A continuation method for monotone variational inequalities

Mathematical Programming 69 (1995) 237-253

A continuation method for monotone variational inequalities "

Bintong Chen a,* Patrick T. Harker b a Department of Management and Systems, College of Business and Economics, Washington State University,

Pullman, WA 99164-4726, USA b Department of Systems Engineering, School ofEngineering andApplied Science, University of Pennsylvania,

Philadelphia, PA 19104, USA

Received 17 August 1992; revised manuscript received 8 April 1994

Abstract

This paper presents a continuation method for monotone variational inequality problems based on a new smooth equation formulation. The existence, uniqueness and limiting behavior of the path generated by the method are analyzed.

Keywords: Variational inequality; Nonlinear complementarity; Nonlinear programming; Continuation method

1. Introduction

The variational inequality problem (VIP) is an important area of applied mathematics

due to its numerous applications. Both complementarity problems (CP) and convex

nonlinear programs (NLP) can be considered as special cases of the VIP. It has been

noted that many social and economic models can be formulated as a VIP or CP. Among

them are the PIES model, traffic equilibria, the prediction of interregional commodity

flows, the solution of Nash equilibria, the prediction of freight transport prices and

services, and the solution of the Walrasian or general equilibrium model of economic

activities. The paper [10] provides an extensive survey on the theory, algorithms and

applications for finite-dimensional VIP and nonlinear complementarity problems (NCP).

In this paper, a new continuation method is proposed to solve the monotone VIP. The

method is motivated by the recent progress made in interior-point path-following

algorithms for linear and nonlinear complementarity problems [12-15]. Given a CP, the

This work was supported by the National Science Foundation Presidential Young Investigator Award ECE-8552773 and by a grant from the Burlington Northern Railroad.

* Corresponding author, e-mail: [email protected].

0025-5610 © 1995 - The Mathematical Programming Society, Inc. All rights reserved SSDI 0025-5610(94)00034-Q

238 B. Chen, P.T. Harker / Mathematical Programming 69 (1995) 237-253

algorithms for linear and nonlinear complementarity problems [12-15]. Given a CP, the path-following algorithm creates a strictly feasible path, called the path of centers, by relaxing the complementarity condition while maintaining the feasibility condition. It is shown that by following the path properly, the algorithm will converge a solution of the CP. However, most of path-following algorithms for NCP have the following shortcom- ings. First, the dimension of the problem is usually enlarged in order to obtain an initial point, unless a good estimate close to the path is available. As a result, a subproblem with larger size has to be solved at each iteration. Second, each intermediate iterate is required to stay inside the feasible region which sometimes restricts the choice of step size. Finally, as we have shown in [4], the Newton's direction generated at each iteration is not "efficient".

The main effort of the most path-following algorithms is spent on tracking the feasible path, or the path of centers. Let the infeasible paths be those that satisfy the same conditions as the feasible path except for the feasibility condition. Each intermediate iterate is forced to stay close enough to the feasible path or else these points may be "trapped" by other infeasible paths. This phenomenon can occur because the feasible path has the same attraction domain as other paths. The natural way to overcome the above difficulties is to increase the attraction domain of the feasible path. The method proposed herein actually identifies the feasible path, which means, under appropriate assumptions, the attraction domain of the feasible path is all of ~ n, while those of the infeasible paths are empty. Therefore, the algorithm can start at any initial point, whether feasible or not and, in addition, each intermediate point does not have to stay feasible.

The remainder of this paper is structured as follows. The next section will provide a brief review of the relevant literature. Section 3 presents the basic results concerning this new algorithm. Section 4 applies the result to nonlinear programs, and conclusions are drawn in Section 5.

The following notation will be used throughout the paper. Vectors are denoted by boldface lower-case letters, such as x. All vectors are column vectors, unless explicitly stated otherwise. Row vectors are the transpose of column vectors, for example, x T denotes the row vector (Xa, . . . , xn). For notational simplicity, (x~, x~) T is sometimes simplified as (x 1, x2) T. ~", ~+ and ff~++ denote, respectively, n-dimensional Eu- clidean space, the nonnegative orthant of ~n, and the strictly positive orthant of ~". Matrices are denoted by boldface capital letters, such as M. The /th row of the n × n matrix M is denoted by M/ , and the jth column is given as M.~. Scalar-valued flmctions are denoted by Roman or Greek letters, such as f and 0. Vector-valued functions are denoted by boldface Roman or Greek letters, such as F and g. Scalars are denoted by lower-case Roman or Greek letters, such as k and A.

2. Literature review

In this section, the VIP, CP and NLP are defined mathematically and the related literature is briefly reviewed. To begin, let us define the problem under investigation.

