WEIGHTED COMPLEMENTARITY PROBLEMS - A NEWPARADIGM FOR COMPUTING EQUILIBRIA

FLORIAN A. POTRA∗

Abstract. This paper introduces the notion of a weighted Complementarity Problem (wCP),which consists in finding a pair of vectors (x, s) belonging to the intersection of a manifold with acone, such that their product in a certain algebra, x ◦ s, equals a given weight vector w. Whenw is the zero vector, then wCP reduces to a Complementarity Problem (CP). The motivation forintroducing the more general notion of a wCP lies in the fact that several equilibrium problems ineconomics can be formulated in a natural way as wCP. Moreover, those formulations lend themselvesto the development of highly efficient algorithms for solving the corresponding equilibrium problems.For example, Fisher’s competitive market equilibrium model can be formulated as a wCP that canbe efficiently solved by interior-point methods. Moreover, it is shown that the Quadratic Program-ming and Weighted Centering problem, which generalizes the notion of a Linear Programming andWeighted Centering problem recently proposed by Anstreicher, can be formulated as a special linearmonotone wCP. The main contribution of the paper is to introduce and analyze two interior-pointmethods for solving general monotone linear wCPs.

Key words. weighted complementarity, interior-point, path-following, Fisher equilibrium

AMS subject classifications. 90C51, 90C33

1. Introduction. The aim of this paper is to introduce the notion of a weightedComplementarity Problem (wCP), to study its theoretical properties, and to con-struct numerical methods for its numerical solution. This notion significantly extendsthe notion of a complementarity problem (CP). Generally speaking, wCP consists infinding a pair of vectors (x, s) belonging to the intersection of a manifold with a cone,such that their product in a certain algebra, x ◦ s, equals a given weight vector w.When w is the zero vector, wCP reduces to a Complementarity Problem (CP). Withnonzero weight vectors, the theory of wCP becomes more complicated than the theoryof CP. However, many of the essential properties of CP extend to wCP. Also, manyinterior-point methods for CP can be extended to efficient algorithms for solving wCP.

We have been motivated to introduce the notion of wCP by the fact that wCPcan be used for modeling a larger class of problems from science and engineering.Even when a problem can also be modeled by CP, the wCP model leads to a moreefficient numerical solution method. For example, we will show that the Fisher marketequilibrium problem, which can be modeled as a nonlinear CP, can also be modeleda linear wCP. The latter can be solved more efficiently than the former. Ye’s al-gorithm [25] for solving the Fisher problem with linear utilities can be viewed as anumerical method for solving a particular instance of a linear wCP. It consists of twophases. First, the potential reduction method from [24, page 106] is used to constructa starting point satisfying a restrictive centering condition [25, (5)]. Then a modi-fied primal-dual path-following algorithm is employed to compute an ε-approximatesolution of the Fisher market equilibrium problem. We note that the path followedby Ye’s algorithm is nonsmooth, and that the primal-dual path-following algorithm,which is a modification of the standard primal-dual path-following algorithm for lin-ear complementarity problems of Kojima et al. [14], is a short-step algorithm thathas optimal computational complexity but rather poor practical performance [22]. Ye

∗Department of Mathematics and Statistics, University of Maryland Baltimore County, 1000Hilltop Circle, Baltimore, MD 22150 (potra@umbc.edu).

1

2 FLORIAN A. POTRA

[25, Corollary 1] mentions that the predictor-corrector method of Mizuno-Todd-Ye(MTY) can be used instead of the short-step algorithm, but gives no details abouthow MTY is to be modified for this problem.

Recently the computational complexity results for the Fisher problem from [25]have been improved by Anstreicher [2]. He proposes a generalization of the Eisenberg-Gale formulation of the Fisher problem [6], called Linear Programming and WeightedCentering (LPWC) problem and shows that it possesses a natural dual problemDPWC. He obtains a lower iteration complexity for DPWC by utilizing a combi-nation of the volumetric [1, 23] and logarithmic [17] barriers. It turns out that LPWCgeneralizes both linear programming (LP), and the problem of finding the weightedanalytic center of a polytope [3, 10]. In the present paper we consider a generalizationof LPWC, called Quadratic Programming and Weighted Centering (QPWC) and weshow that this problem and its dual lead to a monotone linear wCP.

The main contribution of our paper is to propose and analyze two interior-pointmethods for solving general monotone linear wCPs. The first method can be inter-preted as an extension of the largest-step path-following method of McShane [15].The algorithm requires one matrix factorization per iteration, has the same computa-tional complexity as the short-step algorithm, but is much more efficient in practice.We also propose an extension of the MTY predictor-corrector method which requirestwo matrix factorizations at each iteration. Both algorithms follow a smooth centralpath. When applied to the Fisher problem, they can use the standard starting pointdescribed in [25] without having to use another method for obtaining a better centeredstarting point.

Although, for the sake of generality, we present the notion of a nonlinear wCP ina Jordan algebra setting, we only study the case of a linear wCP over the nonnegativeorthant of IRn. In this case all the components of the weight vector w are nonnegative,which is denoted by w ≥ 0. When w = 0 this problem reduces to a mixed horizontallinear complementarity problem, while in the case when all components of w arestrictly positive (w > 0) we obtain a weighted analytic centering problem. These twoproblems have been extensively studied and their properties are well understood. Themore general case when some of the components of w are equal to zero and the othercomponents are strictly positive (w ≥ 0) appears to be more difficult to analyze, andgeneralization of algorithms developed for the cases w = 0 and w > 0 is nontrivial.

Conventions. We denote by IN the set of all nonnegative integers. IR, IR+,IR++ denote the set of real, nonnegative real, and positive real numbers respectively.The symbol e represents the vector of all ones, with dimension given by the context.We denote by log t the natural logarithm of t.

We denote component-wise operations on vectors by the usual notations for realnumbers. Thus, given two vectors u, v of the same dimension, uv, u/v, etc. willdenote the vectors with components uivi, ui/vi, etc. This notation is consistentas long as component-wise operations always have precedence in relation to matrixoperations. Note that Auv = A(uv) 6= (Au)v. Also, if f is a scalar function and v is avector, then f(v) denotes the vector with components f(vi). For example if v ∈ IRn

+,then

√v denotes the vector with components

√vi, and 1− v denotes the vector with

components 1 − vi. Traditionally the vector 1 − v is written as e − v, where e is thevector of all ones. Inequalities are to be understood in a similar fashion. For exampleif v ∈ IRn, then v ≥ 3 means that vi ≥ 3, i = 1, . . . , n. Traditionally this is writtenas v ≥ 3 e. For a vector v ∈ IRn we denote max v = max{vi : i = 1, . . . , n} andmin v = min{vi : i = 1, . . . , n}. If ‖ . ‖ is a vector norm on IRn and A is a matrix,

WEIGHTED COMPLEMENTARITY PROBLEMS 3

then the operator norm induced by ‖ . ‖ is defined by ‖A ‖ = max{‖Ax ‖ ; ‖x ‖ = 1}.As a particular case we note that if U is the diagonal matrix defined by the vector u,then ‖U ‖2=‖u ‖∞.

