a mizuno–todd–ye type predictor–corrector algorithm for sufficient linear complementarity...

European Journal of Operational Research 181 (2007) 1097–1111

www.elsevier.com/locate/ejor

A Mizuno–Todd–Ye type predictor–corrector algorithmfor sufficient linear complementarity problems

Tibor Illes *, Marianna Nagy

Department of Operational Research, Eotvos Lorand University of Sciences, Pazmany Peter setany 1/C, 1117 Budapest, Hungary

Received 5 October 2004; accepted 10 August 2005Available online 2 May 2006

Abstract

We analyze a version of the Mizuno–Todd–Ye predictor–corrector interior point algorithm for the P�ðjÞ-matrix linearcomplementarity problem (LCP). We assume the existence of a strictly positive feasible solution. Our version of theMizuno–Todd–Ye predictor–corrector algorithm is a generalization of Potra’s [F.A. Potra, The Mizuno–Todd–Ye algo-rithm in a larger neighborhood of the central path, European Journal of Operational Research 143 (2002) 257–267] resultson the LCP with P�ðjÞ-matrices. We are using a kv�1 � vk proximity measure like Potra to derive iteration complexityresult for this algorithm . Our algorithm is different from Miao’s method [J. Miao, A quadratically convergentO�ðjþ 1Þ ffiffiffinp L

�-iteration algorithm for the P*(j)-matrix linear complementarity problem, Mathematical Programming

69 (1995) 355–368] in both the proximity measure used and the way of updating the centrality parameter. Our analysisis easier than the previously stated results. We also show that the iteration complexity of our algorithm is O

�ð1þ jÞ

32ffiffiffinp

L�.

� 2006 Elsevier B.V. All rights reserved.

Keywords: Linear complementarity problem; Sufficient matrix; P�ðjÞ-matrix; Interior point method; Mizuno–Todd–Ye predictor–cor-rector algorithm

1. Introduction

Consider the linear complementarity problem (LCP): find vectors x; s 2 Rn, which satisfy the constraints

0377-2

doi:10.

* CoE-m

�Mxþ s ¼ q; xs ¼ 0; x; s P 0; ð1Þ
where M 2 Rn�n and q 2 Rn.
The linear complementarity problem belongs to the class of NP-complete problems, since the feasibilityproblem of linear equations with binary variables can be described as an LCP problem [6]. Therefore, we can-not expect an efficient solution method for the linear complementarity problem without special property of thematrix M.

217/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

1016/j.ejor.2005.08.031

rresponding author. Tel.: +36 1 2090555; fax: +36 1 3812156.ail addresses: [email protected] (T. Illes), [email protected] (M. Nagy).

mailto:[email protected]

mailto:[email protected]

1098 T. Illes, M. Nagy / European Journal of Operational Research 181 (2007) 1097–1111

We assume that the matrix M is a P�ðjÞ-matrix and we generalize the well-known Mizuno–Todd–Ye pre-dictor–corrector interior point algorithm for this class of the linear complementarity problems.

One of the first version of the predictor–corrector interior point algorithm for linear programming prob-lems was initiated by Sonnevend et al. [16]. This algorithm needs more corrector steps after each predictor stepin order to return to the appropriate neighborhood of the central path. Mizuno et al. [8] published such a pre-dictor–corrector interior-point method for the linear programming problem in which each predictor step isfollowed by a single corrector step and whose iteration complexity is the best known in the linear program-ming literature. Anstreicher and Ye [18] extended this result to the linear complementarity problem with apositive semidefinite matrix with the same iteration complexity.

In one of the best papers on interior-point algorithms, Kojima et al. [5] offered a polynomial primal-dualinterior point method for the positive semidefinite matrix linear complementarity problem. The properties of amore general matrix class can be formulated in a natural way from the iteration complexity analysis of theiralgorithm. This class is called a P�ðjÞ-matrix by Kojima et al. [6]. The primal-dual interior point algorithm ofKojima et al. is generalized to P�ðjÞ-matrices. This algorithm is also polynomial if a j P 0 is known, suchthat the matrix of the problem is P�ðjÞ-matrix. The iteration complexity is a polynomial of j, the dimensionn and the bit length L of the problem.

Since Kojima et al. published their book [6] on interior point methods for LCPs, the quality of a variant ofan interior point algorithm is measured by the fact whether it can be generalized to the P�ðjÞ-matrix linearcomplementarity problem or not.

The natural outcome of this was the emergence of different interior-point algorithms for the P�ðjÞ-matrixlinear complementarity problem in the mid-90s.

Several variants of the Mizuno–Todd–Ye type predictor–corrector interior-point algorithm are known inthe literature. First Miao [7], later Potra and Sheng [11] gave a generalization of Mizuno–Todd–Ye predic-tor–corrector algorithm with the P�ðjÞ-matrix linear complementarity problem assuming the existence of astrictly positive solution. Miao updated the central path parameter l in such a way that xTs/n = l equalityholds throughout. Therefore, the updating of l is more complicated than in the skew-symmetric case, wherel 0 = (1 � a)l and a is the length of the Newton-step in the predictor phase. Further generalization has beenestablished: Ji et al. [4] extended the algorithm to the infeasible linear complementarity problem, Potra andSheng [12,13] to the infeasible and degenerate problem. In these methods the parameter l is updated byl 0 = (1 � a)l, thus xTs/n 5 l in their cases.

The Mizuno–Todd–Ye type predictor–corrector algorithm for the skew-symmetric or positive semidefinitelinear complementarity problem (horizontal linear complementarity problem, HLCP) of [14] is the basis of thispaper. As it is explained in [14] the Mizuno–Todd–Ye predictor–corrector method based on a very simple andelegant idea that is used in various other fields of computational mathematics such as the numerical methods ofdifferential equations and continuation methods. We already mentioned that there exists NP-complete linearcomplementarity problems. The NP-completeness of linear complementarity problems are related to the prop-erties of the matrix M. Therefore, our aim is to generalize the Mizuno–Todd–Ye algorithm – which is one of themost remarkable interior point method for linear programming and quadratic programming – for the widestpossible matrix class where the method is polynomial. This is the P� matrix class defined by Kojima et al. [6].