B. Chen, P.T. Harker / Mathematical Programming 69 (1995) 237-253 239

Definition 2.1. Let X be a nonempty subset of E" and let F be a mapping from E n to

itself. The variational inequality problem (VIP), denoted by VIP(X, F) , is to find a vector x ~ X such that

F ( x ) T ( y - - x ) > ~ O , V y ~ X . (1)

It is not difficult to see that when X = ~~_, the VIP is reduced to the NCP defined as follows.

Definition 2.2. Let F be a mapping from E ~ to itself. The nonlinear complementarity problem (NCP), denoted by NCP(F) , is to find a vector x ~ E n such that

x>~O, F(x)>~O and F ( x ) T x = O . (2)

The first two inequalities are called the feasibility conditions, and the equality is called the complementarity condition.

The NLP is also closely related to the VIP. Specifically, if F ( x ) is a gradient function of some real-valued function f : ~n ~ E, then the VIP(X, F ) is equivalent to the stationary condition of the following mathematical program:

min f ( x ) , s.t. x ~ X . (3)

In addition, if f ( x ) is a pseudo-convex function and X is a convex set, any solution of VIP(X, F ) is a global minimum of the mathematical program (3).

In this paper, we restrict the feasible set X of VIP(X, F ) to the following form:

X={X~~n: gi(x) ~0» i = 1 . . . . . m; h j ( x ) = 0 , j = 1 , . . , l } , (4)

where gi and hj are assumed to be concave and affine real-valued functions, respectively. In addition, we assume that X satisfies some standard constraint qualifications similar to those imposed in nonlinear programming [3].

The following definitions (see, e.g., [10]) will be needed in what follows.

Definition 2.3. The mapping F : E ~ --* E ~ is said to be a (1) monotone function over a set X if

[ F ( x ) - F ( y ) ] ~ ( x - y ) > t 0 , Vx, y ~ X ;

(2) strictly monotone function over a set X if

[ F ( x ) - F ( y ) ] T ( x - y ) > O , Vx, y ~ X , x ~ y ;

(3) strongly monotone function over a set X if there exists an a > 0 such that

[ F ( x ) - - F ( y ) ] T ( x - - y ) > ~ « I I x - - y l [ 2, Vx, y ~ X .

Based on the above definitions, strong monotonicity implies strict monotonicity which in turn implies monotonicity.

The paper [10] presents a fairly complete survey of the results for the VIP. Hence, we shall not reiterate what can be found in that paper. Rather, the methods directly related to the continuation method will be reviewed.


Continuation methods study the relationship between a problem P, called the original problem, and the perturbed problems P ( e ) associated with P. In this context, « is the perturbation parameter, which equals to zero for the original problem P. The content of continuation methods is rich. On the one hand, perturbation and parametric analysis discuss the behavior of the problem P ( e ) based on information of P, depending on whether « is small or large, respectively. On the other hand, penalty-related methods, proximal point algorithrns and homotopy methods alm at finding a solution of P by successively solving the perturbed problems P(«) .

Theoretical results concerning local perturbations (sensitivity analysis) for VIP and NCP were surveyed in [10,16]. Parametric analysis concerning the NCP has been studied extensively in the literature. Given an NCP(F) and /x > 0, define the Perturbed NCP (PNCP), denoted by PNCP(~), as orte of finding an x such that

w = F ( x ) > O , x > 0 , w i x i = [ z , i = 1 , 2 , . . , n .

For both monotone complementary problems and complementarity problems with maximal monotone functions, McLinden [21] presented results concerning the existence, characterization, and continuous dependence of the solutions of PNCP(/x). These results were then extended to both monotone and general variational inequalities [22]. For the monotone NCP, Kojima et al. [13] extended McLinden's results and analyzed the limiting behavior of PNCP(/x) as /x approaches zero. Similar results were also obtained for the NCP with uniform P-function [12] and the NCP with P0-function [14]. For a monotone LCP, Kojima et al. [15] first developed a polynomial path-following interior- point algorithm based on the above idea. More recently, Pang and Wang [25] designed an embedding method to solve variational inequality and nonlinear complementarity problems. The approach is different from those techniques mentioned above in that it solves a family of one-parameter problems of the same type.

Proximal point algorithms can also be regarded as continuation methods. In the context of the NLP, the algorithm can be understood as a type of perturbation or convexification of the objective function. Rockafellar [27] studied the proximal point algorithm applied to the maximal monotone operator. In the context of optimization problems, proximal point algorithms have been applied to nondifferentiable convex programs in [1,8,20,26]. These algorithms were then specialized to NLP and combined with interior and exterior penalty algorithms in [11] and [2], respectively. They were applied to LP and QP in [18], where the algorithm takes the form of Tihonov regularization and the subproblems are solved by SOR techniques. Continuing along this line, De Leone and Mangasarian [6], and more recently Wright [29] implemented the algorithm by incorporating augmented Lagrangians. Computational experiments by these authors show that the above algorithms are competitive with the simplex method for the LP and amenable to large-scale problems for both serial and parallel implementation.