Throughout this paper we use the MATLAB-like notation [u ; v ; w ] to denotethe column vector [uT vTwT ]T . Given a matrix P , we denote by RanP its range (orcolumn space) and by KerP its kernel (or null space).

2. The weighted complementarity problem. We first present the notion ofa wCP in a general Jordan algebra setting [7], since this general framework reveals theessential geometric features of a wCP and will be used in subsequent papers. If (J , ◦)is an Euclidean Jordan algebra, we denote by K = {x◦x : x ∈ J } the cone formed bythe squares of its elements. It turns out that the cone K is symmetric, in the sense thatit is self-dual, K = K∗, and its automorphism group acts transitively on its interior.Symmetric cones are intimately related to Euclidean Jordan algebras, since it can beproved that a cone is symmetric if and only if it is the cone of squares of some EuclideanJordan algebra. Interestingly, symmetric cones are also intimately related to self-scaled barriers, since a cone is symmetric if and only if it is self-scaled, i.e., it admitsa self-scaled barrier. Interior-point methods for solving linear programming problemsover self-scaled cones were studied in [18, 19]. The relation between the self-scaledcones and the symmetric cones (also called homogenous self-dual cones) was notedin [12]. Classical results in Euclidean Jordan algebras show that every symmetriccone can be decomposed into five irreducible symmetric cones (see also [13] and theliterature cited therein). The most important symmetric cones used in mathematicalprogramming are the positive orthant, the cone of positive semidefinite matrices,and the second-order (Lorentz) cone. The extension of interior-point algorithms to aJordan algebra setting, was first detailed in [9, 8]. In [8] one considers the problemof minimizing a convex quadratic programming problem over the intersection of alinear manifold and a symmetric cone, and it is shown that the monotone linearcomplementarity problem over symmetric cones can be reduced to such a problem.In fact, the reverse is also true. The linear manifold considered in [8] was a subset ofJ × J that possessed a certain monotony property.

Since the Lagrange multipliers y ∈ IRm corresponding to equality constraints playa special role in equilibrium problems, we consider a (possibly nonlinear) manifoldM⊂ J × J × IRm. Given such a manifold and a vector w ∈ K, we define a wCP asthe problem of finding

(x, s, y) ∈M∩K ×K × IRm, such that x ◦ s = w.(2.1)

If M is defined as

M = {(x, s, y) ∈ J × J × IRm : F (x, s, y) = 0},(2.2)

where F : J ×J ×IRm → J ×IRm is a given nonlinear map, then (2.1) can be writtenas the problem of finding a solution of the following nonlinear system

x ◦ s = wF (x, s, y) = 0

x, s ∈ K(2.3)

With a zero weight vector w = 0, this reduces to a mixed nonlinear complementarityproblems over the Euclidean Jordan algebra J . Such problems were considered, forexample, in [27]. There are no existence results for the solution of (2.3) in the general

4 FLORIAN A. POTRA

case. When w = 0 an existence result follows from [27, Theorem 3.10] (see also[27, Corollary 4.4]). The uniqueness of the solution is not guaranteed even for theparticular case of a monotone linear complementarity problem.

2.1. The linear wCP over the nonnegative orthant. In this subsection wewill analyze the case where J = IRn, x◦s = xs, K = IRn

+, and F is an affine mapping.Then (2.3) can be written as

xs = wPx+Qs+Ry = a

x, s ≥ 0(2.4)

Here P ∈ IR(n+m)×n, Q ∈ IR(n+m)×n, R ∈ IR(n+m)×m are given matrices, a ∈ IRn+m

is a given vector, and w ∈ IRn+ is a given weight vector (the data of the problem). The

matrix R is assumed to have full column rank. In the first equation above, xs denotesthe vector having as components the product of the corresponding components ofx and s. In other words, xs is the componentwise product of the vectors x and s.The notation x ≥ 0 means that all components of the vector x are nonnegative, i.e.,x ∈ IRn

+. Similarly, x > 0 means that all components of the vector x are positive, i.e.,x ∈ IRn

++. wCP (2.4) is called monotone if

P∆x+Q∆s+R∆y = 0 implies ∆xT ∆s ≥ 0 ,(2.5)

and it is called skew-symmetric if

P∆x+Q∆s+R∆y = 0 implies ∆xT ∆s = 0 .(2.6)

If (2.4) is monotone and strictly feasible then it has a solution. This statement willbe proved in a more general setting in [21].

In the next subsection we will show that the Fisher problem reduces to a skew-symmetric wCP.

2.2. The Fisher equilibrium problem as a wCP. We consider a marketcomposed of nc ≥ 2 consumers and np ≥ 2 producers. Consumer i has a budgetwi > 0 to spend on buying goods from the producers in such a way that an individualutility function is maximized. The price equilibrium is an assignment of prices togoods, so that when every consumer buys a maximal bundle of goods then the marketclears, meaning that all the money is spent and all the goods are sold. Without lossof generality, we assume that producer j has one unit of some good to sell. Let theindividual utility function of consumer i be of the form

ui =np∑

j=1

uijxij ,(2.7)

where uij is the utility coefficient of consumer i for the good produced by producerj, and xij represents the amount of good bought by consumer i from producer j. Weassume that the following inequalities are satisfied for all i and j:

wi > 0, uij ≥ 0,nc∑

k=1

ukj > 0,np∑

k=1

uik > 0 .(2.8)

WEIGHTED COMPLEMENTARITY PROBLEMS 5

Under these assumptions Eisenberg and Gale [6] proved that the market clearing pricesare given by the optimal Lagrange multipliers for the last np equality constraints ofthe following convex optimization problem:

maximizeui, xij

nc∑i=1

wi log ui(2.9)

subject to

ui −np∑

j=1

uijxij = 0 , i = 1, . . . , nc

nc∑i=1

xij = 1, j = 1, . . . , np

ui ≥ 0, xij ≥ 0 , i = 1, . . . , nc , j = 1, . . . , np.

This optimization problem can be written under the form

maximizex

n∑i=1

wi log xi(2.10)

subject to Ax = b

x ≥ 0 ,

where x is a n-dimensional vector, with n = nc(np +1), having its first nc coordinatesformed by u1, ..., unc

, and the remaining ncnp coordinates consisting of the variablesxij ,

x = [u1, ..., unc, x1 1, . . . , x1 np

, x2 1, . . . , x2 np, . . . , xnc 1, . . . , xnc np

]T .

