24]. On the other hand, the method defined by a Z-function splitting is interesting because of its global convergence property and also of the fact that the subproblems can be solved by some specialized algorithm, [50].

In a way, the convergence of the (nonlinear) Jacobi method has been established by Ahn [3] because of the following relationship between the variational inequality problem VI(K,f) and the market equilibrium problem (studied by Ahn) of finding a vector z* E P(Q(z*)) where P and Q are given point-to-set and point-to-point mappings respectively. The latter problem includes the PIES model.

Proposition 5.1. Suppose that the mapping f is invertible. Then the vector x* solves the variational problem VI(K, f ) if and only if the vector z* = - f ( x * ) satisfies

z* ~ a~/'Kff-~( - z*))

where a~r,(') is the subgradient map of the indicator function of K:

gt~(y)= {0, i f y E K ,

oo, otherwise.

For the sake of completeness, we give a detailed convergence proof of the Jacobi method applied to the variational inequality problem. Our proof does not rely on the invertibility of the mapping f. Nevertheless, it uses the same line of argument as Ahn [3]. See also [13, 23].

Given a vector z E R", we define the Jacobi approximation mapping jz( . ) of f at z by

JZ(x ) i = f i ( z l . . . . . z i - l , xi, Zi+l . . . . . z,), i = 1 . . . . . n.

Note that jz( . ) is separable and JZ(z) = f(z) . The (nonlinear) Jacobi method for solving VI(K,f) operates by solving the sequence of subproblems VI(K, jk) where j k ( . )= jxk(.). We assume that each x ~÷l exists. The following is a local convergence result for the method.

Theorem 5.2. Let x* solve the problem VI(K, f). Suppose that the conditions below are satisfied:

(i) The mapping f is differentiable and c3fi/cgxi >- 0 for each i everywhere; the set K is convex.

(ii) The mapping f is continuously differentiable in a neighborhood of x*, afi(x*)/axi > 0 for all i, and

IID-~/2BD-'/21I= < 1

where D and B are respectively the diagonal and off-diagonal parts of Vf(x*).

J.S. Pang and D. Chart/Variational and complementarity problems 307

Then, provided that the initial vector x ° is chosen in a suitable ne ighborhood o f

x*, the sequence {x k} generated by the Jacobi me thod will converge to x*.

Proof. By the differentiabili ty assumpt ions o n / , the fo l lowing hold

lim f ( x ) - / ( y ) - V f ( x * ) ( x - y) = 0 ~x,r~*.x*~ IIx - x*l l + Ily - x*ll

= lim f ( x ) - J r ( x ) - B ( x - y )

H e n c e for each • > 0, there exists a ~ > 0 such that IIx - x*llo -< ~ and IlY - x*llo -< imply

I l D - ~ ( f ( x ) - f ( y ) - V f ( x * ) ( x - y)) l lo -< • ( l lx - x* l l o ÷ Ily - x* l lo ) ,

I I D - ' ( f ( x ) - J Y ( x ) - B ( x - Y))llo -< • ( l lx - x* l lo + IlY - x* l lo ) .

Le t t = rain(l , ~/llx'- x*llo). Then x t = t x ' + (1 - t )x* satisfies IIx' - x*llo -< ~. Suppose that t < 1. Then x* - x t = t (x* - x 1) = t (x ' - xl)/(1 - t). We have

( x * - x ' ) L r ° ( x ' ) = i x * - x t )T ( / ° (X ' ) -- I ° ( X ' ) ) + (X* -- x ' ) T J ° ( X ')

-> (x* - x')T(J°(X t) -- J°(x 1)) (because x 1

--- 0 solves VI (K, j0))

because Ofl/axi >- 0 for each i eve rywhere .

Le t x ° be chosen such that IIx ° - x*ll~ - 8. Since x* solves VI (K, f ) and xt E K, we have (x ' - x*)Tf(x *) >-- O. Thus

0 ~" (X t - - x * ) T ( f ( x *) -- J ° ( x t ) ÷ V f ( x * ) ( X t - - X * ) - - D ( x t - x*)

-B(x'-x*)) = ( x ' - x * ) T [ - - t y ( X ' ) -- f ( X * ) -- V f ( x * ) ( X ' -- X*))

+ i f ( X ' ) -- J ° (X , ) -- B ( x ' - x° ) ) - B ( x ° - x * ) - D ( x ' - x * ) ]

which implies

IIx' - x* l lo -< • l l x ' - x* l l~ + • ( l l x ' - x* l lo + IIx ° - x* l lo )

+ IIO-'~2BO-'J=ll211x°l- x ' l iP .