The method presented herein solves a sequence of perturbed VIP, where each is reformulated as a system of smooth nonlinear equations. In contrast to the equations used in the interior-point path-following algorithms, they do not require that all points be strictly interior. Other equation-based methods have appeared in the literature. Man-


gasarian [17] first transformed the NCP as a system of differentiable equations. Watson [28] explores the use of continuation methods to solve the NCP based on Mangasarian's formulation. More recently, Fukushima [9] formulated asymmetric VIP as differentiable optimization problems. The result was further extended in [19] to formulate NCP as unconstrained optimization problems. Nondifferentiable equation formulations of VIP- related problems have attracted much attention in recent years due to the theoretical development of Newton's method when applied to a system of non-Fréchet differentiable equations. See [24] for a brief survey.

3. The continuation method

This section presents the new continuation algorithm for monotone VIP. Section 3.1 presents the algorithm and each subproblem of the algorithm is formulated as a system of nonsingular nonlinear equations. In Section 3.2, we prove that each subproblem has a solution. In Section 3.3, we show that each subproblem encountered during the course of the algorithm has a unique solution and the path leading to a solution of the VIP is continuous. In Section 3.4, we show that if the VIP has at least one solution, then the algorithm converges one of the solutions of the VIP. In addition, if some restrictions are imposed on the continuation parameters, the algorithm converges the least two-norm solution of the VIP.

3.1. The basic algorithm

Consider the problem VIP(X, F ) with X being defined by (4). Assume also that the necessary constraint qualifications hold (e.g., linear independence of the binding constraint gradients). The problem can be reformulated as a mixed NCP as follows (see [1o]):

F ( x ) - Vg(x)Ty - Vh(x)Tz=O, (5)

g(x)>~O, y>~0, g(x)Ty=O, (6)

h ( x ) = 0, (7)

where y and z are dual variables of constraints g(x) >1 0 and h(x) = 0, respectively. Instead of solving the VIP directly, we consider the following mixed NCP representation of the Perturbed VIP, PVIP(e, /x):

F( x) + 8 1 X - - Vg( x)Ty -- Vh( x)Tz = O,

g(x) q-E2Y>O , y > 0 , ( g i ( x ) - ~ o ° 2 Y i ) Y i ~ ~ - ] . L , i = l , . . . , m ,

h( x) + e3z - - 0 ,

(8)

(9)

(lo)


where ~ > 0 and «l > 0 for all l --- 1, 2, 3. Solving (gi(x) + «2Yi)Yi = t* for Yi and taking the positive root, we have

Yi=~l [~/g2(x)+4e21L--gi(x)], i = l , . . . ,m ,

o r

2 « 2 Y i + g i ( x ) - - ~ g 2 ( x ) + 4 e 2 1 . L = O , i = 1 . . . . . m.

Using this transformation, orte has the following result.

Theorem 3.1. (x, y, z) T is a solution of PVIP(e, /.L) if and only if it solves the following system of nonlinear equations, denoted by J( x, y, z, s , / . t ) = 0:

F( x) + e l x - Vg( x ) T y - Vh( x )Tz = O, ( l l )

2 e 2 Y i + g i ( x ) - - ~ g 2 ( x ) +4~2/z = 0 , i = 1 . . . . ,m, (12)

h ( x ) + e3Z = 0. (13) Proof. It suffices to show that (x, y)T solves (9) if and only if it solves (12).

(9) ~ (12): Let

J2i( x, Y, ~, t*) = 2e2Yi + gi( x) - ~/g2( x) + 4e2/,t ,

Jfi ( x, y, ~, It) = 2 e 2 Yi + gi( x ) + ~g2( x) + 4s 2/*.

Then by factorization, (gi(x) + e 2 Yi)Yi =/.t can be rewritten as

J] , ( x , y, «, ~)J2i(X, y, ,9, I t ) : 0 . However, J~i(x, y, «, /~) > 0 since y > 0 from (9). This implies that Jzi(x, y, «, I~) = 0 for all i = l , . . . , m .