At choosing the proximity measure we followed Potra (kv�1 � vk) in contrast to the previous works(kv � ek, where v ¼

ffiffiffiffiffiffiffiffiffiffixs=l

p). The reason was that in practice interior point algorithms make longer step in

wide neighborhood, therefore their practical performance might even be better than the theoretical one. Fur-thermore Mizuno–Todd–Ye algorithm for P�ðjÞ linear complementarity problems was not generalized earlierfor large neighborhoods.

Summarizing the previous, we may state, that we present a new variant of the Mizuno–Todd–Ye predictor–corrector algorithm for P�ðjÞ linear complementarity problems that uses self-regular proximity measurekv�1 � vk, and therefore the iterates lies in a wider neighborhood of the central path than in the earlier pub-lished Mizuno–Todd–Ye type algorithms for this class of problems. Our algorithm’s iteration complexity isO�ð1þ jÞ

32ffiffiffinp

L�.

The following section deals with the fundamental properties of P�ðjÞ-matrices and with some well-knownresults. Section 3 describes the predictor–corrector algorithm, and the following sections analyze the method.Section 3.2 deals with the predictor step and determines the length of the Newton-step. The next part examines

T. Illes, M. Nagy / European Journal of Operational Research 181 (2007) 1097–1111 1099

the corrector step and the relationship between s and s 0 proximity parameters. The last section provides theiteration complexity of the algorithm.

Finally our notations:

Rnþ the set of n dimensional positive vectors,

Rn� the set of n dimensional nonnegative vectors,

x, xi vectors are thick letters, while scalars are normal letters,xs the componentwise product (Hadamard product) of vectors x and s,xa n-dimensional vector whose ith component is xa

i ,xTs the scalar product of two vectors,M the coefficient matrix of the LCP, M 2 Rn�n,X the diagonal matrix from x vector, so X = diag(x),k Æk the Euclidean norm,k Æk1 the infinity norm,e the n-dimensional vector of ones,I identity matrix of size n · n.

2. The P*(j)-matrices and their properties

The P*(j)-matrices were introduced by Kojima et al. [6], and can be considered as a generalization of thepositive semidefinite matrices.

Definition 1. Let j P 0 be a nonnegative number. A matrix M 2 Rn�n is called P�ðjÞ-matrix if

ð1þ 4jÞX

i2IþðxÞxiðMxÞi þ

Xi2I�ðxÞ

xiðMxÞi P 0 for all x 2 Rn;

where I+(x) = {1 6 i 6 n : xi(Mx)i > 0} and I�(x) = {1 6 i 6 n : xi(Mx)i < 0}.

The nonnegative real number j denotes the weight that need to be used at the positive terms so that theweighted ‘scalar product’ be nonnegative for each vector x 2 Rn. Therefore, naturally the P�ð0Þ is the positivesemidefinite matrix class (if we set aside the symmetry of the matrix M).

Definition 2. A matrix M 2 Rn�n is called a P�-matrix if it is a P�ðjÞ-matrix for some j P 0, i.e.,

P� ¼[jP0

P�ðjÞ.

Another matrix class, the class of sufficient matrices was introduced by Cottle et al. [1].

Definition 3. A matrix M 2 Rn�n is a column sufficient matrix if for all x 2 Rn

X ðMxÞ 6 0 implies X ðMxÞ ¼ 0;

and row sufficient if MT is column sufficient. The matrix M is sufficient if it is both row and column sufficient.

Kojima et al. [6] proved that a P� matrix is column sufficient and Guu and Cottle [2] proved that it is rowsufficient, too. Therefore, each P� matrix is sufficient. Valiaho proved the other direction of inclusion [17], sothe class of P�-matrices is equal to the class of sufficient matrices.

For further use we recall some well-known results. The reader may consult the book of Kojima et al. [6] forproofs and details.

Proposition 4. If M 2 Rn�n is a P�ðjÞ-matrix then

M 0 ¼�M I

S X

� �is a nonsingular matrix

for any positive diagonal matrices X ; S 2 Rn�n.


Corollary 5. Let M 2 Rn�n be a P�ðjÞ-matrix, x; s 2 Rnþ. Then for all a 2 Rn the system

�MDxþ Ds ¼ 0;

sDxþ xDs ¼ a;

has a unique solution (Dx,Ds).

We use the symbols F, Fþ, F� to denote the feasible region of the (LCP), its interior and the solution setof the LCP, respectively,

F :¼ ðx; sÞ 2 R2n� : �Mxþ s ¼ q

� �;

Fþ :¼ ðx; sÞ 2 R2nþ : �Mxþ s ¼ q

� �;

F� :¼ ðx; sÞ 2Fþ : xs ¼ 0f g;

and the central path is denoted by

C :¼ ðx; sÞ 2Fþ : xs ¼ le; for some l > 0f g.

Theorem 6 [10]. Let a linear complementarity problem with a P�ðjÞ-matrix M be given. Then the following

statements are equivalent:

1. Fþ 6¼ ;,2. 8w 2 Rn

þ, 9!ðx; sÞ 2Fþ : xs ¼ w,

3. 8l > 0, 9!ðx; sÞ 2Fþ : xs ¼ le, i.e., the central path C exists and it is unique.

Kojima et al. [6] showed that the central path C is a one-dimensional smooth curve that converges to a solu-tion of (LCP) as l! 0, so F� 6¼ ;.