The n-dimensional weight vector w has its first nc coordinates equal to w1, . . . , wnc,

and the remaining coordinates equal to zero. A is a full rank m × n-matrix, withm = nc + np, and b is an m-dimensional vector having its first nc coordinates equalto zero, and the remaining np coordinates equal to 1, i.e., b = [0; e]. By introducingthe vectors

a1 = −[u1 1, . . . , u1 np ]T , a2 = −[u2 1, . . . , u2 np ]T , . . . , anc = −[unc 1, . . . , unc np ]T ,

we can write

A =

1 0 · · · 0 a1 T 0 · · · 00 1 · · · 0 0 a2 T · · · 0· · · · · · · · · · · ·0 0 · · · 1 0 0 · · · anc T

0 0 · · · 0 eT1 eT

1 · · · eT1

0 0 · · · 0 eT2 eT

2 · · · eT2

· · · · · · · · · · · ·0 0 · · · 0 eT

npeTnp

· · · eTnp

=(Inc A1 A2 · · · Anc

0 InpInp

· · · Inp

).

Let us now consider a general optimization problem of the form (2.10), with arbitraryw ∈ IRn

+, A ∈ IRm×n, b ∈ IRm. We assume that A is full rank. When all componentsof the weight vector w are positive, i.e. w > 0, (2.10) is known as a Weighted Analytic

6 FLORIAN A. POTRA

Center Problem [3, 10]. In the present paper we consider the more general case wheresome of the components of w may vanish. This makes the problem more difficult. Wenote that if x is the solution of (2.10), then we must have xi > 0 whenever wi > 0,so that in this case the product wix

−1i is well defined. If wi = 0, then we take by

definition wix−1i = 0 for any value of xi. With this convention, the KKT conditions

for (2.10) can be written as a nonlinear CP:

xv = 0wx−1 + v −AT y = 0

Ax = bx, v ≥ 0

(2.11)

We note that the second equation above is defined for any v ∈ IRn+, y ∈ IRm, and

any x ∈ IRn+ such that xi 6= 0 whenever wi > 0. By denoting s = AT y, we have

s = wx−1 + v ≥ 0. Multiplying this equation by x we obtain the following linearwCP:

xs = ws−AT y = 0

Ax = bx, s ≥ 0

(2.12)

This a particular case of wCP (2.4) with

P =(A0

), Q =

(0I

), R =

(0−AT

), a =

(b0

),(2.13)

which is easily shown to be skew-symmetric. Indeed, if the right-hand-side of (2.6) issatisfied then ∆x ∈ KerA and ∆s ∈ RanAT , so that ∆xT ∆s = 0.

We note that (2.12) was first obtained by Ye [25], who used the Eisenberg andGale [6] formulation of the Fisher problem.

2.3. The Quadratic Programming and Weighted Centering problem.In this section we introduce a more general convex optimization problem that leadsto a monotone linear wCP. Given an n × n symmetric positive semidefinite matrixM , a full rank m× n matrix A, with m < n, and vectors f ∈ IRn, w ∈ IRn

+, b ∈ IRm,we consider the following optimization problems:

minimizex

ϕ(x) :=12xTMx+ fTx−

n∑i=1

wi log xi(2.14)

subject to Ax = b

x ≥ 0 ,

and

maximizeu,s,y

ψ(u, s, y) := −12uTMu+ bT y +

n∑i=1

wi log si +n∑

i=1

wi(1− logwi)(2.15)

subject to s = Mu−AT y + f

s ≥ 0.

In (2.14) and (2.14) we have tacitly made the convention that if wi = 0 then thecorresponding terms wi log xi, wi log si, wi(1− logwi) are set to zero. By denoting

I = {i ∈ {1, . . . , n} : wi > 0} ,(2.16)

WEIGHTED COMPLEMENTARITY PROBLEMS 7

we haven∑

i=1

wi log xi =∑i∈I

wi log xi,

n∑i=1

wi log si =∑i∈I

wi log si,

n∑i=1

wi logwi =∑i∈I

wi logwi.

We call (2.14) a Quadratic Programming and Weighted Centering (QPWC) problem.For M = 0 it reduces to the recently introduced notion of a Linear Programmingand Weighted Centering (LPWC) problem [2]. We say that x is strictly feasible (oran interior point) for (2.14) if Ax = b and x > 0. Similarly (u, s, y) is called strictlyfeasible (or an interior point) for (2.15) if s = Mu − AT y + f > 0. We note thatin the special case M = 0, Anstreicher [2] considers a more general notion of strictfeasibility requiring only that xi > 0, si > 0 for all i ∈ I. Since in this paper weare concerned with interior-point methods, we will only consider the more restrictivenotion of strict feasibility defined above.

Theorem 2.1.1. (weak duality) If x is feasible for (2.14) and (u, s, y) is feasible for (2.15) then

ϕ(x) ≥ ψ(u, s, y);

2. (optimality conditions) x is an optimal solution for (2.14) and (u, s, y) is anoptimal solution for (2.15) if and only if

xs = w, Mx = Mu.(2.17)

Moreover, in this case (x, s, y) is also an optimal solution for (2.15), and wehave ϕ(x) = ψ(u, s, y) = ψ(x, s, y);

3. (strong duality) If (2.14) and (2.15) are strictly feasible then they have optimalsolutions x∗, (x∗, s∗, y∗) with ϕ(x∗) = ψ(x∗, s∗, y∗).

Proof. Let x be feasible for (2.14) and (u, s, y) be feasible for (2.15). If xi = 0for some i ∈ I then ϕ(x) = +∞, and if si = 0 for some i ∈ I then ψ(u, s, y) = −∞.Therefore in what follows, we can assume without loss of generality that xi > 0 andsi > 0 for all i ∈ I. Using the feasibility assumption and the inequality

ν − σ log ν ≥ σ − σ log σ,(2.18)

we deduce that

ϕ(x)− ψ(u, s, y)

=12xTMx+

12uTMu+ fTx− bT y −

n∑i=1

wi log(xisi)−n∑

i=1

wi(1− logwi)

≥ 12xTMx+

12uTMu+ fTx− bT y −

n∑i=1

xisi

=12xTMx+

12uTMu+ (s−Mu+AT y)Tx− (Ax)T y − xT s

=12

(x− u)TM(x− u) ≥ 0.