(12) ~ (9): From (12), we have

ge(X) q- ,92Yi~- ~ g2( X) q-4E2/.L + gi( x) > 0 ,

1 2 yi:'~-'--2 [\/gi(x) -~t-4821A, --«i(X)] >0,

and in addition, (gi(x) + Æ2Yi)Yi =/* by algebraic calculation. []

Denote ri(x, «, I~) = 1 - g i ( x ) / ~/g2i ( x) + 4e 2/*,

R ( x , e, t*)= diag{ri(x, e, /~)},

and

L ( x , y, Z, e , ) = F ( x ) + e , x - V g ( x ) T y - V h ( x ) T z . The Jacobian of J (x , y, z, «, ~) is obtained through algebraic manipulation:

V j ( x , y, z, «, ~ ) (~ o o) = R(x , e » ~) 0

0 I

VxL( X, y, z, el)

x V g ( x )

V b ( . )

- V g ( x ) ~ - V b ( x ) ~~

2teR(x, Æ2' /"L) -1 0

0 «3I


Proposition 3.2. VJ(x , y, z, «, Ix) is nonsingular for all (x , y, z) T ~ R n X R"~× R 1 and e, /x>0 .

Proof. First we observe that 0 < r i ( x , «» I t )<2 , i = 1 . . . . . m, for all /x, e 2>0 . Therefore, R(x , e2, /x) is a positive diagonal matrix, and so is its inverse. Clearly, the matrix on the left of Jacobian is positive definite. Now

l

VxL(X, y, z, e l) = V F ( x ) + e l I - ~ yiI72gi(x) - E zj~72hj(x) • i = 1 j = l

By assumption, F is monotone, gi is concave, hj is affine and e 1 > 0, yi >~ 0. Therefore, VxL(x, y, z, «~) is positive definite. Taking into consideration the matrix structure of the second part of the Jacobian, it is easy to see that the second part is also positive definite. Consequently, the Jacobian is nonsingular. []

An immediate consequence of the above result is the following corollary.

Corollary 3.3. Let (x* , y*, z*) T be a solution of J ( x , y, z, e, Ix)= O. Suppose Newton's method is applied to solve equation system (11)-(13) with the initial iterate ( x o, yO, zO)T located in a small neighborhood of ( x *, y*, z* )T. Then this method will converge (x *, y *, z * )T at a quadratic rate.

Due to the introduction of the dual variables, the dimension of the problem is increased beyond its original formulation. A nonlinear equation system with a smaller dimension can be derived by eliminating the dual variables from PVIP(e, /x). Solving for y and z in (12) and (13) and substituting them into (11), we obtain

m Vg i ( x ) 1 t F ( x ) + e l x - 2 1 x E + - - E h i ( x ) V h j ( x ) = O .

i=1gi(X)-t-~/g2i(X) +4821 ~ '~3 j = l

(14) To conclude this section, we present the following continuation algorithm for

VIP(X,F) based on the equation formulation given by Theorem 3.1.

Initiation Step. Let e be a given stopping tolerance. Choose any initial point (x 0, y0, Z 0 ) ~ ~ N n × R m × N1 and sequences /x ~, e ~ > 0 , 1 = 1 , 2, 3, k =

0, 1, 2 . . . . . such that

Bk >/xk+ ~ and lim/x k = 0, k--~~

e ~ > e { +t and l i m e ~ - - 0 , I - - 1 , 2 , 3 . k--~m

Set k = 0 and go to Main Step. Main Step.

(1) Starting with (x k, y~, zk) T, mately for (x k+l, yk+l, zk+l)r.

(2) If err(x k+ 1, yk+ 1, z~+ 1) < e, stop. Otherwise, set k = k + 1, and go to Step (1).

solve equation J ( x , y, z, e k, /x k) = 0 approxi-

244 B. Chen, P.T. Harker /Mathematical Programming 69 (I995) 237-253

Here, err(.) is the error function which measures the distance between the current iterate and the solution of the VIP. However, no error function for the monotone VIP is known in the current literature. The closest result is the error function for the strongly

monotone VIP with linear constraints [23]. Denote

el (x , y, z ) = [ I C ( x ) - V g ( x ) T y - Vh(x)Tz[[ 2,

e2(x, y, z) = [ [min{g(x) , y)[[2,

e3(x, y, z) = [[ h ( x ) Il 2, where "min" is taken component-wise. A possible error function could be

3

err(x , y, Z) = E «i(x, y, Z). i = 1

However, theoretical justification of this error function remains an open question, which will be studied in the future.

3.2. Existence of a solution to PVIP(«, I ~)

We first introduce the concept of the Open set VIP (OVIP) defined as follows.

Definition 3.4. Let X o c ~n be an open set. The OVIP, denoted by OVIP(Xo, F) , is to

find a vector x ~ X o such that

F(x)T(y-x) >~0, Vy~Xo. It is not difficult to see that OVIP(X0, F ) is equivalent to finding an x ~ X 0 such

that F ( x ) = 0, a system of nonlinear equations to be solved in an open set. The following result, to be used later, follows immediately from the above observation.