Throughout the paper we make the following assumptions:

1. Fþ 6¼ ;,2. moreover we have an initial point ðx0; s0Þ 2Fþ,3. matrix M is a P�ðjÞ-matrix.

3. The predictor–corrector algorithm

As the first step of solving the (LCP) the problem is relaxed to the central path problem as it follows:

�Mxþ s ¼ q;

x; s > 0;

xs ¼ le:

9>>=>>; ðCPPlÞ

As l! 0 the sequence (x(l), s(l)) solutions of the central path problems approach the solution (x, s) of the(LCP).

Suppose ðx; sÞ 2Fþ and seek the unique solution ðxðlÞ; sðlÞÞ of central path problem with respect to l inthe form xðlÞ ¼ xþ Dx, sðlÞ ¼ sþ Ds. After substituting and neglecting the quadratic term we obtain the fol-lowing Newton-system:

�MDxþ Ds ¼ 0;

SDxþ XDs ¼ le� xs:

)ð2Þ

Since M is a P�ðjÞ-matrix, the solution of the above system exists and it is unique (see Corollary 5). Thepositivity condition of ðxðlÞ; sðlÞÞ is ensured with the right choice of the Newton-step length.


Let the proximity measure be w(x, s,l) = w(v) :¼ kv�1 � vk (where v ¼ffiffiffixsl

q). The w meets our requirements:

proximity must be zero for the point (x, s) on the central path, i.e., xs = le for some l > 0. Moreover w pun-ishes a point approaching the boundary, as some coordinates of x or s tends to 0 or infinity, the proximitymeasure approaches infinity.

Let 0 < s < s 0 be suitable proximity parameters defining the neighborhood of the central path (their rela-tionship will be determined later).

In predictor1 step we solve the Newton-system eagerly with l = 0 from a given initial point (x0, s0,l0),where w(x0, s0,l0) < s and we denote the solution by (Dx,Ds). Let our new point be

1 Som

xp ¼ x0 þ aDx; sp ¼ s0 þ aDs; lp ¼ ð1� aÞl0;

where a 2 (0, 1] is the maximal real number for which w(xp, sp,lp) 6 s 0 is satisfied, i.e., we allow certain amountof deviation from the smaller neighborhood after the predictor step.

The aim of the corrector step is to return to the s-neighborhood of the central path. By solving the Newton-system with l = lp we get ðD~x;D~sÞ. Let the new point be

xc ¼ xp þ D~x; sc ¼ sp þ D~s; lc ¼ lp;

so in contrast with the predictor step we do a full Newton-step here. The point (xc, sc,lc) will be in the s neigh-borhood again, so we can continue the iteration with this point as an initial point (x0, s0), while the duality gapis not small enough.

Now let us outline our new algorithm as it follows:

Mizuno–Todd–Ye predictor–corrector algorithm

Input:

an accuracy parameter e > 0;proximity parameters s, s 0, 0 < s < s 0;an initial point (x0, s0,l0), w(x0, s0,l0) 6 s;

begin

x :¼ x0, s :¼ s0, l :¼ l0

while xTs P e do

Predictor stepSolve (2) with l = 0 and letxp = x + aDx, sp = s + aDs, lp = (1 � a)l;Determine a: maxfa > 0 : ðxp; spÞ 2Fþ and wðxp; sp; lpÞ 6 s0g

Corrector step

Solve (2) with x = xp, s = sp, l = lp and letxc ¼ xp þ D~x, sc ¼ sp þ D~s, lc = lp;x = xc, s = sc, l = lc;

endend.

In the rest of the paper we deal with the analysis of the previous algorithm. At the analysis of the predictorstep we will give conditions to an a step-length and at the corrector step to the proximity parameters s, s 0.

3.1. Scaling

Let us first introduce the following notations:

v ¼ffiffiffiffiffixs

l

r; d ¼

ffiffiffix

s

r; dx ¼

d�1Dxffiffiffilp ¼ vDx

x; ds ¼

dDsffiffiffilp ¼ vDs

s.

etimes this kind of predictor step is called the affine scaling step in the literature [15].


After rescaling the Newton-system we have

�Mdx þ ds ¼ 0;

dx þ ds ¼ v�1 � v;

)

where M ¼ DMD and D = diag(d).

3.2. The predictor step

Let ðx; sÞ 2Fþ and wðx; s; lÞ < s. We solve the Newton-system with parameter l = 0 that means we get thefollowing scaled predictor system:

�Mdx þ ds ¼ 0;

dx þ ds ¼ �v;

)

which has the following solution:

dx ¼ �ðI þMÞ�1v and ds ¼ �MðI þMÞ�1

v.

We choose the step-length a so that the following two conditions are satisfied:

1. ðxp; spÞ 2Fþ, i.e., xpsp > 0,2. w(xp, sp,lp) < s 0,

where xp = x + aDx, sp = s + aDs, and lp = (1 � a)l, i.e., the positivity condition is satisfied at the new pointand we do not deviate from the central path too much.

We can examine the first condition as it is usual in the interior point literature, thus

xpsp ¼ ð1� aÞlv2 þ a2ldxds; and so ðvpÞ2 :¼ xpsp

ð1� aÞl ¼ v2 þ a2

1� adxds > 0

is necessary for ðxp; spÞ 2Fþ. In details, it means if (dxds)i P 0, then obviously (vp)i > 0 is true, but if

(dxds)i < 0, then the condition a2

1�a < �v2

iðdxdsÞi

must hold for the step-length.

Let us denote u ¼ min � v2i

ðdxdsÞi: ðdxdsÞi < 0

n o. Let u ¼ a2

1�a, then u 2 ½0; uÞ is the necessary condition so thatðxp; spÞ 2Fþ is satisfied.

After that examine the second condition for choosing step length a. To do so, compute the proximity mea-sure of vector v

wðvpÞ2 ¼ kðvpÞ�1 � vpk2 ¼ wðvÞ2 þ eT e

v2 þ udxds� e

v2þ udxds

.

