An O(√

nL) iteration primal-dual path-following method, based on

wide neighborhoods and large updates, for monotone LCP

Wenbao Ai ∗ Shuzhong Zhang †

February 18, 2005

Abstract

In this paper we propose a new class of primal-dual path-following interior point algorithmsfor solving monotone linear complementarity problems. At each iteration, the method wouldselect a target on the central path with a large update from the current iterate, and then theNewton method is used to get the search directions, followed by adaptively choosing the stepsizes, which are e.g. the largest possible steps before leaving a neighborhood that is as wideas the N−

∞ neighborhood. The only deviation from the classical approach is that we treat theclassical Newton direction as the sum of two other directions, corresponding to respectively thenegative part and the positive part of the right-hand-side. We show that if these two directionsare equipped with different and appropriate step sizes then the method enjoys the low iterationbound of O(

√n log L), where n is the dimension of the problem and L = (x0)T s0

ε with ε therequired precision and (x0, s0) the initial interior solution. For a predictor-corrector variant of themethod, we further prove that, besides the predictor steps, each corrector step also reduces theduality gap by a rate of 1− 1/O(

√n). Additionally, if the problem has a strict complementarity

solution then the predictor steps converge Q-quadratically.

Keywords: Monotone LCP, primal-dual interior point method, wide neighborhood.

Mathematics Subject Classification: 90C33, 90C51, 90C05.

∗School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China.

Email: wenbaoai@vip.sina.com†Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong,

Shatin, Hong Kong. Email: zhang@se.cuhk.edu.hk. Research supported by Hong Kong RGC Earmarked Grant

CUHK4174/03E.

1

1 Introduction

In this paper we consider the following monotone linear complementary problem (LCP):

(LCP )

{s = Mx + q

x ≥ 0, s ≥ 0, xT s = 0,

where q ∈ <n and M ∈ <n×n is a monotone matrix, i.e. M + MT is positive semidefinite, orequivalently, xT Mx ≥ 0 for any x ∈ <n.

A particular choice of M is a block skew symmetric matrix, namely M =

[0 A

−AT 0

]. In that case,

the corresponding monotone LCP problem is nothing but a linear programming problem.

The primal-dual interior point method for linear programming was first introduced by Kojima,Mizuno and Yoshise [5] and Megiddo [7], which essentially aims at solving the following parame-terized problem by Newton’s method, for shrinking values of the parameter µ > 0,

(LCP )µ

{s = Mx + q

xi > 0, si > 0, xisi = µ, i = 1, ..., n.

The exact solution of the above problem is known as the analytic central path, with the varyingpath parameter µ > 0. At each iteration, the method would choose a target on the central path andapply the Newton method to move closer to the target, while confining the iterate to stay within acertain neighborhood of the analytic central path. This method was found to be not only elegant inits simplicity and symmetricity, but also extremely efficient in practical implementations. There hasbeen, however, an inconsistency between theory and practice: fast algorithms in practice may actuallyrender worse complexity bounds. In their first paper [5], Kojima, Mizuno and Yoshise proposed thatthe iterates reside in a wide neighborhood of the central path, known as the N−∞-neighborhood(details of the notion will be discussed later), and the targets on the central path are shifted towardsthe origin by a large update (percentage reduction) at each iteration. The worst case iterationbound was proved to be O(nL), where n is the larger dimension of a standard linear programmingproblem, and L is its input-length. Then, in a subsequent paper, [6], the same authors proposeda variant of the method, where the iterates are restricted to a much smaller neighborhood, knownas the N2-neighborhood, and at each step the target is shifted with a small update. The algorithmbecame too conservative to be efficient in practice. However, the worst case iteration bound of thevariant was improved to O(

√nL). In fact, many early primal-dual interior point methods either use

a narrow neighborhood, or take small step sizes; see e.g. the primal-dual method by Monteiro andAdler [9, 10]. The first practically efficient O(

√nL) primal-dual interior point algorithm was the

celebrated predictor-corrector algorithm of Mizuno, Todd and Ye [8]. In the predictor step of thealgorithm, an adaptive step size is taken, ensuring its practical efficiency. The iteration bound is still

2

retained to be O(√

nL) since the N2 small neighborhoods are used to control the centrality of theiterates. Gonzaga [2] proposed to compute and combine the predictor and corrector steps based onthe information of the same iterate, thus reducing the effort required by the Cholesky factorization.Along a related but different line, Ye, Guler, Tapia and Zhang [18] proved that the predictor stepin the predictor-corrector scheme reduces the duality gap with a quadratic convergence rate. Thisresult was extended by Ye and Anstreicher in [17] to the monotone LCP problem, assuming a strictcomplementary solution exists. We refer to Wright [14] for an excellent exposition on the primal-dualinterior point method for linear programming and LCP problems.

The issue of the neighborhood size in the method has generated some research interests on its own.It is believed that in its original form, the primal-dual interior point algorithm of Kojima, Mizunoand Yoshise [5] may indeed not enjoy the low iteration bound of O(

√nL). However, it is possible to

modify the algorithm to gain both the theoretical and the practical advantages. A first such attemptwas made by Xu [15], who proposed an O(

√nL) method, in which the small neighborhood was only

used as a safeguard, and the iterates are allowed to go far beyond. However, the new neighborhood,though much larger than the small neighborhood, does not necessarily contain the wide neighborhood:the neighborhood is still much narrower than the N−∞ wide neighborhood. Hung and Ye [4] proposedto use higher-order corrections on the Newton method, and showed that the iteration bound of theirhigh order primal-dual interior point method with the N−∞ wide neighborhood can be reduced toO(n

n+12n L). Sturm and Zhang [13] proposed to follow a central region, instead of the central path.

The central region is defined to be precisely an N−∞ wide neighborhood of the central path. Then,they introduced a (narrow) neighborhood of the central region (the whole area is thus wider thanthe central region which is a wide neighborhood itself) in which all the iterates reside. By choosingthe direction towards a target in the central region properly, Sturm and Zhang [13] managed toshow that their algorithm has an iteration bound of O(

√nL). Since the iterates are required to take

adaptive (thus long) steps, maximum possible within the wide neighborhood, the algorithm is highlyefficient in practice. Later, Sturm generalized the method to solve Semidefinite Programming (SDP)problems, and the method has become one of the pillars for his famous SDP solver SeDuMi [12].Ai [1] proposed a new wide neighborhood interior point algorithm with O(

√nL) iteration bound.