Proposition 3.5. OVIP(X 0, F) has a solution if and only if there exists a closed set X c X o such that VIP(X, F) has a solution x* ~ int(X).

To show the existence of a solution for PVIP(e, /z), observe that (9) is equivalent to

g ( x ) + ~ 2 Y > O , y > O , ( g i ( x ) + e 2 y i - ~ i i t Y i = O , i = l . . . . . n.

Therefore, it suffices to prove that the following conditions have a solution:

F(x) + « l x - Vg(x)Ty - Vh(x)~z=O,

g ( x ) + «2Y - I £ Y - l e : O, y > O,

h ( x ) + e~z = o,

where Y = diag{y/} and e is a vector whose elements are all ones. By the definition of OVIP and the observation thereafter, one can verify that the above conditions are

equivalent to OVIP(~2++, J ) , where

( F ( x ) T T ' B C1 x - V g ( x ) y - V h ( x ) z

J ( x, y, z, 6, I~) = g( x) + e2y - tzY- le ,

h( x) + e3Z


and g2++ = ~n × Qm++ × Rt. An immediate result which will prove useful in the sequel

is the following,

Proposition 3.6. J ( x , y, z, «, tx) is strongly monotone over £2++ for all e, I.t > O.

Prooi. Under the assumption that F is monotone, g is concave and h is affine, it suffices to show that - i x y - l e is a monotone function of y for all y ~ ~m++. Let y~,

in Y2 E R++. We have

m 1 T

( - » Y { l e + lxY 2 e) ( Y a - Y 2 ) = / x ~ i = 1

This completes the proof. []

(Yl i --Y2i) 2 >~0.

YliY2i

If O++ were a closed set, the above strong monotonicity result would have been enough to show OVIP(O++, at), and thus PVIP( ~, e) has a solution. We now prove that OVIP(g2++, J ) has a solution for the open set g2++. Consider the associated closed set VIP, VIP(g2 c, J ) , where

J2« = {(x, y, z) T ~ . ~ n x ~ m x ~ l : y > / c e > O } . (15)

Since O c c O++ is a closed and convex set and ,I is strongly monotone over g2++, VIP(g2 c, J ) has a unique solution [10].

Lemma 3.7. Assume F is strongly monotone over set X, which could be an open set.

Let X c = { x ~ X : x i>~~>O, V i ~ I } be a closed convex set, where ! c { 1 . . . . . m}. Suppose x c is a solution of VIP(Xc, F). Then for any c > 0 such that X c is nonempty,

there exists a constant rlc < + ~ such that Il x« II < rlc for all 0 < ~ <~ c.

Proof. Suppose X is bounded. By assumption, X~ is a closed and convex set and F is strongly monotone over X D Xt. Therefore, x; ~ X~ c X exists and is bounded. Suppose now that X is unbounded. I.et x o ~ X c. Since F is strongly monotone over X, it is

possible to find a T > Il F(xo)l l and ~« > Il Xo Il such that

( F ( x ) - F ( x o ) ) T ( x - x o ) >~V{Ix-xoll , Vllx[I >~~c, x ~ X .

Thus,

F( x)T( x - Xo) >1 r II x - x 0 Il +F( xo)T( x - Xo)

~> T Il x - x 0 Il F ( x 0 ) JJ II ( x - x0) II

>~(~- lIF(xo)ll)(llxll-lqXotl) >0, VIIxll>~n~, x ~ x . (16)

Since x o ~ x« c__x c for all 0 < ~ ~< c and by assumption xc is a solution of VIP(Xc, F), we have

F(X«)T(X« - x0) = - F ( x « )T( X0 -- X~- ) ~ 0. (17)

Comparing (16) and (17), we conclude that JJ x~ I] < ~1« for all 0 < ~ ~< c. []

246 B. Chen, P.T. Harker /Mathematical Programming 69 (1995) 237-253

We now prove the main result of this section.

Theorem 3.8. PVIP(e, /x) has at least one solution for all «, I ~ > O.

Proof. It suffices to show that OVIP(O++, J ) has a solution. By Proposition 3.6, 37 is strongly monotone over O++. Given a c > 0, suppose (x t , Yc, zc) ~ is a solution of VIP(~2«, . ] ) where ~2 c is defined by (15). By Lemma 3.7 there exists an ~/c such that

[l(xc, Y«, Zc)l[ < *Ic. Therefore, Il Xc Il < ne. Choose a Q« > 0 such that

> Q c > m a x max g i ( x ) . i x:ll x I1.« ~«

Choose a scalar 0 < ~ ~< c such that

/ z < 0 " Qc+«2¢-y Now let ( x t , yc, z¢) T be the solution of VIP(Oc, J ) . We show that yc> ffe. Suppose, on the contrary, that Y~i <~ ( for some i = 1 . . . . , m. By Lemma 3.7, I[ x; [[ < ~c, hence gi(x;) < Q« for all i by the construction of Q«. Therefore,

/z g i ( x~ ) + «2Y~i -- - - <0.