Analogous to [14] let f : ½0; uÞ ! R be the function, defined as the difference of the square of new and oldproximity measures as function of the u, in terms of the step length, therefore

f ðuÞ ¼ wðvpÞ2 � wðvÞ2 ¼ eT e

v2 þ udxds� e

v2þ udxds

.

It is easy to show that the function f is strictly convex on the interval ½0; uÞ, f(0) = 0 and limu!uf ðuÞ ¼ 1.From the properties of function f it follows that the equation

f ðuÞ ¼ s02 � wðvÞ2

has a unique solution on interval I ¼ ½0; uÞ. We can define u� ¼ ða�Þ2

1�a�, and then the step length can be computed

as a� ¼ �u�þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðu�Þ2þ4u�p

2.

Proposition 7. The a* is the maximal feasible step length such that wðvpÞ 6 s0 and ðxp; spÞ 2Fþ.


Proof. If a is larger, then w(vp) > s 0 holds. For the feasibility the following must be satisfied:

2 Fro

qþMxða0Þ ¼ sða0Þ;xða0Þ; sða0Þ > 0; 8a0 2 ½0; a��:

Suppose on the contrary that $a 0 2 [0,a*] : x(a 0)s(a 0) = 0. Then lima!a0wðvðaÞÞ ¼ 1. This contradicts tow(v(a))2 = w(v)2 + f(u) <1. h

3.3. The corrector step

In the predictor step the point ð�x;�s; �l ¼ ð1� aÞlÞ ¼ ðxp; sp;lpÞ is obtained. Our aim now is to return fromthe s 0-neighborhood to the s-neighborhood of the central path. For this, we solve the following Newton-system:

�MDxþ Ds ¼ 0;

�xDsþ �sDx ¼ �le� �x�s:

�

After scaling one has

�M�dx þ �ds ¼ 0;�dx þ �ds ¼ �v�1 � �v;

)

where

�v ¼ffiffiffiffiffi�x�s

�l

r; �dx ¼

�vDx

�x; �ds ¼

�vDs

�s; �d ¼

ffiffiffi�x

�s

r; M ¼ DMD.

Let

xþ ¼ �xþ Dx; sþ ¼ �sþ Ds; lþ ¼ �l ¼ ð1� aÞl; then ðvþÞ2 ¼ eþ �dx�ds.

We will see that ðvþÞ2 ¼ eþ �dx�ds > 0, namely the corrector point is feasible if s parameter is in the given inter-

val (Proposition 10), because in that case k�dx�dsk1 < 1.

By similar computation we get ððvþÞ�1Þ2 ¼ eeþ�dx�ds

. Then the proximity measure is

wðvþÞ2 ¼ kvþ � ðvþÞ�1k2 ¼ eT ð�dx�dsÞ2

eþ �dx�ds

.

By introducing �q ¼ DxDs�l we get the following expression:

wðvþÞ2 ¼ eT �q2

eþ �q¼Xn

i¼1

�q2i

1þ �qi¼Xi2Iþ

�q2i

1þ �qiþXi2I�

�q2i

1� j�qij;

where Iþ ¼ f1 6 i 6 n : �qi > 0g and I� ¼ f1 6 i 6 n : �qi < 0g.Taking into consideration that the matrix M is the P�ðjÞ-matrix, the further analysis of corrector step is

different from the analysis of [14], because in our case j > 0 is possible, too, but Potra dealt only with the casej = 0. Because M is a P�ðjÞ-matrix the following estimation2 holds:
X
i2I�

j�qij 6 ð1þ 4jÞXi2Iþ

�qi. ð3Þ

We can estimate the distance of the new point and the central path by applying the property of infinitynorm and (3) as it follows:

m inequality (3) follows that maxjqj > 0 (so maxj(Dx)j(Ds)j > 0). This consequence will be used for example in Proposition 12.


wðxþ; sþ;lþÞ2 ¼Xn

i2Iþ

�q2i

1þ �qiþXn

i2I�

�q2i

1� j�qij6

Xn

i2Iþ

k�qk1�qi

1þ k�qk1þXn

i2I�

k�qk1j�qij1� k�qk1

6k�qk1

1þ k�qk1

Xi2Iþ

�qi þk�qk1

1� k�qk1ð1þ 4jÞ

Xi2Iþ

�qi ¼2k�qk1

1� k�qk21

1þ 2jþ 2jk�qk1� �X

i2Iþ

�qi.

Now we can state the following lemma.

Lemma 8. Let matrix M be a P�ðjÞ matrix and let (Dx,Ds) be the solution of

MDx ¼ Ds;

�xDsþ �sDx ¼ a;

�

then

Xi2Iþ

DxiDsi 61

4

affiffiffiffiffixsp��

��2

;

kDxDsk1 61

4þ j


��2

;

kDxDsk1 61

2þ j


��2

;

kDxDsk2 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4þ j

1

2þ j

saffiffiffiffiffixsp��

��2

:

The first statement is proved in [14], and the second statement’s proof is similar to Lemma 5.1 by Illes et al.[3]. The third estimation is followed from the first statement in addition to property of P�ðjÞ-matrices and thelast estimation is a corollary of the second and third statements using the property of norms.

By applying Lemma 8 with a ¼ �le� �x�s one has

k�qk1 6 ð1þ 4jÞ 14wð�x;�s; �lÞ2 and

Xi2Iþ

�qi 61

4wð�x;�s; �lÞ2. ð4Þ

Using these results we can continue the estimation of w(x+, s+,l+) as it follows:

wðxþ; sþ;lþÞ2 6 2k�qk11� k�qk2

1ð1þ 2jþ 2jk�qk1Þ

Xi2Iþ

�qi 6ð1þ 4jÞw4

16� ð1þ 4jÞ2w4ð2þ 4jþ jð1þ 4jÞw2Þ;

where w :¼ wð�x;�s; �lÞ.We would like that the step length a satisfy w(x+, s+,l+)2 < s2. Then we require a stronger condition, i.e.,

ð1þ 4jÞw4

16� ð1þ 4jÞ2w4ð2þ 4jþ jð1þ 4jÞw2Þ < s2.