Since (2.18) holds with equality if and only if σ = ν, it follows that ϕ(x) = ψ(u, s, y)if and only if (2.17) is satisfied. Obviously, in this case x is an optimal solution for(2.14), and (u, s, y), (x, s, y) are optimal solutions for (2.15). Assume now that x is

8 FLORIAN A. POTRA

an optimal solution for (2.14). Since ϕ is convex and the constraints are linear, theKKT conditions must hold. Therefore there are vectors y , v such that

v = Mx+ f − w

x−AT y ≥ 0.

We note that if x is optimal then xi > 0 whenever wi > 0, since otherwise the objectivefunction is +∞. Therefore the ith component of the vector w

x is well defined wheneverwi > 0 . All the other components of w

x are set to 0 by definition. If we define

s = Mx+ f −AT y = v +w

x≥ 0,

then (x, s , y ) is feasible for (2.15), and we have xs = w. Hence (x, s , y ) is an optimalsolution for (2.15), and (2.17) must hold for any other optimal solution (u, s, y) of(2.15). This finishes the proof of points 1) and 2) of our theorem. In order to completethe proof of 3), we have to show that under the assumption of strict feasibility (2.14)has an optimal solution. Let x , (u, s, y) be strictly feasible for (2.14) and (2.15)respectively, i.e.,

Ax = b, s = Mu−AT y + f, x > 0, s > 0,

and consider the level set

L = {x : Ax = b, x ≥ 0, ϕ(x) ≤ ϕ(x )}.

For any x ∈ L we have xi > 0, ∀i ∈ I, and

ϕ(x ) ≥ 12xTMx+ fTx−

n∑i=1

wi log xi

=12xTMx+ xT s+ yTAx− uTMx−

n∑i=1

wi log xi

=12xTMx− uTMx+ bT y + xT s+

n∑i=1

wi log xi .

It follows that

ξ := ϕ(x )− 12uTMu− bT y −

n∑i=1

wi log si

=12

(x− u)TM(x− u) + xT s−n∑

i=1

wi log(xisi)

≥∑i/∈I

xisi +∑i∈I

xisi − wi log(xisi) .(2.19)

It is easy to show that for any i ∈ I there is ζi > 0 such that t−wi log t ≥ .5t, ∀t ≥ ζi.By denoting

ζ = maxiζi, I1 = {i ∈ I : xisi ≥ ζ}, I2 = {i ∈ I : xisi < ζ}, σ = min

isi ,

WEIGHTED COMPLEMENTARITY PROBLEMS 9

and using (2.18), we deduce that

ξ ≥∑i/∈I

xisi +12

∑i∈I1

xisi +∑i∈I2

xisi − wi log(xisi)

≥ σ∑i/∈I

xi +σ

2

∑i∈I1

xi +∑i∈I2

wi − wi log(wi).

It follows that

xj ≤ min{ζ, 2σ

(ξ −∑i∈I2

wi − wi log(wi))}, j = 1, . . . , n .

Hence, the level set L is compact. Since ϕ is convex and bounded below on L, it willattain its minimum on L.

The optimality conditions from the above theorem give rise to a linear wCP ofthe form (2.4) with

P =(

AM

), Q =

(0−I

), R =

(0−AT

), a =

(b−f

).

It can be shown that this wCP is monotone. In the particular case when M = 0, thewCP is skew-symmetric.

3. Ye’s method for solving the Fisher equilibrium problem. In thissection we review the interior-point method developed by Ye [25] for the Fisher equi-librium problem. In order to be able to better compare his method with the twomethods proposed in this paper we will present Ye’s method as a method for solvingthe linear wCP (2.4) for the case (2.13). For this problem Ye considered a startingpoint z0 = [x0; s0; y0] where the first nc coordinates of x0 are given by

u0i =

1np

np∑k=1

uik, i = 1, . . . , nc(3.1)

and the remaining ncnp coordinates are equal to 1/np,

x0ij =

1np, i = 1, . . . , nc , j = 1, . . . , np.(3.2)

We have clearly Ax0 = b and x0 > 0. The dual vector y0 = [q0; p0] has components

q0i =β

u0i

, i = 1, . . . , nc ; p0j = 2npβ, j = 1, . . . , np.(3.3)

Finally, the components of the slack vector s0 = Ay0,

s0 = [v01 , ..., v

0nc, s01 1, . . . , s

01 np

, s02 1, . . . , s02 np

, . . . , s0nc 1, . . . , s0nc np

]T ,

are given by

v0i = q0i , i = 1, . . . , nc ; s0i j = p0

j − q0i ui j , i = 1, . . . , nc , j = 1, . . . , np .(3.4)

10 FLORIAN A. POTRA

It follows that

u0i v

0i = β , i = 1, . . . , nc ,

x0ijs

0ij = 2β − β uij∑np

k=1 uik

∈ [ β, 2β ] , i = 1, . . . , nc , j = 1, . . . , np ,(3.5)

µ0 =x0 T s0

nc + npnc=

2np β

np + 1·

Let us denote the feasible set of wCP (2.4) by

F = {z = [x; s; y] ∈ IR2n+m : Px+Qs+Ry = a, x ≥ 0, s ≥ 0} .(3.6)

Its relative interior

F0 = {z = [x; s; y] ∈ IR2n+m : Px+Qs+Ry = a, x > 0, s > 0}(3.7)

is called the set of strictly feasible (or interior) points. The central path of F is thecurve given by the set of all points [t; z] = [t;x; s; y], with t > 0, satisfying

xs = tePx+Qs+Ry = ax > 0, s > 0

.(3.8)

If [t; z] is on the central path, then obviously

µ = µ(z) =xT s

n= t .

Therefore, a good proximity measure of a point z = [x; s; y] ∈ F to the central path(3.8) is given by

δ2(z) =∥∥∥∥ xs

µ(z)− e

∥∥∥∥2

, µ(z) =xT s

n·(3.9)

Using this proximity measure we can define the following neighborhood of the centralpath (3.8):

N2(α) = {z ∈ F0 : δ2(z) ≤ α}.(3.10)

For the starting point z0 = [x0; s0; y0] we have

µ20δ2(z0)2 = β2

(np − 3)nc

np + 1+

nc∑i=1

np∑j=1

(uij∑np

k=1 uik

)2 ≤ 2(np − 1)ncβ

2

np + 1·

It follows that

δ2(z0) ≤

√(n2

p − 1)nc

√2np

<

√nc√2.(3.11)

In [25] Ye uses the starting point z0 with

β =np + 1

2np‖w‖∞ ,(3.12)

WEIGHTED COMPLEMENTARITY PROBLEMS 11

which implies

µ0 = µ(z0) = ‖w‖∞.(3.13)