Y~i

However, since x¢ and y¢ are assumed to be a solution of VIP(O; , J) , we have

/x gi(xC)+~2Y~i---->~O, V i = I . . . . ,m.

Yci

This leads to a contradiction. Therefore, ( x ; , Yc, z¢)T ~ int(~2c). Applying Proposition 3.5, we conclude that OVIP(O++, J ) has at least one solution. This completes the proof. []

3.3. Continuity of the solution to PVIP(e, tx)

The following lemma from [12] is useful in the subsequent proof.

L e m m a 3.9. Suppose that a h, b h, k = 1, 2 are nonnegative numbers. Then,

( a 1 - a2)(b 1 - b e ) <~ l a~b I _ a2b2[.

We remark that the inequality holds strictly if a h, b h, k = 1, 2, are positive and either a 1 ~ a z or b 1 :~ b 2. The following theorem is the main result of this section.

Theorem 3.10. Suppose ( x 1, ya, zl)T and (X 2, y2, z2)T are solutions of PVIP(e, /z 1) and PVIP(«, /x2), respectively, with «, tx l, I~ 2 > O. Then,

«1[[xl--x2[I2+e2[[yl--y2[[2+g31[Zl--Z2[[2<~mltzl--1~2[. (18)


Proof. From (8), we have

F(Xl)T(X 1 --X 2) + ,~I(X 1 -- x2)Tx 1 -- [Vg(x l ) (X 1 -- X2)]T71

- [ V h ( x l ) ( X l - X 2 ) ] T z l = O ,

F(x2)T(x 2 --X 1) -k- «1(/2 - - /1)Tx2 -- [Vg(x2) (x 2 - / 1 ) ] T y 2

- [ V h ( x 2 ) ( x 2 - xl)]Tz2=O.

From (10), we have

h(x k )+e3z k=O, k = l , 2.

By assumption, F is monotone, g is concave and h is affine. Note that y k > 0 for k = 1, 2 since they are solutions of PVIP(e, /z 1) and PVIP(«, /x2), respectively. There- fore, we have

[ F ( x 1) - F ( X2)] T ( X 1 - X 2) » 0,

Ire(/1)( X 1 -- / 2 ) ] T71 ~ [ g ( / 1 ) -- g ( X2)]T71,

[Vg(x2)(X2--Xl)]Ty 2 <~ [g (x 2 ) - g ( x l ) ] T y 2,

Vh( x ' ) ( x 1 - x 2) = h ( / 1 ) _ _ h(x2 ) ,

V h ( x 2 ) ( x 2 - x ' ) = h ( x 2) - h(x l ) .

Adding the first two equalities and substituting the rest of the equalities and inequalities, we obtain

~1 Il x 1 - x 2 II 2 + ~3 Il z 1 - z 2 Il 2 < [ g ( x 1) - g ( x 2 ) ] T ( 7 1 _ 7 2 ) .

Therefore,

~1 Il x 1 - x 2 Il 2 + ~2 Il 7 1 - 7 2 Il 2 + ~3 Il z 1 - z 2 Il 2

~<[ (g (x l ) + ~ 2 y l ) - ( g ( x 2) + e2y2)]T ( yl - y 2)

: ~ [(gi(xl) + e 2 y l ) - - ( g i ( x 2) + e2y~)](y]--y 2) i=1

~ < m l /~1 --/z2 l,

where the last inequality follows from Lemma 3.9 since, by assumption, (x 1, ya, zl)T and (x 2, y2, z2)T are solutions of PVIP(«, /x 1) and PVIP(e, /x2), respectively. []

Note that Theorem 3.10 implies that the solution of PVIP(e, /x) is continuous in the parameter/x, and in addition, it is unique, as stated in the following corollary.

Corollary 3.11. PVIP(c, /x) has a unique solution for all ~, tx > O.


Proof. The result follows by setting /z 1 =/x 2. []

3.4. Limiting behavior of the solution trajectory

In the previous sections we have discussed the existence and uniqueness properties of PVIP(e, /z). As e and /x go to zero, one hopes that the solution of PVIP(e, /z) will approach that of VIP(X, F). This section will give a condition under which this is true and, in addition, if the algorithm converges a solution, will characterize the properties this solution possesses. These results are extensions of those of [13] for monotone NCP.