The current paper is inspired by [1], in the definition of the new wide neighborhood; however, theydiffer greatly in both the scope and the results to be achieved. Recently, Peng, Terlaky and Zhao [11]introduced a variant of the corrector-predictor approach, based on a self-regular function to definethe neighborhood, which is wide. They showed that their algorithm enjoys the iteration bound ofO(√

n log nL) for linear programming problems.

As far as we know, in the context of path-following approach, none had succeeded in retaining theO(√

nL) complexity while allowing a large update (meaning a reduction by a universal percentage,independent of the problem parameters) of the target along the central path at all iterations, even ifone is allowed to stay within narrow neighborhoods. Indeed, deriving a path-following interior point

3

method with the O(√

nL) iteration bound while working with large-updates and wide neighborhoodat each iteration is one of the objectives to be achieved in this paper. In other words, the current paperaims at modifying the original primal-dual interior point method with minimum changes, maintaininglarge updates and working with wide neighborhood at all iterations, to gain the O(

√nL) iteration

bound and retain practical efficiency. We organize the paper as follows. Our new methodology willbe introduced in Section 2, where the main underlying ideas will be explained. In Section 3, wepresent the technical lemmas that will be important for the subsequent analysis, and in Section 4 wediscuss some easily implementable variants of the general method and show the low computationalcomplexity status. In a similar spirit, we discuss another variant in Section 5, based on the predictor-corrector methodology. Novel properties of the algorithm will be discussed, including its progressiveproperty even for the corrector steps. The low complexity bound will be proven for this variant, andthe superlinear convergence property will be shown, provided that a strictly complementary solutionexists.

The notation used in this paper is fairly standard: the i-th component of vector x ∈ <n is denoted byxi; e is the all one vector with an appropriate dimension; if d ∈ <n then we denote D to be an n× n

diagonal matrix with d as the diagonal components; for x, y ∈ <n, xy is the component product in<n, and so is true for other operations, e.g. 1/(xy) and (xy)−0.5; x ≥ (>) y means that the inequalityholds component-wisely; for any a ∈ <, a+ denotes its nonnegative part, i.e. a+ := max{a, 0}, anda− denotes its nonpositive part, i.e. a− := min{a, 0}; the same notation is used for vector x ∈ <n,namely x+ is the nonnegative part of x and x− is the nonpositive part of x; the Lp-norm of x ∈ <n

is denoted by ‖x‖p, and in particular we write ‖x‖ for ‖x‖2 — the Euclidean norm.

2 Separating large and small components: a new paradigm

Let us denoteF++ := {(x, s) | s = Mx + q, x > 0, s > 0},

which is assumed to be nonempty throughout this paper.

The central path for (LCP) is defined as

C := {(x, s) ∈ F++ | xs = µe}

and its small neighborhood is defined as

N2(β) := {(x, s) ∈ F++ | ‖xs− µe‖ ≤ βµ}

where β ∈ (0, 1) is a given constant and µ := xT s/n. The so-called wide neighborhood is defined asfollows:

N−∞(1− τ2) := {(x, y, s) ∈ F++ | xs ≥ τ2µe}

4

where 0 < τ2 < 1.

Before proceeding, let us recall the classical primal-dual interior point method with wide neighborhoodand large update of the targets. Let 0 ≤ τ ≤ 1 and 0 < τ2 < 1 be two given parameters. Supposethat the current iterate is (x, s) ∈ N−∞(1 − τ2). The search direction (∆x,∆s) is the solution of thefollowing system of linear equations:

{∆s = M∆x,

s∆x + x∆s = τµe− xs,(1)

whereµ = xT s/n.

Then the next iterate will be given by

(x + α∆x, s + α∆s)

where α is the solution of the subproblem

minimize (x + α∆x)T (s + α∆s)subject to (x + α∆x, s + α∆s) ∈ N−∞(1− τ2)

α ∈ [0, 1].

Naturally, all the iterates are contained in the wide neighborhood N−∞(1− τ2).

An important ingredient of this paper is to introduce a new neighborhood for the central path, definedas

N (τ1, τ2, η) := N−∞(1− τ2)

⋂{(x, s) ∈ F++

∣∣ ∥∥(τ1µe− xs)+∥∥ ≤ η(τ1 − τ2)µ

}, (2)

where η ≥ 1 and τ1 satisfying 0 < τ2 < τ1 < 1, are two more parameters.

The above defined neighborhood is itself a wide neighborhood, since one can easily verify that

N−∞(1− τ1) ⊆ N (τ1, τ2, η) ⊆ N−

∞(1− τ2).

Moreover, if τ1 − η(τ1 − τ2)/√

n− 1 > τ2, then

N−∞

(1− τ1 + η(τ1 − τ2)/

√n− 1

) ⊆ N (τ1, τ2, η),

and if τ1 − η(τ1 − τ2)/√

n− 1 ≤ τ2, then

N (τ1, τ2, η) = N−∞(1− τ2).

Specially, if we choose η = 1, the neighborhood can be expressed more simply as follows.

N (τ1, τ2, 1) = {(x, s) ∈ F++ |∥∥(τ1µe− xs)+

∥∥ ≤ (τ1 − τ2)µ} ⊇ N−∞(1− τ1). (3)

5

In this paper, the newly introduced neighborhood N (τ1, τ2, η) will play an important role. The reasonfor working with N (τ1, τ2, η) is that the measure for the components in xs that are ‘dangerously’close to zero is captured by the quantity ‖(τ1µe− xs)+‖, and we are less concerned about the ‘large’components of xs present in (τ1µe − xs)−. In fact, we will see later that this separation is crucial.The part (τ1µe− xs)+ is used to control the centrality, and the other part (τ1µe− xs)− is importantfor the progress towards optimality.

Suppose that our current iterate is (x, s). Let 0 ≤ τ ≤ 1. Another key ingredient of our method is todecompose the Newton step, from xs to the target on the central path τµe (large update), into twoequations: {

∆s− = M∆x−s∆x− + x∆s− = (τµe− xs)−

(4)

and {∆s+ = M∆x+

s∆x+ + x∆s+ = (τµe− xs)+.(5)

Since τµe − xs = (τµe − xs)− + (τµe − xs)+, the usual Newton direction is simply (∆x−,∆s−) +(∆x+,∆s+). In this paper, however, we propose to treat these two directions separately. Essentiallythose are what we need to modify the original large update and wide neighborhood path-followingmethod. The payoff for the changes will become clear later. At this stage, we only remark thatthe extra computational effort is very marginal, compared to the computation of a single Newtondirection. For reference, we shall call (∆x−,∆s−) and (∆x+,∆s+) the Newton constituent directions.