Starting from this point, the potential reduction method from [24, page 106] is used toconstruct a point z ∈ N2(α) for some α < 1. This is achieved in O(nc) iterations. Wenote that at page 324 of [25] it is claimed that only O(log(nc np)) iterations are needed.This is clearly a missprint, as acknowledged by Ye in a private communication. Then,the so-called modified primal-dual path-following algorithm, starting from z, is usedto follow the modified central path{

xs = w(t)Px+Qs+Ry = a

, w(t) = max{te, w}, t > 0 .(3.14)

Here x, s ∈ IRn++, w ∈ IRn

+, with n = nc(np + 1), but only nc components of w arenonzero. If t ≥ ‖w‖∞ then (3.14) reduces to (3.8), and we have t = µ = µ(z) = xT s/n.On the other hand for any [t; z] satisfying (3.14) and t ≤ min{wi : wi > 0} we have

µ =eTw + ncnpt

nc(np + 1)= t+

eTw − nct

nc(np + 1),

so that µ→ eTw/n as t→ 0.The l2-neighborhood of the modified central path (3.14) is defined by

N2(w,α) = {[t; z] ∈ IR++ ×F0 : ‖xs− w(t)‖2 ≤ αt} .

The point z produced by the potential reduction method satisfies [t; z] ∈ N2(w,α),with t = µ(z) ≥ ‖w‖∞ and α < 1. At each iteration, the so-called modified primal-dual path-following algorithm from [25] starts with a point [t; z] ∈ N2(w,α) andproduces a point [t+; z+] ∈ N2(w,α) with t+ = (1− α/

√n ) t. Let us denote the set

of ε-approximate solutions of wCP (2.4) by

Sε = {z = [x; s; y] ∈ F : ‖xs− w‖2 ≤ ε} .(3.15)

If [t; z] ∈ N2(w,α), then

‖xs− w‖2 ≤ ‖xs− w(t)‖2 + ‖w(t)− w‖2 ≤ αt+ ‖te‖2 ≤(α+√n)t.(3.16)

Therefore, the modified primal-dual path-following algorithm from [25] produces anε-approximate solution in at most O (

√n log (‖w‖∞

√n/ε)) iterations. If we add the

cost of the potential reduction method, it follows that an ε-approximate solution forthe Fisher problem is obtained in at most

O(√ncnp log ((nc + np) ‖w‖∞/ε) +O(nc)(3.17)

iterations. Since at each iteration we have t+ = (1− α/√n ) t, the algorithm be-

longs to the class of short-step methods, and therefore its practical performance isclose to the worst case bounds reflected in the above iteration complexity result. Inwhat follows we show that it is not necessary to first find a point in N2(α) withα < 1. Instead, we start with z0, define a new (smooth) central path emanatingfrom it, and consider a new neighborhood of this central path. We will show thatour algorithms find an ε-approximate solution for the Fisher problem in at mostO(√ncnp log ((nc + np) ‖w‖∞/ε) iterations. Our algorithms are long step algorithmsand therefore their practical performance are much better than indicated by this it-eration complexity result. Moreover our algorithms work for any monotone linearwCP.

12 FLORIAN A. POTRA

4. Two interior-point methods for solving monotone linear wCPs. Inthis section we present a long step path-following method and a predictor-correctormethod for solving a general linear wCP of the form (2.4) that is monotone in thesense of (2.5). The long step method may be interpreted as a generalization of Mc-Shane’s largest step algorithm [15]. The name “largest step algorithm” was givenby Gonzaga [11] in the case of monotone complementarity problems (see also [4, 5]).The predictor-corrector method can be considered as a generalization of the Mizuno-Todd-Ye method [16], which was the first algorithm for solving linear programmingproblems having both polynomial complexity and superlinear convergence. The maindifference between the two algorithms to be introduced in this section is that the longstep path-following algorithm uses only one matrix factorization per iteration whilethe predictor-corrector method uses two factorizations.

4.1. A long step path-following method. Let us consider the notations from(2.4), (3.6) and (3.7). Given a strictly feasible starting point z0 = [x0, s0; y0] ∈ F0,we denote

t0 = µ(z0), c = x0s0, γ =min c

t0, w(t) = (1− t/t0)w + (t/t0)c, t ∈ (0, t0],(4.1)

where min c = min{cj : j = 1 . . . , n}. We define the central path of wCP (2.4)emanating from z0 as the set of all points [t; z] = [t;x; s; y], with t ∈ (0, t0], satisfying

xs = w(t)Px+Qs+Ry = ax > 0, s > 0

.(4.2)

By construction [t0; z0] belongs to this path. We note that for the Fisher problem wehave t0 = ‖w ‖∞ and γ > 1/2, as indicated by (3.5), (3.12) and (3.13).

Before describing our interior-point methods let us make some remarks about themodified central path (3.14) used by Ye [25] and the central path (4.2). First, as notedabove the starting point belongs by construction to the central path (4.2). This isnot the case with the modified central path (3.14). In fact, as mentioned in Section 3,a potential reduction method, which can be considered a “Phase I” algorithm, isused to produce a point in a certain neighborhood of the modified central path.The difference between the two central paths consists in the fact that right-handside of the first equation in (3.14) is given by w(t), while the right-hand side of thecorresponding equation in (4.2) is w(t). Since the wCP is monotone, strictly feasible,and w(t) > 0, w(t) > 0, ∀t ∈ (0, t0], it follows that both (3.14) and (4.2) haveunique solutions for any t ∈ (0, t0]. However, while w(t) is smooth on (0, t0], w(t) isnot smooth at the points in the set {wi : wi > 0}.

Given a parameter α such that

0 ≤ γ

3≤ α ≤ 2γ

3,(4.3)

we define the following neighborhood of the above central path:

N2(w, c, α) = {[t; z] = [t;x; s; y] ∈ (0, t0]×F0 : ‖xs− w(t)‖ ≤ αt}.