Consequen t ly , we obtain

IIx' - x*llo -< rllx ° - x*llo

where r = (IID-l~2BD-l/2112+ •)/(1- 2•). By choos ing • > 0 small enough, we have

r < 1. Then,

I Ix ' - x*IID = I Ix* - x'llo/t <- r8/ t < 8It.

Thus t = 1 and I Ix ' - x*llD -< rllx ° - x*llo. Consequen t ly , it fo l lows f rom a con t rac - t ion a rgumen t tha t the sequence {x k} conve rges to x*. This comple tes the p r o o f

of the theorem.

To establ ish the next conve rgence result , we recall tha t a mapp ing h :R" ~ R"

is said to be ant i tone if h ( x ) <- h(y) w h e n e v e r x -> y.

308 J.S. Pang and D. Chan] Variational and complementarity problems

Theorem 5.3. Let f (x ) = g(x) + h(x) be a splitting of the mapping f with both g and h continuous. Suppose that g is a Z- funct ion and h is antitone. Consider the following algorithm for solving the nonlinear complementarity problem (1). Given x k feasible to the problem (1), let x k÷~ be the least-element solution to the (nonlinear) complementarity subproblem

x -> 0, h(x k) + g(x) >- 0 and xT(h(x k) + g(x)) = 0, (17)

i.e., x k÷~ satisfies x k+l <--x for any x feasible to (17). Then provided that x ° is feasible to the problem (1), the sequence {x k} is well defined and converges to a solution of (1).

Proof. We show by induction that the sequence {x k} is well defined, nonincreasing and feasible to (1). Suppose that x k is well defined and feasible to (1). Then x k+~ is well defined (see, e.g., [50]) and xk+I<--X k by the choice of x k+~. Moreover, since h is antitone, we have

f ( x k+l) = g(x k+z) + h(x k+l) >-- (x k+l) + h(x k) >- O.

Thus x k+~ is feasible to (1). This completes the inductive step. Since x k -> 0 for each k, the sequence {x k} therefore converges to some vector x*. By continuity, it follows readily that x* solves the problem (1).

Remark. The assumption on the splitting of the mapping f implies that f itself is a Z-function.

6. A special variational inequality problem

In this section, we investigate the convergence of the linear approximation methods for solving the special variational inequality problem (4). The reason why we need to single out this problem from the general theory is because all the results developed in the last few sections are not directly applicable to it. Referring to the statement of (4), we assume throughout the following discussion that the set K1 is compact. In the context of the traffic equilibrium problem, this compactness assumption is made only for the sake of convenience, [5].

The linear approximation mapping fk has the form:

fk(y, Z)= f(yk, Zk)+ ( ( B T G ( B y k ) B H ( O z k ) ) + E ) ( Y z - : : )

{ BT g(Byk) + BTG(Byk)B(y + Ydk) + c) + E(Yz) = l, h ( z k ) + H ( ? ) ( z - z k)

where G : R m ~ R '~×m and H : W ~ W ×t. To state the convergence results, we first give a lemma dealing with the existence of solution to each subproblem.

Lemma 6.1. Let K~ and K2 be nonempty closed and convex subsets of R" and R ~


respectively with K~ compact. Let

\A21 A 2 : / \ z ]

be an affine function with A:2 positive definite. Then the variational inequality problem VI(K1 x K:, ]) has a solution.

Proof. We quote the following result from Asmuth [5] and Fang and Peterson [20]: Let K be a closed and convex subset of R N and C a compact subset of K. Le t f :R N ~ R N be continuous. Suppose that for all x E K ~ C, there exists a w E C such that (x - w)Tf(x) >- O. Then the problem VI(K, f ) has a solution. To apply this result to the problem VI(KI x K2, f) , fix a vector v ~ K2. Then since A22 is positive definite and K~ is compact, we have

0 = lim sup [ZTAEEZ + (z - v)T(q + m~,y)]/llzll= z ~ y ~ K I

z ~ . K 2

= l i m s u p [ ( : ) - y,z)/Hzll2. z ---~ y E K 1

z E K 2

Thus by letting C = K1 × (K2 ;q ~ ) where ~ is a large enough ball containing the vector v, it follows that for every x = (y, z) E K ~. C, (x - w)T[(x) >- 0 where w = (y, v) ~ C. This proves the lemma.