Let α := (α1, α2) ∈ <2+ be the step sizes taken along (∆x−,∆s−) and (∆x+,∆s+) respectively. The

step is(x(α), s(α)) := (x, s) + α1(∆x−,∆s−) + α2(∆x+,∆s+).

The best α can be obtained by solving the following two-dimensional optimization problem:

minimize x(α)T s(α)subject to (x(α), s(α)) ∈ N (τ1, τ2, η)

0 ≤ α1 ≤ 1, 0 ≤ α2 ≤ 1.

(6)

Remark that due to the monotonicity, the above objective function x(α)T s(α) is convex in α.

Below we describe a generic framework for our wide-neighborhood and large-update primal-dualpath-following method.

6

Algorithm 2.1.

Input parameters: required precision ε > 0, neighborhood parameters ηk ≥ 1, 0 < τk2 < τk

1 < 1, andtarget parameters 0 ≤ τk ≤ 1, k = 0, 1, ..., and the initial solution (x0, s0) ∈ N (τ0

1 , τ02 , η0).

Output: a sequence of iterates {(xk, sk) | k = 0, 1, 2, ...}.

Step 0 Set k = 0.

Step 1 If (xk)T sk ≤ ε then stop.

Step 2 Solve (∆xk−,∆sk−) and (∆xk+,∆sk

+) based on (4) and (5).

Find step size vector αk ∈ <2++, such that (x(αk), s(αk)) ∈ N (τk

1 , τk2 , ηk).

Step 3 Set (xk+1, sk+1) := (x(αk), s(αk)).

Let k := k + 1 and go to Step 1.

We remark here that the optimal step sizes according to (6) may be used in Step 2, and theparameters ηk, τk

1 , τk2 and τk may be set to constants. It is however convenient to allow for the

flexibilities at this stage.

The main result of this paper is to prove that the above generic method can be specified intoeasy implementable variants with given parameters, in such a way that the iteration bound willbe O(

√n log (x0)T s0

ε ). Moreover, the method can also be implemented in the predictor-correctorstyle. In that case, in addition to the above iteration bound one also obtains a quadratic convergencerate for the predictor steps, provided that a strict complementary solution exists. These specificimplementations will be discussed in Sections 4 and 5 respectively. To facilitate the analysis, we needto study the properties of the two separated Newton constituent directions. This will be the topic ofthe next section.

3 Technical lemmas

In this section we choose to set τ = τ1.

First, it is useful for our subsequent analysis to note the following triangle inequalities for the ‘minus’and ‘plus’ operations on the vectors.

Proposition 3.1. For any u, v ∈ <n and p ≥ 1, we have

‖(u + v)+‖p ≤ ‖u+‖p + ‖v+‖p

7

and‖(u + v)−‖p ≤ ‖u−‖p + ‖v−‖p.

Proof. As 0 ≤ (u + v)+ ≤ u+ + v+ we conclude that

‖(u + v)+‖p ≤ ‖u+ + v+‖p ≤ ‖u+‖p + ‖v+‖p.

Similarly, we have‖(u + v)−‖p ≤ ‖u−‖p + ‖v−‖p.

2

The next proposition is concerned with the feasibility of the iterates along given Newton directions.It would be undesirable if the iterates would leave the feasible region and then return to it again.

Proposition 3.2. Suppose that (x, s) ∈ F++ and z + xs ≥ 0. Let (∆x,∆s) be the solution of∆s = M∆x, s∆x + x∆s = z. If (x + t0∆x)(s + t0∆s) > 0 for some 0 < t0 ≤ 1, then x + t∆x > 0and s + t∆s > 0 for all 0 ≤ t ≤ t0.

Proof. Let (x, s) := (x + t0∆x, s + t0∆s).

We have

(x + δt0∆x)(s + δt0∆s)

= xs + δt0(s∆x + x∆s) + δ2t20∆x∆s

= xs + δt0z + δ2(xs− t0z − xs)

= (1− δ)xs + δ(1− δ)(t0z + xs) + δ2xs > 0 (7)

for all 0 ≤ δ ≤ 1.

If there are 0 < t1 ≤ t0 and 1 ≤ i ≤ n with either (x + t1∆x)i < 0 or (s + t1∆s)i < 0, then, since(x, s) > 0, by continuity there must exist 0 < t2 < t1 ≤ t0, such that (x + t2∆x)i(s + t2∆s)i = 0.Letting δ = t2/t0, we have (x+δt0∆x)i(s+δt0∆s)i = 0, which would contradict (7). The propositionis thus proven. 2

The term z + xs is sometimes called the target to be tracked, and it is naturally nonnegative formost interior point methods. In particular, for Algorithm 2.1, this property boils down to verifyingxs + α1(τ1µe− xs)− + α2(τ1µe− xs)+ ≥ 0.

Let us denote

h(α) := xs + α1(τ1µe− xs)− + α2(τ1µe− xs)+ (8)

8

and

I+ := {i | τ1µ− xisi > 0}. (9)

Since (x, s) ∈ F++ we have

hi(α) =

{xisi + α2(τ1µ− xisi) = (1− α2)xisi + α2τ1µ > 0, i ∈ I+,

xisi + α1(τ1µ− xisi) ≥ xisi + τ1µ− xisi = τ1µ > 0, i 6∈ I+,(10)

for all α ∈ [0, 1]2. Proposition 3.2 thus asserts that (x(α), s(α)) ∈ N (τ1, τ2, η) if and only if x(α)s(α) ≥τ2µ(α)e and

∥∥(τ1µ(α)e− x(α)s(α))+∥∥ ≤ η(τ1 − τ2)µ(α), where

µ(α) := x(α)T s(α)/n. (11)

We further have

µ(α) := (x + ∆x(α))T (s + ∆s(α))/n

=(xT s + sT ∆x(α) + xT ∆s(α) + ∆x(α)T ∆s(α)

)/n

= µ + α1eT (τ1µe− xs)−/n + α2e

T (τ1µe− xs)+/n + ∆x(α)T ∆s(α)/n, (12)

where ∆x(α) = α1∆x− + α2∆x+ and ∆s(α) = α1∆s− + α2∆s+.

Lemma 3.3. It holds thatµ(α) ≥ (1− α1)µ.