For our starting point we have x0s0 = c = w(t0), so that [t0, z0] ∈ N2(w, c, α).At a typical iteration of our algorithm we have a point [t; z] ∈ N2(w, c, α), for

some t ≤ t0. Since c ≥ γt0 e, it follows that

xs ≥ w(t)− αte ≥ (t/t0)c− αte ≥ (γ − α)te = βte, β = γ − α ≥ γ

3≥ α

2·(4.4)

WEIGHTED COMPLEMENTARITY PROBLEMS 13

Let us denote

t(θ) = (1− θ)t, z(θ) = [x(θ); s(θ; y(θ))] = [x+ u(θ); s+ v(θ); y + d(θ)] ,(4.5)

where u(θ), v(θ), d(θ) are the solutions of the following linear system{su(θ) + xv(θ) = w(t(θ))− xs

Pu(θ) +Qv(θ) +Rd(θ) = 0 .(4.6)

Using the stepsize

θ+ = max{θ ∈ [0, 1] : [t(θ); z(θ)] ∈ N2(w, a, α), ∀θ ∈ [0, θ ]},(4.7)

we obtain the new point

[t+; z+] := [t(θ+); z(θ+)] ∈ N2(w, c, α) ,(4.8)

and we can begin a new iteration.In order to efficiently compute the stepsize defined in (4.7) we first solve the

following two linear systems

{su+ xv = w(t)− xs

P u+Qv +Rd = 0,

{su+ xv = w − xs

P u+Qv +Rd = 0.(4.9)

The solution of the linear system (4.6) can be written under the form

u(θ) = (1− θ)u+ θu, v(θ) = (1− θ)v + θv, d(θ) = (1− θ)d+ θd.(4.10)

From (4.5) and (4.6) we have

x(θ)s(θ) = w(t(θ)) + u(θ)s(θ) = w(t) + θ(t/t0)(w − c) + u(θ)s(θ) .(4.11)

By denoting

ψ(θ) = ‖x(θ)s(θ)− w(t(θ))‖22 − α2t(θ)2,

we deduce that

[t(θ); z(θ)] ∈ N2(w, c, α) if and only if ψ(θ) ≤ 0.

From (4.5),(4.6),(4.9),(4.10),(4.11) we have:

ψ(θ) = ‖u(θ)v(θ)‖22 − (1− θ)2t2α2

= ‖(1− θ)2uv + θ(1− θ) (uv + vu) + θ2uv‖22 − (1− θ)2t2α2.

Finally, by developing the square of the norm of the sum of three vectors we obtainan explicit form of ψ(θ) as a quartic in θ:

ψ(θ) = (1− θ)4 ‖ uv ‖22 + 2θ(1− θ)3eT (u2vv + uv2u)

+θ2(1− θ)2(‖ uv + vu ‖22 + 2eT (uvuv)

)(4.12)

+2θ3(1− θ)eT (uu2v + vvu2) + θ4 ‖ uv ‖22 − (1− θ)2t2α2.

14 FLORIAN A. POTRA

In what follows we will use the following technical result, which is well known inthe interior-point literature (see for example [20, Lemma 3.1]).

Lemma 4.1. If (2.5) is satisfied then the linear system{su+ xv = g

Pu+Qv +Rd = 0

has a unique solution for any x ∈ IRn++, s ∈ IRn

++, g ∈ IRn, and the following inequalityholds

‖uv ‖2 ≤1√8

∥∥∥ (xs)−1/2g∥∥∥2

2.

At the current iteration we have [t; z] ∈ N2(w, c, α), and by using (4.4) we obtain

‖ uv ‖2 ≤1

βt√

8‖xs− w(t) ‖22 ≤

α2t

β√

8≤ 2αt√

8< αt ,

so that ψ(0) = ‖ uv ‖22 − t2α2 < 0 and ψ(1) = ‖ uv ‖22 ≥ 0. Therefore the quarticequation ψ(θ) = 0 has at least one root in the interval (0, 1] and our steplength canbe computed as

θ+ = the smallest root of the quation ψ(θ) = 0 from the interval (0, 1].(4.13)

Our algorithm can be therefore formally defined as

Algorithm 1 (Largest Step)Given a starting point z0 = [x0; s0; y0] ∈ F0:

Consider the notation from (4.1);Choose a parameter α satisfying (4.3);Set k ← 0 ;repeat

Set z = [x; s; y] ← zk , t ← tk ;Solve the linear systems (4.9);Compute steplength θ+ from (4.12) and (4.13);Compute t+, z+ from (4.5), (4.8), and (4.10);Set θk ← θ+ , zk+1 ← z+ , tk+1 ← t+;Set k ← k + 1.

continue

We note that the linear systems from (4.9) have the same matrix, so that thealgorithm requires only one matrix factorization per iteration.

In the remainder of this section we will establish the computational complexityof Algorithm 1. For convenience we introduce the following notation:

ρ = 1 +‖c− w‖2

t0·(4.14)

Theorem 4.2. If wCP (2.4) is monotone, then Algorithm 1 is well defined andgenerates an iteration sequence satisfying the following properties:

[tk; zk] ∈ N2(w, c, α) ;tk+1 = (1− θk)tk ;

θk ≥α

9ρ·

WEIGHTED COMPLEMENTARITY PROBLEMS 15

Proof. The first two properties hold from construction. In order to prove thelast property we first note that if [t(θ); z(θ)] is given by (4.5) then according to (4.4),(4.11), and Lemma 4.1 we have

‖x(θ)s(θ)− w(t(θ))‖2 = ‖u(θ)s(θ)‖2 ≤‖w(t(θ))− xs‖22√

8 minxs≤ ‖w(t(θ))− xs‖22√

2αt·

In order to majorize the last term above we note that

‖w(t(θ))− xs‖2 = ‖w(t) + θ(t/t0)(w − c)− xs‖2 < t(α+ θρ).

If 0 < θ ≤ α/(9ρ) < 1/9 then

‖x(θ)s(θ)− w(t(θ))‖2αt(θ)

≤ (α+ θρ)2√2α2(1− θ)

≤ (α+ α/9)2√2α2(1− 1/9)

< .99 .

It follows that [t(θ); z(θ)] ∈ N2(w, c, α), ∀θ ∈ (0, α/(9ρ)], which implies θ+ ≥ α/(9ρ),where θ+ is the stepsize (4.7) used by our algorithm.

Corollary 4.3. If wCP (2.4) is monotone, then Algorithm 1 finds an ε-appro-ximate solution for this problem (i.e., a point z ∈ Sε, where Sε is defined in (3.15))in at most

O

(x0 T s0/n+

∥∥x0s0 − w∥∥

2

minx0s0log

x0 T s0/n+∥∥x0s0 − w

∥∥2

ε

)

iterations.Proof. From Theorem 4.2, (4.1), (4.3), and (4.14) we have∥∥xksk − w

∥∥2≤∥∥xksk − w(tk)

∥∥2

+ ‖w(tk)− w ‖2 ≤ tk(α+ ‖w − c ‖2 /t0)

< tkρ ≤(

1− α

9ρ

)k

t0ρ ≤(

1− γ

27ρ

)k

t0ρ ≤(

1− minx0s0

27t0ρ

)k

t0ρ ,

and the complexity result follows from a standard argument.Corollary 4.4. When applied to the wCP generated by the Fisher problem (see

(2.13)) with starting point z0 given by (3.1)-(3.4) and (3.12), Algorithm 1 finds anε-approximate solution for this problem in at most

O

(√ncnp log

(nc + np) ‖w‖∞ε

)iterations.