After simple computation we get

jð1þ 4jÞ2w6 þ ½ð2þ 4jÞð1þ 4jÞ þ ð1þ 4jÞ2s2�w4 < 16s2. ð5Þ

We know that w ¼ wð�x;�s; �lÞ < s0, so that by suitable choice of s and s 0, the inequality (5) has a solution interms of w. Using these results we can estimate the relationship between s and s 0 as follows:

jð1þ 4jÞ2s06 þ ½ð2þ 4jÞð1þ 4jÞ þ ð1þ 4jÞ2s2�s04 < 16s2. ð6Þ


Proposition 9. If

s0 <2ffiffiffispffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 4jp ffiffiffiffiffiffiffiffiffiffiffiffiffi

2þ s24p ; ð7Þ

then the inequality (6) is true for all s 0 > s > 0 and for all j P 0.

Proof. By inequality (7) one has

s06 <64s3

ð1þ 4jÞ3ð2þ s2Þffiffiffiffiffiffiffiffiffiffiffiffiffi2þ s2p and s04 <

16s2

ð1þ 4jÞ2ð2þ s2Þ.

Then using inequality (6) we obtain

jð1þ 4jÞ2s06þ ½ð2þ 4jÞð1þ 4jÞ þ ð1þ 4jÞ2s2�s04 < 16s2 � 1

1þ 4j4sj

ð2þ s2Þffiffiffiffiffiffiffiffiffiffiffiffi2þ s2p þ 2þ 4jþ ð1þ 4jÞs2

2þ s2

" #

< 16s2 � 1

1þ 4j� 2ð1þ 4jÞ þ ð1þ 4jÞs2

2þ s2¼ 16s2.

This completes the proof. h

Using the relationship between corrector and predictor proximity parameters s and s 0 given in Proposition9, we can determine the suitable value of s depending on the parameter j.

Proposition 10. If s <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 16

ð1þ4jÞ2qr

, then there exists a s 0 that satisfies (7) and the full Newton step is

feasible at corrector step.

Proof. By Proposition 9 we have

s < s0 <2ffiffiffispffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 4jp ffiffiffiffiffiffiffiffiffiffiffiffiffi

2þ s24p ;

which implies

s4 <16s2

ð1þ 4jÞ2ð2þ s2Þ.

After trivial restructuring, we get

ð1þ 4jÞ2s4 þ 2ð1þ 4jÞ2s2 � 16 < 0.

The zero points of the quadratic expression on the left side of the previous inequality are

ðs2Þ1;2 ¼�2ð1þ 4jÞ2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ð1þ 4jÞ4 þ 4� 16ð1þ 4jÞ2

q2ð1þ 4jÞ2

¼ �1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 16

ð1þ 4jÞ2

s.

Considering the signs we get the first statement.The condition ðxþ; sþÞ 2Fþ, namely x+s+ > 0 is equivalent with ðvþÞ2 ¼ eþ �dx

�ds > 0. From inequality (4)follows,

k�dx�dsk1 ¼ kqk1 6

1

4ð1þ 4jÞwð�x;�s; �lÞ2 6 1

4ð1þ 4jÞs2.

Using the given upper bound on s

k�dx�dsk1 <

1

4ð1þ 4jÞ �1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 16

ð1þ 4jÞ2

s !<

1

4ð1þ 4jÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16

ð1þ 4jÞ2

s¼ 1.

This complete the proof. h


Obviously the value of j effects the size of the s and s 0 neighborhood parameters. The larger the value of jthe smaller the s and s 0 neighborhoods which ensures that the Mizuno–Todd–Ye predictor–corrector algo-rithm take one predictor and one corrector step alternatively.

4. Iteration complexity analysis

Thereinafter we deal with the iteration complexity of the algorithm following the analysis of [14]. For thiswe need a lower bound of the step length of the Newton-step. We determine it in three steps. At first, we givean upper bound for the range of function f, using this we determine a lower bound for u. Finally, the bound iscomputed for a.

We will use the following lemma for further estimation.

Lemma 11 [14]. Let x; s 2 Rnþ and l > 0 number and let w2(x, s,l) 6 2g. Then

1

1þ gþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gþ g2

p 6xisi

l6 1þ gþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gþ g2

pfor all i ¼ 1; . . . ; n.

Applying Lemma 11 with g ¼ s2

2gives

1

mðsÞ 6 mini

v2i 6 kv2k1 6 mðsÞ; where mðsÞ ¼ 1þ s2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ s4

4

r. ð8Þ

First we divide the feasible set of u into two parts: 0; 38

minixisimaxjðDxÞjðDsÞj

h iand 3

8minixisi

maxjðDxÞjðDsÞj; u

i. Then we give a

lower bound on u separately. In the second interval the 38

mðsÞkqk1

is a trivial lower bound (where q = DxDs/l).

In the first interval the function f is finite because 38


6 u (the constant 3/8 will be analyzed in Section

5). Therefore, we can get an upper bound on the function value of f on the first interval.

Proposition 12. For any u 2 0; 38


h ione has

f ðuÞ 6 ð1þ 4jÞuhðsÞXi2Iþ

qi; ð9Þ

where hðsÞ ¼ 8 m2ðsÞ5� 1

11m2ðsÞ

�and q ¼ DxDs

l ¼ dxds.