Proof. By the monotonicity we have ∆x(α)T ∆s(α) ≥ 0. Therefore from (12) we have

µ(α) ≥ µ + α1eT (τ1µe− xs)−/n

≥ µ + α1eT (−xs)/n

= (1− α1)µ.

2

We note the following simple but useful relationships:

eT (τ1µe− xs)− = −(1− τ1)xT s− eT (τ1µe− xs)+,

‖(xs)−0.5(τ1µe− xs)−‖ = ‖(√xs− τ1µe/√

xs)+‖ ≤ ‖√xs‖ = xT s,

eT (τ1µe− xs)+ ≤ √n‖(τ1µe− xs)+‖.

(13)

For convenience we set

β :=τ1 − τ2

τ1. (14)

9

Obviously we have β ∈ (0, 1), τ1 − τ2 = βτ1 and τ2 = (1− β)τ1. Let

η = max{‖(τ1µe− xs)+‖

βτ1µ, 1

}(15)

It follows that η ≤ η if (x, s) ∈ N (τ1, τ2, η).

Lemma 3.4. If µ(α) ≤ µ, then it holds that

‖(τ1µ(α)e− h(α))+‖ ≤ (1− α2)ηβτ1µ(α).

Proof. As µ(α) ≤ µ it follows from (10) that

τ1µ(α)− hi(α) ≤{

τ1µ(α)− µ(α)µ hi(α) = µ(α)

µ (1− α2)(τ1µ− xisi), if i ∈ I+,

0, else,

which implies that

‖(τ1µ(α)e− h(α))+‖ ≤ µ(α)µ

(1− α2)‖(τ1µ− xs)+‖ ≤ (1− α2)ηβτ1µ(α).

2

Lemma 3.5. Let u, v ∈ <n be such that uT v ≥ 0, and let r = u + v. Then, we have

‖(uv)−‖1 ≤ ‖(uv)+‖1 ≤ 14‖r‖2.

Proof. Let the index set J beJ := {i | uivi > 0}.

As uT v ≥ 0 we have

‖(uv)−‖1 ≤ ‖(uv)+‖1 =∑

i∈J

uivi ≤ 14

∑

i∈J

(ui + vi)2 =14

∑

i∈J

(ri)2 ≤ 14‖r‖2.

2

Lemma 3.6. Suppose β ≤ 12 and α1 = tα2η

√βτ1n for some t ≥ 0. Then we have

‖(∆x(α)∆s(α))−‖1 ≤ ‖(∆x(α)∆s(α))+‖1 ≤ (t2 + 1)α22η

2βτ1µ/4.

Proof. We have

s∆x(α) + x∆s(α) = α1(τ1µe− xs)− + α2(τ1µe− xs)+.

10

Multiply both sides of the above equality by (xs)−0.5. Denote u := x−0.5s0.5∆x(α), v := x0.5s−0.5∆x(α)and r := (xs)−0.5(α1(τ1µe − xs)− + α2(τ1µe − xs)+). So we have u + v = r. Notice that uT v =∆x(α)T ∆s(α) ≥ 0. Therefore, by Lemma 3.5 we have

‖(∆x(α)∆s(α))−‖1

≤ ‖(∆x(α)∆s(α))+‖1

≤ 14‖(xs)−0.5α1(τ1µe− xs)− + (xs)−0.5α2(τ1µe− xs)+‖2

=14

(α2

1‖(√

xs− τ1µe/√

xs)+‖2 + α22‖(xs)−0.5(τ1µe− xs)+‖2

)

≤ 14

(α2

1‖√

xs‖2 + α22‖(τ1µe− xs)+‖2/(τ2µ)

)

≤ 14

(t2α2

2η2βτ1µ + α2

2η2βτ1µ

)

= (t2 + 1)α22η

2βτ1µ/4. (16)

2

Lemma 3.7. Suppose that τ1 ≤ 1/4 and β ≤ 1/2. If α1 = α2η√

βτ1n and α2 ≤ 1/η2, then we have

µ(α) ≤ (1− η√

βτ1

10√

nα2)µ.

Proof. By (12), (13) and Lemma 3.6 we have

µ(α) ≤ µ + α1eT (τ1µe− xs)−/n + α2e

T (τ1µe− xs)+/n + ‖(∆x(α)∆s(α))+‖1/n

≤ µ− α1(1− τ1)µ + α2‖e‖‖(τ1µe− xs)+‖/n + ‖(∆x(α)∆s(α))+‖1/n

≤ µ− α1(1− τ1)µ + α2ηβτ1µ/√

n + α22η

2βτ1µ/2n

≤ µ− 3α1µ/4 + 3α2ηβτ1µ/2√

n

= µ− α23η√

βτ1(1− 2√

βτ1)µ4√

n

≤ µ− α23η√

βτ1(1− 1/√

2)µ4√

n

≤ (1− α2η√

βτ1

10√

n)µ.

2

Lemma 3.8. Suppose that (x, s) ∈ N (τ1, τ2, η), τ1 ≤ 1/4 and β ≤ 1/2. If α1 = α2η√

βτ1n and

α2 ≤ 1/η2, then (x(α), s(α)) ∈ N (τ1, τ2, η).

11

Proof. By Lemma 3.7 it follows that µ(α) ≤ µ. Further, it follows from (10) and Lemma 3.6 that

x(α)s(α) = h(α) + ∆x(α)∆s(α)

≥ (τ2µ + α2(τ1 − τ2)µ)e− ‖(∆x(α)∆s(α))−‖e≥ (τ2µ + α2βτ1µ)e− (α2

2η2βτ1µ/2)e

≥ (τ2µ + α2βτ1µ− α2βτ1µ/2)e

≥ τ2µe

≥ τ2µ(α)e,

which also implies (x(α), s(α)) > 0 according to Proposition 3.2. Therefore, (x(α), s(α)) ∈ N−∞(1−τ2).

At the same time, by Lemma 3.3 we have

µ(α) ≥ (1− α1)µ ≥ (1− η√

βτ1√n

)µ ≥ (1−√

βτ1)µ ≥ µ/2.