Let el(t), /x(t), ~ + ~ ~+, l = 1, 2, 3, be continuous functions such that

e t ( t ) , / x ( t ) ~ ~ + + , ' d t>O, (19)

e, (o) = 0, ~ ( o ) = o. (20)

Let (x( t ) , y(t), z(t)) T be the solution of J ( x , y, z, e(t), /x(t)) = 0 or equivalently, of PVIP(«(t), /x(t)). By the definition of el(-), /x(.) and the results of the last two sections, (x( t ) , y(t), z( t ) ) T exists for all t > 0. Therefore, the trajectory

T = { ( x ( t ) , y ( t ) , z ( t ) , t ) : t > 0 }

is well-defined.

Theorem 3.12. Let S be the solution set of VIP(X, F). Suppose the trajectory T is bounded as t ~ 0; then it approaches a limit point (x(O), y(O), z(0)) T ~ S.

Proof. Since T is bounded as t approaches zero, it has at least one limit point (x(O), y(O), z(0)) x. That the solution (x(0), y(O), z(0)) x ~ S follows from the continuity of F, g, h, Vg, Vh, e t , /~ with respect to t in (8)-(10), and the fact that PVIP(e, /x) approaches VIP(X, F ) as e and /x approach 0. []

To establish the condition under which the trajectory T is bounded as t ~ 0, we need the following lemma.

Lemma 3.13. Ler (x( t ) , y(t), z(t), t ) ~ T. I f the NCP (5)-(7) has a solution ( ~ , ) , £)T, then

e l ( t ) ( x ( t ) -- ~)Tx( t ) + e2( t ) ( y ( t ) -- J3)Ty(t) + e3 ( t ) ( z ( t ) -- ~)Tz(t)

<~mtx(t). (21)

Proof. Following the same procedure as in the proof of Theorem 3.10, we obtain

[ g ( x ( t ) ) + e2Y(t)] T) + g ( ~ ) T y ( t ) _ g(:~)T)

+ e l ( t ) ( x ( t ) _ ~ ) T x ( t ) + e 2 ( t ) ( y ( t ) _ ) ) T y ( t ) + e 3 ( t ) ( z ( t ) ^ T -- Z) z ( t )

< m ~ ( t ) .

However, by assumption, g(x ( t ) ) + eeY(t) > 0, y(t) > O, g (~ ) >~ O, ) >~ O, and g(~)z ) = 0. Therefore, the first three terms in the inequality are nonnegative and the proof is completed. []

B. Chen, P,T. Harker / Mathematical Programming 69 (1995) 237-253 249

One set of conditions to ensure the boundedness of T as t ~ 0 is given by the following theorem.

Theorem 3.14. Suppose the assumptions of Lemma 3.13 and the following conditions hold:

e2(t ) «3(t) /x(t) lim - - = C 2 > 0, lim - - - C 3 > 0, lim - - = C >~ 0. t--0 e l ( t ) t--0 « l ( t ) t--,0 e l ( t )

Then T is bounded as t -~ O.

Proof. Substituting the definition of e ( t ) and /x(t) into the inequality (21) in Lemma 3.13 and reorganizing terms, we obtain

ez( t) 8 3 ( 0 l[ x ( t ) - ½~ Il 2 + ~ Il y ( t ) - ½~ Il z + ~ II z ( t ) - ½~ I[ 2

~<¼(11.~[12_F s2( t ) 83(t ) ) /z( t ) EI-~-~ 1[ Y Il 2 "1- E I ' ~ - 1[ Z [I 2 + m el(t----3- •

By the definition of limit, given a 6 > 0, there exists a } such that for all r ~ t,

1 e 2 ( T ) 1 E3(T) 3 j[Z(T)

for some 8 > 0. Therefore, for all r ~ t we have

II x ( r ) - l ^ 2 1 1^ 2 1 ~X I[ "[- ~-C 2 1] y(r) - ~ y I[ + ~-C 3 Il z ( T ) _l~_z ̂ 1[ 2

3 2 3 2) ~< ¼(Il ~112 + ~C21l ~ll + ~C3 Il £,ll + m(C + a ) ,

which is true only if T is bounded as t ~ 0. []

Corollary 3.15. Let el(t) , 1 = 1, 2, 3, and tx(t) be defined as in Theorem 3.14. Then T is bounded as t --> 0 if and only if VIP(X, F) has a solution.

Proof. Necessity is a direct consequence of Theorem 3.14, since the NCP (5)-(7) is equivalent to VIP(X, F) . Sufficiency follows from Theorem 3.12. []

The trajectory T has now been shown to lead to a solution of VIP(X, F) . The following result shows that the limit point of this trajectory possesses some special properties.