Proof. By writing

‖c− w‖22 = ‖w‖22 + ‖c‖22 − 2wT c = ‖w‖22 + ‖c‖22 −np + 1np

‖w‖∞‖w‖1 ,

and using the relation ‖w‖2∞ ≤ ‖w‖22 ≤ ‖w‖∞‖w‖1 we deduce that

‖c− w‖22 ≤ ‖c‖22 −‖w‖22np

≤ ‖c‖22 −‖w‖2∞np

.

16 FLORIAN A. POTRA

On the other hand we have

‖c‖22 = ncβ2 +

∑i,j

(s0iju

0ij

)2= β2

nc +nc∑i=1

np∑j=1

(2− uij∑np

k=1 uik

)2

= β2

nc +nc∑i=1

np∑j=1

(4− 2

uij∑np

k=1 uik

+u2

ij(∑np

k=1 uik

)2)

= β2

(4ncnp − nc +

nc∑i=1

∑np

j=1 u2ij(∑np

k=1 uik

)2)

=(np + 1

2np

)2(

4ncnp − nc +nc∑i=1

∑np

j=1 u2ij(∑np

k=1 uik

)2)‖w ‖2∞ .

Since∑np

j=1 u2ij ≤

(∑np

j=1 uij

)2 we deduce that for all nc, np ≥ 1 there holds

‖c− w‖ ≤ ‖c‖ ≤ np + 1np

√ncnp ‖w ‖∞ ≤ 2

√ncnp ‖w ‖∞ .

Finally, from (3.5), (3.12), (3.13) we have

min c ≥ ‖w ‖∞ /2 , t0 = x0 T s0/n = ‖w ‖∞ ,

and the desired complexity result follows Corollary 4.3 .

4.2. A predictor-corrector method. As mentioned in the previous subsec-tion, Algorithm 1 requires only one matrix factorization per iteration. At a costof two matrix factorizations per iteration, we can generalize the Mizuno-Todd-Yepredictor-corrector algorithm to our setting. The purpose of the predictor is to im-prove as much as possible the optimality measure t while not departing too muchfrom the central path (4.2). The algorithm depends on two parameters α and α , suchthat

γ

3≤ α < α ≤ 2γ

3,

4α3≤ α ≤

√2α .(4.15)

The above relations are satisfied for example by α =√

2γ/3, and α = 2γ/3.

4.2.1. The predictor. At the beginning of the predictor step we are given apoint [t; z] ∈ N (w, c, α) and we compute the predictor direction [u; v; d] as the solutionof the linear system {

su+ xv = w − xsPu+Qv +Rd = 0 .(4.16)

We define

x(θ) = x+ θu, s(θ) = s+ θv, y(θ) = y + θd, t(θ) = (1− θ)t,z(θ) = [x(θ) ; s(θ) ; y(θ) ] .(4.17)

The stepsize along this direction is taken as

θ = max{θ ∈ [0, 1] : [ t(θ) ; z(θ) ] ∈ N2(w, c, α ), ∀θ ∈ [0, θ ]},(4.18)

WEIGHTED COMPLEMENTARITY PROBLEMS 17

We have

x(θ)s(θ) = (x+ θu)(s+ θv) = (1− θ)xs+ θw + θ2uv ,

w(t(θ)) = w + (1− θ)t(c− w) ,x(θ)s(θ)− w(t(θ)) = (1− θ)(xs− w(t)) + θ2uv .

Therefore the inequality

‖x(θ)s(θ)− w(t(θ))‖2 ≤ α t(θ)

can be written as

β0(1− θ)2 + 2β1(1− θ)θ2 + β2θ4 ≤ 0,(4.19)

where

β0 =‖xs− w(t)‖22

t2− α 2, β1 =

(uv)T (xs− w(t))t2

, β2 =‖uv‖22t2·(4.20)

Since [t; z] ∈ N (w, c, α) we have β0 ≤ α2 − α 2 < 0. If uv = 0 then β1 = β2 = 0, sothat in this case we have θ = 1. Therefore, in what follows we assume β2 > 0. Byusing the substitution φ = (1−θ)/θ2, (4.19) can be reduced to the following quadraticinequality

β0 + 2β1φ+ β2φ2 ≤ 0.(4.21)

The left-hand-side of the above inequality is strictly negative for φ = 0 (since β0 < 0),and strictly positive for φ sufficiently large (since β2 > 0). Therefore the aboveinequality holds for all φ ≤ φ , where

φ =−β0

β1 +√β2

1 − β0β2

·(4.22)

It follows that stepsize θ+ defined in (4.18) is explicitly given by

θ =

1 if uv = 0

2φ

φ +√φ + φ

2if uv 6= 0

.(4.23)

Having computed this steplength we obtain the predicted point[t ; z

]=[t(θ ) ; z(θ )

]= [ (1− θ )t ; x(θ ) ; s(θ ) ; y(θ ) ] ∈ N2(w, c, α ).(4.24)

4.2.2. The corrector. A corrector steps usually follows a predictor step. Itstarts with a point [ t ; z ] ∈ N (w, c, α ) and produces a point [ t ; z+ ] ∈ N (w, c, α).Note that the measure of optimality t remains unchanged, but the measure of prox-imity to the path (4.2) is improved, wherefrom the name corrector. The direction ofthe corrector is computed as the solution of the linear system{

su+ xv = w(t)− xsPu+Qv +Rd = 0 .(4.25)

18 FLORIAN A. POTRA

By taking a unit step along this direction we obtain the points

t+ = t, x+ = x+ u, s+ = s+ v, y+ = y + d, z+ = [x+ ; s+ ; y+ ].(4.26)

Proceeding as in (4.4) with α instead of α and using (4.15) we deduce that xs ≥(γ − α )te ≥ α/2. According to Lemma 4.1 and (4.15) we have

∥∥x+s+ − w(t)∥∥

2= ‖uv ‖2 ≤

‖w(t)− xs ‖22minxs

√8≤ α 2t

α√

2≤ αt.(4.27)

Hence [ t+ ; z+ ] ∈ N (w, c, α).