Using Lemmas 3.4 and 3.6 we obtain

‖ (τ1µ(α)e− x(α)s(α))+ ‖= ‖ (τ1µ(α)e− h(α)−∆x(α)∆s(α))+ ‖≤ ‖(τ1µ(α)e− h(α))+ + (−∆x(α)∆s(α))+‖≤ ‖(τ1µ(α)e− h(α))+‖+ ‖(∆x(α)∆s(α))−‖≤ (1− α2)ηβτ1µ(α) + α2

2η2βτ1µ/2

≤ (1− α2)ηβτ1µ(α) + α2ηβτ1µ(α)

= ηβτ1µ(α)

≤ ηβτ1µ(α),

proving that (x(α), s(α)) ∈ N (τ1, τ2, η). 2

4 The iteration bound and an implementation

Now we are in a position to present our complexity results.

First, let us consider the generic Algorithm 2.1.

Theorem 4.1. Suppose that η ≥ 1, τ = τ1 ≤ 1/4 and β ≤ 1/2 are fixed for all iterations. Further-more, suppose that the plane-search procedure (6) is applied at each iteration of Algorithm 2.1. Then,Algorithm 2.1 terminates in O(

√n log (x0)T s0

ε ) iterations.

12

Proof. By Lemma 3.8, at each iteration, if we let α = (√

βτ1/n/η, 1/η2) then we have

(x(α), s(α)) ∈ N (τ1, τ2, η).

Furthermore, according to Lemma 3.7 we also have

µ(α) ≤ (1−√

βτ1

10η√

n)µ ≤ (1−

√βτ1

10η√

n)µ.

Therefore, the exact plane search would lead to at least the same amount of reduction in µ(α), andhence the theorem is proven. 2

The plane-search subproblem being an optimization problem with a convex objective and only twovariables can be solved relatively easily. However, it is also possible to reduce the number of searchvariables in the subproblem to only one without sacrificing the practical efficiency too much.

The main observation here is that under some mild conditions, the objective function in the subprob-lem (6) is monotone with respect to α1 for any fixed α2 ∈ [0, 1].

More precisely, we have the following result.

Theorem 4.2. Suppose (x, s) ∈ N−∞(1 − τ2), τ = τ1 ≤ 1/4 and τ1 ≤ 2τ2 (i.e. β ≤ 1/2). For anyfixed α2 ∈ [0, 1], x(α)T s(α) is a monotonically decreasing function in α1 for α1 ∈ [0, 1].

Proof. We have

x(α)T s(α) = (x + α1∆x− + α2∆x+)T (s + α1∆s− + α2∆s+)

= xT s + α1(xT ∆s− + sT ∆x−) + α2(xT ∆s+ + sT ∆x+)

+α21∆xT

−∆s− + α1α2

(∆xT

−∆s+ + ∆xT+∆s−

)+ α2

2∆xT+∆s+.

Therefore, since 0 ≤ α1 ≤ 1 and 0 ≤ α2 ≤ 1,

∂(x(α)T s(α))∂α1

= eT (τ1µe− xs)− + 2α1∆xT−∆s− + α2(∆xT

−∆s+ + ∆xT+∆s−)

≤ eT (τ1µe− xs)− + 2∆xT−∆s− +

∣∣(D−1∆x−)T (D∆s+) + (D∆s−)T (D−1∆x+)∣∣

≤ eT (τ1µe− xs)− + 2∆xT−∆s− + ‖(D−1∆x−, D∆s−)‖‖(D−1∆x+, D∆s+)‖ (17)

where D = X1/2S−1/2, and we also used the monotonicity of M (thus ∆xT−∆s− ≥ 0) in the secondstep.

13

By Lemma 3.6 and Lemma 3.5, we have

∆xT−∆s− ≤ ‖(∆x−∆s−)+‖1

= ‖ ((D−1∆x−)(D∆s−)

)+ ‖1

≤ ‖(D−1∆x−)(D∆s−)‖1

≤ 14

(‖D−1∆x−‖2 + ‖D∆s−‖2)

≤ 14‖D−1∆x− + D∆s−‖2

=14‖(τ1µe− xs)−/

√xs‖2

=14‖√

(xs− τ1µe)+√

(xs− τ1µe)+/xs‖2

≤ 14‖√

(xs− τ1µe)+‖2

= eT (xs− τ1µe)+/4, (18)

where we used the monotonicity (D−1∆x−)T D∆s− = ∆xT−∆s− ≥ 0 in the fifth step, and the factthat 0 ≤ (xs− τ1µe)+/xs ≤ e in the eighth step. In fact, one concludes from the chain of inequalitiesin (18) that

‖(D−1∆x−, D∆s−)‖2 ≤ eT (xs− τ1µe)+. (19)

Similarly,

‖(D−1∆x+, D∆s+)‖2 ≤ ‖D−1∆x+ + D∆s+‖2

= ‖(τ1µe− xs)+/√

xs‖2

≤ ‖√

(τ1µe− xs)+‖2 = eT (τ1µe− xs)+,

where the third step is due to the fact that since τ1 ≤ 2τ2 and (x, s) ∈ N−∞(1− τ2) and so

(τ1µe− xs)+ ≤ (2τ2µe− xs)+ = (2(τ2µe− xs) + xs)+ ≤ xs.

Furthermore,

eT (τ1µe− xs)+ ≤ n(τ1 − τ2)µ

≤ nµ/8

≤ (1− τ1)nµ/6

= eT (xs− τ1µe)/6

≤ eT (xs− τ1µe)+/6.

Therefore we have‖(D−1∆x+, D∆s+)‖2 ≤ eT (xs− τ1µe)+/6. (20)

14

Substituting (18),(19) and (20) into (17) finally yields that

∂(x(α)T s(α))∂α1

≤ −(

12− 1√

6

)eT (xs− τ1µe)+ < 0.

2

In view of Theorem 4.1 and Theorem 4.2, we may solve subproblem (6) approximately in the followingway. First, set α2 = 1/η2. Second, find the greatest α1 in [0, 1] such that (x(α1, 1/η2), s(α1, 1/η2)) ∈N (τ1, τ2, η). One may for instance use bisection on α1 for this purpose. Then, set α1 = α1. Theo-rem 4.2 and Lemma 3.8 guarantee that α1 ≥ η−1

√βτ1/n. It is clear that if the plane search procedure

as described in Theorem 4.1 is replaced by this line search procedure then the O(√

n log (x0)T s0

ε ) iter-ation bound still holds. Particularly, if we let ηk ≡ 1, then η ≡ 1 and so we can always choose α2 ≡ 1.Another benefit of ηk ≡ 1 is that the corresponding neighborhoods N (τ0

1 , τ02 , 1) are simply expressed

by (21) and (3). A concrete practical implementation is recommended as follows. Its numericalperformance will be discussed in Section 6.