Proof. Using the properties of the infinity norm and P�ðjÞ-matrices one has

f ðuÞ ¼Xn

i¼1

1

v2i þ uqi

� 1

v2iþ uqi

¼Xn

i¼1

uqi 1� 1

v2i ðv2

i þ uqiÞ

¼Xi2Iþ

uqi 1� 1

v2i ðv2

i þ uqiÞ

�Xi2I�

ujqij 1� 1

v2i ðv2

i � ujqijÞ

6

Xi2Iþ

uqi 1� 1

kvk21ðkvk

21 þ ukqk1Þ

!þXi2I�

ujqij1

mini

v2i ðmin

iv2

i � ukqk1Þ� 1

0@

1A

6

Xi2Iþ

uqi 1� 1

kvk21ðkvk

21 þ ukqk1Þ

!þXi2Iþ

uqið1þ 4jÞ 1

mini

v2i ðmin

iv2

i � ukqk1Þ� 1

0@

1A.

Considering the domain of function f and applying estimation (8)

ukqk1 63

8min

iv2

i 63

8kvk2

1 63

8mðsÞ and min

iv2

i P1

mðsÞ


holds and we can continue the estimation using previous inequalities:

f ðuÞ 6 uXi2Iþ

qi 1� 1118

m2ðsÞ þ ð1þ 4jÞ 158

1m2ðsÞ� 1

!" #¼ u

Xi2Iþ

qi 8m2ðsÞ

5� 8

11m2ðsÞ þ 4j8m2ðsÞ

5� 1

� �;

where m2(s) P 1, therefore 8m2ðsÞ5� 1 6 8m2ðsÞ

5� 8

11m2ðsÞ. This completes the proof. h

Corollary 13. If u 2 0; 38


h i, then

u Pf ðuÞ

ð1þ 4jÞhðsÞP

i2Iþqi

. ð10Þ

Therefore, the uniquely determined u* one has

u� P ~u ¼ mini

s02 � w2ðx; s; lÞð1þ 4jÞhðsÞ

Pi2Iþ

qi

;3

8mðsÞkqk1

( ).

The lower bound of u* is obtained using the first statement of Corollary 13 with substituting the maximalvalue of f(u), considering the restricted domain of function f.

The next lemma gives estimations for further reordering of the inequality (10).

Lemma 14. Let q ¼ DxDsl . Then
X
i2Iþ

qi 6n4

msffiffiffinp

and kqk1 61

4þ j

nm

sffiffiffinp

¼ n4

msffiffiffinp

ð1þ 4jÞ.

Proof. If w(x,s,l)26 2g, then based on Lemma 11 and [14]

1

mffiffiffiffi2gn

q � ¼ 1

1þ gnþ

ffiffiffiffiffiffiffiffiffiffiffiffi2gn þ

g2

n2

q 6xTs

nl6 1þ g

nþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gnþ g2

n2

r¼ m

ffiffiffiffiffi2gn

r !.

By applying Lemma 8 on the vector a = �xs one has
Xi2Iþ
DxiDsi 61

4kffiffiffiffiffixspk2 ¼ xTs

4and

Xi2Iþ

qi 6xTs

4l6

n4

msffiffiffinp

.

Similarly we get the statement on kqk1. h

Now by Lemma 14 one has

u� P ~u P4

nm sffiffinp � min

s02 � s2

ð1þ 4jÞhðsÞ ;3

8mðsÞð1þ 4jÞ

� �¼ 4

nm sffiffinp �

ð1þ 4jÞmin

s02 � s2

hðsÞ ;3

8mðsÞ

� �

¼ 4c

nm sffiffinp �

ð1þ 4jÞ;

where c ¼ min s02�s2

hðsÞ ;3

8mðsÞ

n o.

In reordering the inequality on u* to a* we get a lower bound on a maximal feasible step-length a*.

Theorem 15. In the predictor step the maximal feasible step-length a* satisfies a� P vnffiffinp , where

vn ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

m sffiffinp �

ð1þ 4jÞ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

nm sffiffinp �

ð1þ 4jÞþ 1

s�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

nm sffiffinp �

ð1þ 4jÞ

s0@

1A ð11Þ

is a bounded quantity.


Proof

a ¼ �uþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ 4u

p2

P1

2� 4c

nm sffiffinp �

ð1þ 4jÞþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16c2

n2m2 sffiffinp �

ð1þ 4jÞ2þ 16c

nm sffiffinp �

ð1þ 4jÞ

vuut0B@

1CA

¼ 2


nm sffiffinp �

ð1þ 4jÞ

s�


nm sffiffinp �

ð1þ 4jÞ

sþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

nm sffiffinp �

ð1þ 4jÞþ 1

s0@

1A ¼ vnffiffiffi

np .

The vn expression (11) is bounded because limn!1mð sffiffinp Þ ¼ 1, namely limn!1vn ¼ 2

ffiffiffiffiffiffiffiffic

1þ4j

qis finite. h

It is easy to show that the expression vn monotonically increases in size n of the problem. The vn expression(11) is introduced in [14] for positive semidefinite matrices, corresponding to the case j = 0.

We derive the iteration complexity of the algorithm by applying the previous bound of a*. At first, we deter-mine the lower and upper bound of the duality gap.

Theorem 16. Denote the length of Newton-step in the ith iteration by ai. Then the duality gap of the point

obtained in step k satisfies

nYk

i¼1

ð1� aiÞ !

l0 1� js02

n

� �6 xkT

sk6 n

Yk

i¼1

ð1� aiÞ !

l0 1þ s02

4n

� �.

Proof. Denote the point where we arrive by predictor step from point (xk�1, sk�1) applying the Newton-stepwith step length ak and direction (Dxp,Dsp) by (xp, sp). Let (Dxc,Dsc) be the direction of the corrector Newtonstep. Then

xksk ¼ ðxp þ DxcÞðsp þ DscÞ ¼ lkeþ DxcDsc.