Algorithm 2 (Predictor-Corrector)Given a starting point z0 = [x0; s0; y0] ∈ F0:

Consider the notation from (4.1);Choose parameters α and α satisfying (4.15);Set k ← 0 ;repeat

PredictorSet z = [x; s; y] ← zk , t ← tk ;Solve the linear system (4.16);Compute steplength θ from (4.20), (4.22), and (4.23) ;Compute t , z from (4.17) and (4.24);Set θ k ← θ , z k ← z , tk+1 ← t ;CorrectorSet z = [x; s; y] ← z k , t ← tk+1 ;Solve the linear systems (4.25);Compute z+ from (4.26);Set zk+1 ← z+;Set k ← k + 1.

continue

Theorem 4.5. If wCP (2.12) is monotone, then Algorithm 2 is well defined andgenerates an iteration sequence satisfying the following properties

[tk; zk] ∈ N2(w, c, α) , [tk+1; z k] ∈ N2(w, c, α ) ;tk+1 = (1− θ k)tk ;

θ k ≥2α3ρ·

Proof. The first two properties have already been proved. In order to find a lowerbound for θ we note that

|β1| ≤‖xs− w(t)‖2‖uv‖2

t2=√β0 + α 2

√β2, 0 ≤ β0 + α 2 ≤ α2,

and therefore

φ ≥ −β0

|β1|+√β2

1 − β0β2

≥ −β0(√β0 + α 2 + α

)√β2

≥ −β0

(α+ α )√β2≥ α − α√

β2·

WEIGHTED COMPLEMENTARITY PROBLEMS 19

On the other hand we have

‖xs− w‖2 ≤ ‖xs− w(t)‖2 + ‖w(t)− w‖2 ≤ (α+ ‖c− w‖2)t ≤ ρt ,

‖uv‖2 ≤‖xs− w‖22√

8 minxs<

ρ2t

α√

2,

which leads to the lower bound

φ ≥ α(α − α)√

2ρ2

≥ α2√

23ρ2

>4α2

9ρ2=: φ ·(4.28)

Since α < 2γ/3 ≤ 2/3, and ρ > 1, we have φ < 1/5. Using (4.23) and the fact that

2φ

φ+√φ+ φ2

≥√φ, ∀φ ∈ [0, .5],

we obtain

θ >2φ

φ+√φ+ φ2

≥√φ =

2α3ρ·

The following corollaries are easily proved.Corollary 4.6. If wCP (2.4) is monotone, then Algorithm 2 finds an ε-appro-

ximate solution for this problem (i.e., a point z ∈ Sε, where Sε is defined in (3.15))in at most

O

(x0 T s0/n+

∥∥x0s0 − w∥∥

2

minx0s0log

x0 T s0/n+∥∥x0s0 − w

∥∥2

ε

)

iterations.Corollary 4.7. When applied to the wCP generated by the Fisher problem (see

(2.13)) with starting point z0 given by (3.1)-(3.4) and (3.12), Algorithm 2 finds anε-approximate solution for this problem in at most

O

(√ncnp log

(nc + np) ‖w‖∞ε

)iterations.

4.3. Comparison between the two interior-point methods. In what fol-lows we make some remarks about the similarities and the differences between thelong step method (Algorithm 1) and the predictor-corrector method (Algorithm 2).

First, we note that if the optimal stepsize θ+ as defined by (4.7) was known, thenthe new point z+ = z(θ+) could be obtained by solving the linear system (4.6) withθ = θ+. Definition (4.7), while motivating the name “largest step method” givento Algorithm 1, does not lend itself to a direct computation of θ+. In Algorithm 1we compute θ+ as the smallest root of the quartic (4.12) in the interval (0, 1], whereu, v, u, v are obtained by solving the linear systems (4.9). The first system in (4.9) givesthe so-called centering direction (the Newton direction for the equations defining thecentral path (4.2)), while the second system in (4.9) gives the so-called affine scaling

20 FLORIAN A. POTRA

direction (the Newton direction for the equations defining the wCP (2.4)). Since bothsystems have the same matrix, only one matrix factorization is needed.

Algorithm 2 also uses the centering direction and the affine scaling direction butcomputed at different points. The predictor uses the affine scaling direction at thecurrent point and obtains the predicted point by taking the largest stepsize on thisdirection that keeps the point in the larger neighborhood. The corrector uses thecentering direction computed at the predicted point and obtains the new iterate bytaking a unit stepsize on this direction. It is shown that the corrected point belongsto the original neighborhood. Since the affine scaling direction and the centeringdirections are computed at different points two matrix factorizations are needed ateach iteration.

5. Conclusions. In this paper we have introduced the notion of a weightedcomplementarity problem (wCP) in a general setting. We have shown that the Fishermarket equilibrium problem can be formulated as a skew-symmetric linear wCP. Wehave also shown that the notion of a Linear Programming and Weighted Centering(LPWC) problem recently introduced by Anstreicher [2] reduces to a skew-symmetriclinear wCP. The more general notion of a Quadratic Programming and WeightedCentering (QPWC) problem, introduced in the present paper, reduces to a monotonelinear wCP. We have proposed two interior-point methods for solving general mono-tone linear wCPs and have established their computational complexity. The firstmethod generalizes the largest step interior-point method of McShane [15], while thesecond method generalizes the Mizuno-Todd-Ye predictor corrector method [16]. Ifthe weight vector w is equal to 0, and if the corresponding problems have a strict com-plementarity solution, it is known that the first method is superlinearly convergent,while the second method is quadratically convergent. These asymptotic convergenceresults hold also for nonzero weight vectors. This can be shown by appropriatelymodifying the arguments from [26, 15, 4]. A rigorous proof, in a much more generalsetting, will be given in a subsequent paper [21]. When applied to the wCP generatedby the Fisher equilibrium problem, both algorithms have the same iteration complex-ity as the one obtained by Ye [25]. However our algorithms work directly with thestarting point proposed in [25], without having to use another method to obtain abetter centered starting point. Moreover, the central path followed by our algorithmis smooth, while the central path proposed in [25] is not.

REFERENCES

[1] K. M. Anstreicher. Volumetric path following algorithms for linear programming. Math. Pro-gramming, 76(1, Ser. B):245–263, 1997. Interior point methods in theory and practice(Iowa City, IA, 1994).

[2] K.M. Anstreicher. Interior-point algorithms for a generalization of linear programming andweighted centering. Technical report, Dept. of Management Sciences, University of Iowa,February 2011. www.optimization-online.org/DB HTML/2011/02/2937.html.

[3] D. S. Atkinson and P. M. Vaidya. A scaling technique for finding the weighted analytic centerof a polytope. Math. Programming, 57(2, Ser. B):163–192, 1992.

[4] J. F. Bonnans and C. C. Gonzaga. Convergence of interior point algorithms for the monotonelinear complementarity problem. Mathematics of Operations Research, 21:1–25, 1996.

[5] J. F. Bonnans and F. A. Potra. On the convergence of the iteration sequence of infeasible pathfollowing algorithms for linear complementarity problems. Mathematics of OperationsResearch, 22(2):378–407, 1997.

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WEIGHTED COMPLEMENTARITY PROBLEMS 21

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