Algorithm 4.3.

Input parameters: required precision ε > 0, target parameter and neighborhood parameters 0 < τ =τ1 < 1 and τ2 = 0.5τ1 (i.e. β = 0.5), and the initial solution (x0, s0) ∈ N (τ0

1 , τ02 , 1).

Step 0 Set k = 0.

Step 1 If (xk)T sk ≤ ε then stop.

Step 2 Solve (∆xk−,∆sk−) and (∆xk+,∆sk

+) based on (4) and (5).

Set αk2 = 1 and find the largest αk

1 on the closed interval [√

βτ1/n , 1], such that (x(αk), s(αk)) ∈N (τ1, τ2, 1).

Step 3 Set (xk+1, sk+1) := (x(αk), s(αk)).

Let k := k + 1 and go to Step 1.

5 A predictor-corrector scheme

In this section, we shall slightly change the notation. For simplicity, we shall always choose η = 1,i.e. we consider the neighborhood N (τ1, τ2, 1). We introduce a new notation N (τ1; β) to indicate theset N (τ1, τ2, 1) (see (3)), i.e.,

N (τ1; β) ={(x, s) ∈ F++

∣∣ ‖(τ1µe− xs)+‖ ≤ βτ1µ}

(21)

15

where β = (τ1 − τ2)/τ1, as given in (14).

Below we describe another variant of Algorithm 2.1, which is essentially a predictor-corrector typealgorithm.

Algorithm 5.1.

Input parameters: required precision ε > 0, neighborhood parameters 0 < τ1 ≤ 1/4, 0 < β ≤ 1/2, andthe initial solution (x0, s0) ∈ N (τ1; β/2).

Output: a sequence of iterates {(xk, sk) | k = 0, 1, 2, ...}.

Step 0 Set k = 0.

Step 1 If (xk)T sk ≤ ε then stop. Otherwise, if k is even (including 0), go to Step 2; if k is odd, goto Step 3.

Step 2 (Predictor Step). Set τk = 0. Solve (∆xk−,∆sk−) based on (4).

Find largest step size 0 < αk1 ≤ 1, such that (x(αk

1), s(αk1)) ∈ N (τ1; β). Go to Step 4.

Step 3 (Corrector Step). Set τk = τ1. Solve (∆xk−,∆sk−) and (∆xk+, ∆sk

+) based on (4) and (5).

Find step size vector αk = (αk1 , 1) ∈ [0, 1]2, such that (x(αk), s(αk)) ∈ N (τ1; β/2) and αk

1 ismaximum. Go to Step 4.

Step 4 Set (xk+1, sk+1) := (x(αk), s(αk)).

Let k := k + 1 and go to Step 1.

Remark that in the predictor step, since τk is set to be 0, we have (τkµke− xksk)+ = 0, and so theNewton constituent direction with respect to the positive part is simply zero. In both the correctorand the predictor steps, we only need to search for a single step size. An important feature of theabove algorithm is that in the corrector step we also aim at a large update of the target. In otherwords, the gap function is expected to be reduced for the corrector steps as well.

We shall now prove that Algorithm 5.1 indeed works correctly.

Let us denoteλ := ‖(τ1µe− xs)+‖1/‖(τ1µe− xs)−‖1, (22)

which meansλeT (τ1µe− xs)− + eT (τ1µe− xs)+ = 0.

So when we choose α1 ≥ λ we have

µ(α1, 1) ≤ µ + ∆x(α1, 1)T ∆s(α1, 1)/n. (23)

16

If (x, s) ∈ N (τ1;β), τ1 ≤ 1/4 and β ≤ 1/2, then we derive from (13) that

λ ≤ √n‖(τ1µe− xs)+‖/ (

(1− τ1)xT s)

≤√

βτ1

1− τ1

√βτ1

n

≤√

23

√βτ1

n. (24)

If (x, s) ∈ N (τ1;β), then computing η from (15) would yield

η = 1, (25)

and so by (10) we obtain immediately

h(α1, 1) ≥ τ1µe (26)

for all α1 ∈ [0, 1] and for all (x, s) ∈ F++.

Lemma 5.2. Suppose (x, s) ∈ N (τ1; β), τ = τ1 ≤ 1/4 and β ≤ 1/2. Then, for any α1 ∈[λ,

√βτ12n

]

we have (x(α1, 1), s(α1, 1)) ∈ N (τ1; β/2).

Proof. First of all, observe that (24) guarantees that the interval[λ,

√βτ12n

]is not empty. Due to

(23), (26), (25) and Lemma 3.6, we have

∥∥(τ1µ(α1, 1)e− x(α1, 1)s(α1, 1))+∥∥

=∥∥(τ1µ(α1, 1)e− h(α1, 1)−∆x(α1, 1)∆s(α1, 1))+

∥∥

≤∥∥∥∥∥(

τ1µe− h(α1, 1) +τ1∆x(α1, 1)T ∆s(α1, 1)

ne−∆x(α1, 1)∆s(α1, 1)

)+∥∥∥∥∥

≤∥∥∥∥∥(

τ1∆x(α1, 1)T ∆s(α1, 1)n

e−∆x(α1, 1)∆s(α1, 1))+

∥∥∥∥∥

≤∥∥∥∥τ1∆x(α1, 1)T ∆s(α1, 1)

ne + (−∆x(α1, 1)∆s(α1, 1))+

∥∥∥∥

≤∥∥∥∥τ1∆x(α1, 1)T ∆s(α1, 1)

ne

∥∥∥∥ +∥∥(−∆x(α1, 1)∆s(α1, 1))+

∥∥

≤ ∆x(α1, 1)T ∆s(α1, 1) +∥∥(∆x(α1, 1)∆s(α1, 1))−

∥∥=

∥∥(∆x(α1, 1)∆s(α1, 1))+∥∥

1− ∥∥(∆x(α1, 1)∆s(α1, 1))−

∥∥1+

∥∥(∆x(α1, 1)∆s(α1, 1))−∥∥

≤ ∥∥(∆x(α1, 1)∆s(α1, 1))+∥∥

1

≤ (β/2)τ1(3µ/4).

17

Applying Lemma 3.3 yields

µ(α1, 1) ≥ (1−√

βτ1/2n)µ ≥ (1− 1/4)µ = 3µ/4.