Thus xkTsk = lkn + DxcTD sc. Estimate the scalar product DxcTDsc using Lemma 8 (a = lke � xpsp)

DxcTDsc ¼Xn

i¼1

ðDxcÞiðDscÞi 6Xi2Iþ

ðDxcÞiðDscÞi 61

4

lke� xpspffiffiffiffiffiffiffiffiffixpspp

��

2

¼ 1

4lk

ffiffiffiffiffiffiffiffiffilk

xpsp

r�

ffiffiffiffiffiffiffiffiffixpsp

lk

r��

2

61

4lks

02;

where I+ = {1 6 i 6 n : (Dx)i(Ds)i > 0}. Because of the properties of P�ðjÞ-matrices

DxcTDsc ¼Xn

i2Iþ

ðDxcÞiðDscÞi þXn

i2I�

ðDxcÞiðDscÞi P �4jXn

i2Iþ

ðDxcÞiðDscÞi P �jlke� xpspffiffiffiffiffiffiffiffiffi

xpspp

��

2

P �jlks02.

Substituting the bounds and considering lk ¼ l0

Qki¼1ð1� aiÞ completes the proof. h

Using the estimation of the duality gap we derive the iteration complexity of the algorithm.

Theorem 17. Let the linear complementarity problem for any P�ðjÞ-matrix M be given, where j P 0 and let

l0 = 1. Then the Mizuno–Todd–Ye algorithm generates an (x, s,l) point satisfying xTs < e in at most

ffiffiffinp

v1

log4nþ s02

4e

� �

iterations.

Proof. The Mizuno–Todd–Ye algorithm take only one step both in predictor and corrector steps, then it issufficient to count the updates of parameter l. Since l0 = 1, therefore after iteration k the duality gap is cer-tainly smaller than e if


nYk

i¼1

ð1� aiÞ !

1þ s02

4n

� �< e.

Consider the given lower bound of a* for each iteration

1� vnffiffiffinp

k

<e

n 1þ s024n

� � ;
with simple computing we get
k >log e

n 1þs024n½ �

log 1� vnffiffinp

� ¼ logn 1þs02

4n

� �e

� log 1� vnffiffinp

� .

Using properties of the logarithm �log(1 � h) P h, h 2 (0,1) and monotonicity of vn

logn 1þs02

4n

� �e

� log 1� vnffiffinp

� 6 ffiffiffinp

vnlog

n 1þ s02

4n

h ie

6

ffiffiffinp

v1

logn 1þ s02

4n

h ie

: �

In Theorem 17 the determined number of steps are independent from j only at first sight, because both v1

and s 0 depend on j.

Corollary 18. Let the linear complementarity problem for any P�ðjÞ-matrix M be given, where j P 0 and let

l0 = 1, s ¼ 11þ4j and s0 ¼

ffiffi2p

1þ4j. Then the Mizuno–Todd–Ye algorithm generates an (x, s,l) point satisfying xTs < e

in at most O ð1þ jÞ32ffiffiffinp

log ne

�iterations.

Proof. It is easy to verify the given value of parameter s and s 0 are feasible. After that we estimate on the resultof Theorem 17. For lower estimation of v1 examine the value of m(s), h(s) and c:

1 6 mðsÞ ¼ 1þ s2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ s4

4

r< ð1þ sÞ2;

hðsÞ ¼ 8

5m2ðsÞ � 8

11m2ðsÞ <8

5m2ðsÞ < 8

5ð1þ sÞ4;

c ¼ mins02 � s2

hðsÞ ;3

8mðsÞ

� �> min

s2

8=5ð1þ sÞ4;

3

8ð1þ sÞ2

( )>

s2

8ð1þ sÞ4;

and c < 38mðsÞ <

38. Thus

v1 ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

mðsÞð1þ 4jÞ

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

mðsÞð1þ 4jÞ þ 1

r�


mðsÞð1þ 4jÞ

r

¼ 2


mðsÞð1þ 4jÞ

r1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cmðsÞð1þ4jÞ þ 1

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

mðsÞð1þ4jÞ

q >


mðsÞð1þ 4jÞ

r1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cmðsÞð1þ4jÞ þ 1

q

>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

½3þ 8ð1þ 4jÞ�ð1þ sÞ6

s>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

11ð1þ 4jÞð1þ sÞ6

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ 4jÞ6

11ð1þ 4jÞ3ð2þ 4jÞ6

s>

1

8ffiffiffiffiffi11p ð1þ 4jÞ�

32.

Substitute the lower bound of v1

ffiffiffinp

v1

logn 1þ s02

4n

h ie

< 8ffiffiffiffiffi11pð1þ 4jÞ

32ffiffiffinp

log2ne

.

So the iteration complexity of the algorithm is O ð1þ jÞ32ffiffiffinp

log ne

�. h


5. Conclusion

We showed that a Mizuno–Todd–Ye type predictor–corrector algorithm can solve linear complementarityproblems with a P�ðjÞ-matrix, too. The s and s 0 neighborhood parameters can be chosen in such a way thatafter each predictor step only one corrector step is needed to return into the pre-established neighborhood ofthe central path.

We generalize the analysis of [14] for LCP problems with P�ðjÞ-matrices. Since we are working with P�ðjÞ-matrices our analysis is more complicated, because the acceptable values of parameters s and s 0 depend on j.For larger values of j the values of s and s 0 are decreasing quickly, therefore the constant in the iteration com-plexity is increasing. For example if j = 0.3274, then the feasible value of parameters s and s 0 are less than one,

but if j = 0 the maximal value of s isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ

ffiffiffiffiffi17pp

.In the analysis we used 3/8 as the constant (Proposition 12, Corollary 13) which can be replaced with an

arbitrary number 0 < c < 1. Looking for the best choice of c would make the analysis of the algorithm morecomplicated without notable improvement. The c = 3/8 choice (first used by Potra) can be considered as quiteclose to the upper bound on the optimal value of c in case j = 0. (A slightly better choice is c = 0.27975 butthat would make the computation in several proof much more complicated.)