Theorem 6.2. Let K~ and K2 be as given in Lemma 6.1. Let g : R m ~ R m, h : R I ---> R l, G : R " --* R m×m and H : R I ~ R txt be continuous. Let x* = (y*, z*) be a

solution to the variational problem VI(K~ x KE, f ) where f is given by (4). Suppose that the matrix E is positive semi-definite. I f there exist positive definite matrices X and Y such that G(By*) - X and H (z*) - Y are positive semi-definite and if there exists a neighborhood N of (By*, z*) such that for all (By 1, z 1) and (By z, z 2) in N,

I I~- ' (g (By' ) - g(By 2) - G(By=)B(y ' - y2)ll~ + II~ ' - ' (h(z ' ) - h(z 2) - H(z=)(z ~ - z=))ll~ -< b([lB(y 1 - y=)lk + IIz = - z'll~)

(18)

for some constant b < 1, then provided that the initial vector x °= (y0, z o) is chosen in a suitable neighborhood of x*, the sequence {x k = (yk, zk)} generated by

the linear approximation method is well defined and such that {z k} converges to z*, that {yk} has at least one limit point and if ~ is any such point, (~, z*) also solves the variational problem VI(K~ x K2, f) .

Remark. In the traffic problem, the matrix E is skew-symmetric and therefore positive semi-definite.

Proof of Theorem 6.2. By using the fact that E is positive semi-definite and

310 J.s. Pang and D. Chan/ Variational and complementarity problems

Lemma 6.1 and by following the same argument as in Theorem 2.1, we may easily deduce that the sequence {x k = (yk, zk)} is well defined and satisfies: for each k --- 0

IIB(Y ~+' - y*)ll~ + Ilz k + ' - z ' l i e -< r ( l lB(y ~ - y * ) l k + [I zk - z ' l i e )

for some r < 1. Thus {z k} converges to z* and {By k} to By*. The sequence {yk} is contained in the compact set K1, therefore must have a limit point. Let {yk~+l} be any subsequence converging to some ~. We claim that (~, z*) also solves the variational problem VI(K~ x K2,f) . For each (y, z ) E K~ x K2, and each k~, we have

(Yz - Y '+'aTr/nTg(nY ') + + ~ / I/lYe'+'\1 zk,+v It h(zk')+H(zk')(zk'+~-zk')+d Cl+E,,zk'+~lj>--O"

Since limk~+~By k~+l =l im~+~By k~ = B y * = B~), passing the limit k i ~ in the above inequality we conclude that (~, z*) is a solution to VI(K,f) . This completes the proof of the theorem.

Note that in general, one cannot conclude that {yk} converges. Nevertheless, the sequence {By k} will. This phenomenon is related to the uniqueness result obtained by Asmuth [5] and Aashtiani [1] for the traffic equilibrium problem. Another related result can be found in [36].

Specializing Theorem 6.2 to Newton's method, one obtains:

Corollary 6.3. Let K1, K2, g, h, E and x* be as given in Theorem 6.2. Suppose that both g and h are continuously differentiable and that ~Tg(By*) and ~Th(z*)

are positive definite. Then the conclusion of Theorem 6.2 holds for Newton 's method (G(By) = ~Tg(By) and H(z ) = ~Th(z)).

The next result is the global version of Theorem 6.2. Its proof is omitted.

Theorem 6.4. Let K1, K2, g, h, G, H and E be as given in Theorem 6.2. Suppose that there exist positive definite matrices X and Y such that G ( B y ) - X and H ( z ) - Y is positive semi-definite for all y and z in Kl and K2 respectively and that (18) holds for all y~ and y2 in K~ and z ~ and z 2 in K2 for some scalar b less than 1. Then for all initial vector x ° = (yO, z o) E K1 x K2, the sequence of vectors {x k = (yk, zk)} generated by the linear approximation method is well defined and such that the sequence {z k} converges to some vector z* E K2 and the sequence {yk} has at least one limit point and if ~ is any such point, (~, z*) solves the variational problem VI(K1 × K2, f).

Theorems 6.2 and 6.4 are norm-contraction results. Unfortunately, results of the other two types (i.e., the vector-contraction and monotone approaches) are not so easy to obtain for this special class of variational problems, mainly


b e c a u s e the r equ i r ed cond i t i ons are a b i t too s t rong to be sa t i s f ied fo r f unc t i ons f

of the fo rm (4) tha t ar ise f rom for i n s t ance , the traffic equ i l ib r ium p r o b l e m .

As a final r e m a r k , we po in t ou t tha t l ike Coro l l a r i e s 2.11 and 2.12, T h e o r e m 6.4

can be spec i a l i zed to the p r o j e c t i o n m e t h o d s in w h i c h G ( B y ) = X and H ( z ) -- Y

fo r all y and z w h e r e X and Y are s o m e f ixed bu t o t h e r w i s e a r b i t r a r y s y m m e t r i c

p o s i t i v e defini te ma t r i ce s .

Acknowledgment

T h e first au tho r w o u l d l ike to t h a n k J. B o r w e i n and C. I r w i n fo r s eve ra l

i n t e r e s t i ng c o n v e r s a t i o n s on the s u b j e c t of this pape r .

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Iterative methods for variational and complementarity problems

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