Theorem 3.16. Assume el(t) = e(t) , I = 1, 2, 3, e(t) , /x(t) > 0, Vt > 0, e(0) =/x(0) = 0 and, in addition, lim,_~ o( t ) / e ( t ) = O. Let ( x *, y*, z* )T be an arbitrary solution of VIP(X, F ) and (x(0), y(0), z(0)) T be a limit point as t approaches zero. Then,

I l (x (0 ) , y (0 ) , z(0) ) I I < I l ( x * , y* , z * ) II.


Proof. By Lemma 3.13 and the definitions of e ( . ) and /z(.), we have

( x( t ) -- x* )Tx(t) + (y ( t ) --y* )Ty(t) + ( z(t) --Z*)Xz(t) < ~ m - -

Therefore, as t ---> 0, any limit point (x(0), y(O), z(0)) x satisfies

II (x (0 ) , y(0) , z(0))112< (x (0 ) , y(0) , Z(0))T(x*, y*, Z*)

This completes the proof.

~(t) «(t)

~< ][(x(0), y(O), z(0))II Il(x*, y*, z*) [I. []

It follows that the algorithm will converge to the least two-norm solution of VIP(X, F ) when et(t) and /z(t) are properly defined, for example, e l ( t )= et and B(t) = / z t 2.

4. Applications to nonlinear programming

Consider the following convex NLP:

min f ( x ) , s.t. x ~ X ,

where, as in the VIP case, X is defined by

X = { x ~ ~ " : gi(x)<~O, i=1 . . . . ,m; h j (x )=O, j = l , . . . , / } .

We assume that f is convex, gi is concave, and hj is affine as before. The K-K-T conditions of the NLP are

V f ( x ) - Vg(x)Ty - Vh(x)Tz = O,

g ( x ) > O, y >1 O, g (x )Ty = O,

h(x) = 0 ,

which is the same as VIP(X, F ) except that F(x) is replaced by Vr(x). Since f (x ) is convex, Vr(x) is monotone. Therefore, the algorithm for VIP(X, F ) can be applied directly. At each iteration, we solve a system of nonlinear equations obtained by perturbing the K-K-T conditions. The perturbation parameters are then reduced appropri- ately, and the process continues until the specified accuracy is achieved.

The specialized algorithm for the NLP should be classified as a penalty function algorithm, which can be clearly seen by eliminating perturbed equations. Similar to (14), we have

m ~ T g i ( X ) W(x) + < x - 2~E1

"= gi(x) +~/g2(x) +4e21 a,

the dual variables from the

1 1 + - - E h j ( x ) Vh« (x ) = O,

423 j = l

which are the K-K-T conditions of the following uneonstrained NLP:

1 2 Kl(X, e2, /x) + - - E h}(x) , min f ( x ) + ~ e 1 I1X]I -- ].L 2«3 /=1 x ~ ~ n i=1


where

= g i ( x ) + l n ( g i ( x ) + l / g 2 i ( x ) + 4 8 2 1 , z ) . Ki ( x , 82 , I ,.~) gi( x ) + ~/g2( x) + 482/a~

The algorithm is very similar to the barrier and penalty function method proposed in [7], where f is assumed to be strictly convex. The corresponding unconstrained problem for this latter method is

min f ( x ) - / * In gi( x) q- - - £ h2( x). x~Rn i=1 283 j = l

Therefore, at each iteration, their algorithm solves the following system of nonlinear equations:

~ ~7gi( X) 1 1 v r ( x ) - ~ - - + - - F_, h j ( x ) V h j ( x ) = O.

i=l ge ( x ) 83 j=l

The difference between the two algorithms lies in the second term. Our method uses a penalty function instead of a barrier function at the expense of adding an additional penalty parameter. However, this makes the algorithm capable of starting from any point, whether it is feasible or not. It is interesting to see the role of each perturbation parameter: 81 plays the role of the Tihonov regularization or proximal point parameter, « 3 is the parameter for the quadratic penalty function for the equality constraints, /x is the penalty parameter for the logarithmic barrier function of the inequality constraints, and 8 2 transforms the interior method to an exterior penalty method.

5. Conclusion

In summary, this paper presents a continuation method for the monotone VIP. The method is theoretically interesting and amenable to implementation. Unlike most of the interior-point path-following algorithms for NCP, the new method is more flexible in reducing the continuation parameters and choosing intermediate iterates.

In a series of related papers [4,5], the basic algorithm presented in this paper is specialized to solve the linear complementarity problem, quadratic program, and linear program; extensive numerical results are also reported.

Acknowledgements

The comments and encouragement of Jong-Shi Pang are gratefully acknowledged. The paper has also benefited from a referee's comment, which simplified the proof of Proposition 3.5.

252

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A continuation method for monotone variational inequalities