Therefore, (x(α1, 1), s(α1, 1)) ∈ N (τ1; β/2). 2

Lemma 5.3. Suppose that (x, s) ∈ N (τ1;β), τ = τ1 ≤ 1/4 and β ≤ 1/2. Then we have

µ(

√βτ1

2n, 1) ≤ (1−

√2βτ1

32√

n)µ.

Proof. Let us denote α0 := (√

βτ12n , 1). Notice that η = 1. Due to (13) and Lemma 3.6 we obtain

µ(

√βτ1

2n, 1) = µ +

√βτ1

2n

eT (τ1µe− xs)−

n+

eT (τ1µe− xs)+

n+

∆x(α0)T ∆s(α0)n

≤ µ−√

βτ1

2n(1− τ1)µ + ‖(τ1µe− xs)+‖/√n + ‖(∆x(α0)∆s(α0))+‖1/n

≤ µ− 3√

28

√βτ1

nµ +

βτ1µ√n

+3βτ1µ

8n

≤ µ− 3√

28

√βτ1

nµ +

√βτ1

8nµ +

38√

8

√βτ1

nµ

= µ−√

232

√βτ1

nµ.

The lemma is proven. 2

A remarkable feature of Algorithm 5.1 as revealed by Lemma 5.3 is that, the gap measurement µ isreduced by a rate of 1− 1/O(

√n) even at the corrector steps.

Lemma 5.4. Let (∆xa,∆sa) be the search direction of a predictor step in Algorithm 5.1, and α bethe actual step size taken in that predictor step. Then,

α ≥ 21 +

√1 + 4δ/βτ1

where δ = ‖∆xa∆sa‖ /µ.

Proof. We haveµ(α) = x(α)T s(α)/n = (1− α)µ + α2(∆xa)T ∆sa/n.

Note that (x, s) ∈ N−2 (τ1, β) and

∥∥∥(τ1((∆xa)T ∆sa/n)e−∆xa∆sa

)+∥∥∥

2≤ ∥∥τ1((∆xa)T ∆sa/n)e−∆xa∆sa

∥∥2

= ‖∆xa∆sa‖2 − τ1(2− τ1)((∆xa)T ∆sa)2/n

≤ ‖∆xa∆sa‖2 .

18

Therefore,∥∥(τ1µ(α)e− x(α)s(α))+

∥∥=

∥∥∥((1− α)(τ1µe− xs) + α2τ1((∆xa)T ∆sa/n)e− α2∆xa∆sa

)+∥∥∥

≤ (1− α)∥∥(τ1µe− xs)+

∥∥ + α2∥∥∥(τ1((∆xa)T ∆sa/n)e−∆xa∆sa

)+∥∥∥

≤ (1− α)βτ1µ + α2 ‖∆xa∆sa‖ .

Applying similar reasoning as in Lemma 4.17 of [16], we see that for each α with

0 ≤ α ≤ 21 +

√1 + 4δ/βτ1

we will have∥∥(τ1µ(α)e− x(α)s(α))+

∥∥ ≤ (1− α)βτ1µ + α2 ‖∆xa∆sa‖≤ 2βτ1(1− α)µ

≤ 2βτ1µ(α),

and therefore, (x(α), s(α)) ∈ N (τ1; 2β). Differently put, we have α ≥ 2

1+√

1+4δ/βτ1as the lemma

claims. 2

Since δ ≤ n/2, together with Lemma 5.3, the next theorem follows immediately (see also the proofof Theorem 4.18 in [16]).

Theorem 5.5. Let β = 1/4. Then Algorithm 5.1 will terminate in O(√

n log (x0)T s0

ε ) iterations.

Assume additionally that the LCP problem has a strictly complementary solution; that is, there isa partition B and N , and solution (x∗, s∗), such that B ∪ N = {1, 2, · · · , n}, B ∩ N = ∅, x∗B >

0, x∗N = 0, and s∗B = 0, s∗N > 0. Since the sequence generated by Algorithm 5.1 is contained in awide neighborhood, it satisfies

(1− β)τ1µk ≤ xksk ≤ nµk. (27)

Using Lemma 2 of [3], we know that there exists some constant 0 < ξ < 1 such that

ξ ≤ xkj ≤ 1/ξ for j ∈ B, and ξ ≤ sk

j ≤ 1/ξ for j ∈ N. (28)

For simplicity, we drop the index k. We apply the same proof as for Theorem 3.6 in [17], and sorelations (27) and (28) give rise to

‖∆xa‖ = O(µ) and ‖∆xa‖ = O(µ).

Then due to Lemma 5.4, we have the following result:

19

Theorem 5.6. Let {(xk, sk) | k = 0, 1, 2, ...} be the sequence generated by Algorithm 5.1. Suppose that(LCP ) has a strictly complementary solution. Then, (xk)T sk → 0 Q-quadratically for the predictorsteps.

6 Preliminary numerical tests

We shall test our algorithms on some randomly generated instances, in order to get a feel of how themethod might perform in practice.

To achieve this, we wrote simple Matlab codes for four algorithms: (1) the Mizuno-Todd-Ye typepredictor-corrector algorithm [8]; (2) the classical wide-neighborhood path-following algorithm ofKojima-Mizuno-Yoshise [5]; (3) Algorithm 4.3; and (4) Algorithm 5.1. These four algorithms will bedenoted, respectively, by: (1) Algorithm PC; (2) Algorithm WN; (3) Algorithm New-WN; and (4)Algorithm New-PC. All algorithms do not use Mehrotra’s higher order correction technique. Theneighborhoods are taken to be N2(1/2) in predictor step and N2(1/4) in corrector step for AlgorithmPC, and N−∞(1 − τ/2) for Algorithm WN, and β = 1/2 for Algorithm New-WN and AlgorithmNew-PC. To test the role of the parameter τ , we tried three different values of τ : τ = τ1 = 0.005,τ = τ2 = 0.001, and τ = τ3 = 0.0005, respectively. All algorithms terminate after the relative dualitygap satisfies

xT s

(x0)T s0 + 1≤ 10−8.

For each dimension n, the entry in the column ‘iter’ is the average number of iterations of 10 randomlygenerated monotone LCPs with the same n, and the number in the bracket is the standard deviationof these 10 runs. In case a Matlab numerical warning occurred in the procedure, then we mark asuperscript ∗ next to that corresponding entry.

The first set of testing monotone LCP problems are generated as follows. After one inputs any positiveinteger n, Matlab generates an n × n matrix A = rand(n) randomly. Then we take M = AT A andb = e −Me to obtain a monotone LCP and its initial feasible solution (e, e). The numerical resultsof this set of problems are showed in Table 1.