The update strategy of the central path parameter l is the same l 0 = (1 � a)l as in [14], but it is differentfrom Miao’s, because he updated the parameter l in such a way that the equation xTs/n = l must hold. Sup-

pose we are taking a step from point (x0, s0) to (x, s) with vector (Dx,Ds). Let l :¼ ðx0ÞTs0

n and denote lM theMiao’s and lP the Potra’s updated value of parameter l. Then lM ¼ xTs

n and lP = (1 � a)l, so

lM ¼xTs

n¼ ð1� aÞlþ a2 DxTDs

n¼ lP þ a2 DxTDs

n.

Using the result of Lemma 8 we obtain the following relationship:

lM �1

4a2l 6 lP 6 lM þ ja2l.

Therefore, the Potra’s way of updating is always better in the case j = 0 (the matrix is positive semidefinite)and sometimes is better and sometimes is worse than Miao’s in the case j > 0, but the algorithm and its anal-ysis is easier.

Let us take note that we can hope for further improvements by using some other self-regular proximitymeasures [9] in the algorithm.

5.1. Further research

Unfortunately this version of Mizuno–Todd–Ye interior point algorithm is not suitable to solve practicalproblems, because the parameter j of the matrix is not known usually apriori and furthermore there is nopolynomial algorithm to decide whether a matrix is sufficient or not.

The first step to analyze practical efficiency of interior point algorithms for P�ðjÞ linear complementarityproblems is to generate set of P�ðjÞ-matrices. We would like to learn more about properties and structure ofP�ðjÞ-matrices. We would like to generate P�ðjÞ-matrices of moderate size and compute some initial goodlower bound on the value of j.

We plan to implement some versions of interior point algorithms and test how these work on an arbitrarymatrix coming from some structured application like bimatrix games. These matrices are generally not in P�-class, so it would be interesting to collect all possible failures of the interior point algorithms and find propercertificate that the matrix was not P�-matrix.

Acknowledgements

Tibor Illes acknowledges the Research Professor Fellowship of the MOL, Hungarian Oil Company, inoperations research. The authors research is supported by Hungarian National Research Fund OTKA No.T 049789.


References

[1] R.W. Cottle, J.-S. Pang, V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra and itsApplications 114/115 (1989) 231–249.

[2] S.-M. Guu, R.W. Cottle, On a subclass of P0, Linear Algebra and its Applications 223/224 (1995) 325–335.[3] T. Illes, C. Roos, T. Terlaky, Polynomial affine-scaling algorithms for P

*(j) linear complementarity problems, in: P. Gritzmann, R.

Horst, E. Sachs, R. Tichatschke (Eds.), Recent Advances in Optimization, Proceedings of the 8th French–German Conference onOptimization, Trier, July 21–26, 1996, Lecture Notes in Economics and Mathematical Systems, vol. 452, Springer-Verlag, Berlin,1997, pp. 119–137.

[4] J. Ji, F.A. Potra, R. Sheng, A predictor–corrector method for the P*-matrix LCP from infeasible starting points, Optimization

Methods and Software 6 (1995) 109–126.[5] M. Kojima, S. Mizuno, A. Yoshise, A polynomial-time algorithm for a class of linear complementarity problems, Mathematical

Programming 44 (1989) 1–26.[6] M. Kojima, N. Megiddo, T. Noma, A. Yoshise, A Unified Approach to Interior Point Algorithms for Linear Complementarity

ProblemsLecture Notes in Computer Science, vol. 538, Springer-Verlag, Berlin, Germany, 1991.[7] J. Miao, A quadratically convergent O

�ðjþ 1Þ

ffiffiffinp

L�-iteration algorithm for the P

*(j)-matrix linear complementarity problem,

Mathematical Programming 69 (1995) 355–368.[8] S. Mizuno, M.J. Todd, Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Mathematics of

Operations Research 18 (4) (1993) 964–981.[9] J. Peng, C. Roos, T. Terlaky, Self-Regularity: A New Paradigm for Primal-Dual Interior Point Algorithms, Princeton University

Press, New Jersey, USA, 2002.[10] I. Polik, Novel analysis of interior point methods for linear optimization problems, Ms. Thesis, Eotvos Lorand University of Sciences,

Faculty of Natural Sciences, Budapest, 2002 [in Hungarian]. Available from: <http://www.cs.elte.hu/matdiploma/>.[11] F.A. Potra, R. Sheng, Predictor–corrector algorithm for solving P

*(j)-matrix LCP from arbitrary positive starting points,

Mathematical Programming 76 (1) (1996) 223–244.[12] F.A. Potra, R. Sheng, A large step infeasible interior point method for the P

*-matrix LCP, SIAM Journal on Optimization 7 (2)

(1997) 318–335.[13] F.A. Potra, R. Sheng, Superlinearly convergent infeasible interior point algorithm for degenerate LCP, Journal of Optimization

Theory and Application 97 (2) (1998) 249–269.[14] F.A. Potra, The Mizuno–Todd–Ye algorithm in a larger neighborhood of the central path, European Journal of Operational

Research 143 (2002) 257–267.[15] C. Roos, T. Terlaky, J.-Ph. Vial, Theory and Algorithms for Linear Optimization, An Interior Point Approach, Wiley-Interscience

Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, USA, 1997.[16] Gy. Sonnevend, J. Stoer, G. Zhao, On the complexity of following the central path of linear programs by linear extrapolation,

Methods of Operations Research 63 (1989) 19–31.[17] H. Valiaho, P

*-matrices are just sufficient, Linear Algebra and its Applications 239 (1996) 103–108.

[18] Y. Ye, K. Anstreicher, On quadratic and Oð ffiffiffinp LÞ convergence of a predictor–corrector algorithm for LCP, MathematicalProgramming 62 (1993) 537–551.

http://www.cs.elte.hu/matdiploma/

a mizuno–todd–ye type predictor–corrector algorithm for sufficient linear complementarity...

Documents