To test the influence from the skewness of matrix M , in the next set of test problems we let M =AT A + m(B − BT ), where B = rand(n) and m = 1. The numerical results are showed in Table 2.It turns out that the number of iterations actually decreases on average as compared with the casewhen M is purely positive semidefinite. In our experiences, we found that the numbers of iterationsfor all algorithms tested always decrease if m increases.

Based on the numerical results we have generated so far, Algorithm New-WN is the fastest, andAlgorithm New-WN and Algorithm New-PC are faster than Algorithm WN, while Algorithm PC

20

PC WN New-WN New-PC

τ = τ1 τ = τ2 τ = τ3 τ = τ1 τ = τ2 τ = τ3 τ = τ1 τ = τ2 τ = τ3

n iter iter iter iter iter iter iter iter iter iter

100 34.0 (1.70) 16.8 (0.70) 16.7 (0.87) 16.7 (1.33) 10.7 (0.43) 10.6 (0.45) 10.6 (0.45) 12.5 (0.47) 12.2 (0.49) 12.1 (0.50)

200 36.8 (0.61) 16.1 (0.57) 16.9 (0.37) 16.0 (0.66) 10.6 (0.20) 10.7 (0.22) 10.7 (0.22) 12.9 (0.23) 12.9 (0.23) 12.8 (0.22)

300 38.6 (0.61) 17.1 (0.30) 17.9 (0.49) 19.4 (0.64) 11.0 (0.12) 11.1 (0.10) 11.2 (0.11) 13.2 (0.18) 13.3 (0.16) 13.3 (0.16)

400 41.6 (0.65) 20.0 (0.42) 19.0 (0.45) 21.3 (0.57) 11.7 (0.20) 11.7 (0.19) 11.7 (0.19) 13.6 (0.23) 13.7 (0.21) 13.7 (0.21)

500 40.6∗ (0.50) 19.4 (0.54) 20.8 (0.40) 19.8 (0.63) 11.2 (0.12) 11.4 (0.13) 11.4 (0.13) 13.5 (0.17) 13.4 (0.18) 13.4 (0.18)

600 43.2 (0.45) 20.6 (0.34) 21.2 (0.44) 23.8 (0.45) 11.3 (0.10) 11.5 (0.12) 11.5 (0.12) 14.1 (0.16) 14.2 (0.17) 14.2 (0.17)

700 43.6 (0.30) 22.5 (0.49) 22.8 (0.54) 22.1 (0.40) 12.0 (0.10) 12.0 (0.10) 12.1 (0.10) 14.8 (0.13) 14.9 (0.15) 14.8 (0.16)

800 44.0∗ (0.34) 22.3 (0.41) 23.3 (0.47) 23.0 (0.26) 12.3 (0.07) 12.4 (0.09) 12.4 (0.09) 15.0 (0.14) 15.0 (0.14) 14.9 (0.15)

900 43.4∗ (0.32) 22.3 (0.29) 23.1 (0.34) 21.3 (0.41) 12.0 (0.12) 12.0 (0.12) 12.0 (0.12) 14.4 (0.18) 14.4 (0.18) 14.5 (0.17)

1000 44.4 (0.28) 25.0 (0.41) 23.3 (0.49) 23.8 (0.60) 12.2 (0.10) 12.1 (0.10) 12.1 (0.10) 15.3 (0.13) 15.4 (0.13) 15.4 (0.13)

Table 1: Iteration Numbers on monotone LCPs with M = AT A

PC WN New-WN New-PC

τ = τ1 τ = τ2 τ = τ3 τ = τ1 τ = τ2 τ = τ3 τ = τ1 τ = τ2 τ = τ3

n iter iter iter iter iter iter iter iter iter iter

100 11.0 (0.00) 6.2 (0.46) 5.7 (0.28) 5.1 (0.22) 4.8 (0.13) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00)

200 9.0 (0.00) 5.0 (0.17) 4.9 (0.21) 5.1 (0.19) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00)

300 9.0 (0.00) 4.5 (0.12) 5.3 (0.18) 4.9 (0.19) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00)

400 9.0 (0.00) 4.6 (0.10) 5.4 (0.16) 5.3 (0.16) 4.0 (0.00) 4.0 (0.00) 3.8 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00)

500 9.0 (0.00) 4.7 (0.09) 4.7 (0.11) 4.4 (0.09) 4.0 (0.00) 4.0 (0.00) 3.2 (0.00) 4.0 (0.00) 4.0 (0.00) 4.0 (0.00)

600 9.0 (0.00) 4.5 (0.06) 4.6 (0.12) 4.5 (0.09) 4.0 (0.00) 3.4 (0.06) 3.1 (0.04) 4.0 (0.00) 4.0 (0.00) 3.8 (0.05)

700 9.0 (0.00) 4.7 (0.08) 3.8 (0.07) 4.3 (0.09) 4.0 (0.00) 3.0 (0.00) 3.0 (0.00) 4.0 (0.00) 4.0 (0.00) 3.3 (0.05)

800 9.0 (0.00) 4.5 (0.06) 3.2 (0.07) 4.4 (0.11) 4.0 (0.00) 3.0 (0.00) 3.0 (0.00) 4.0 (0.00) 3.7 (0.05) 3.1 (0.03)

900 9.0 (0.00) 4.4 (0.05) 3.7 (0.07) 4.0 (0.16) 4.0 (0.00) 3.0 (0.00) 3.0 (0.00) 4.0 (0.00) 3.5 (0.05) 3.0 (0.00)

1000 9.0 (0.00) 4.0 (0.00) 3.4 (0.05) 3.9 (0.08) 4.0 (0.00) 3.0 (0.00) 3.0 (0.00) 4.0 (0.00) 3.5 (0.05) 3.0 (0.00)

Table 2: Iteration Numbers on monotone LCPs with M = AT A + m(B −BT )

appears to be the slowest. Moreover, Algorithm WN, Algorithm New-WN, and Algorithm New-PChad always run smoothly, but Algorithm PC got 3 warnings due to badly scaled matrices. Certainly,our implementations are very coarse. For instance, we did not fine-tune the parameters, nor did weuse any higher order corrections. In the future, we plan to study the performance of the method forpractical problems with more refined linear algebras and careful implementations.

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21

[4] P. Hung and Y. Ye, An asymptotical O(√

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