Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 340192, 12 pages
doi:10.1155/2011/340192
Research Article
An Improved Predictor-Corrector Interior-Point
Algorithm for Linear
√ Complementarity
Problems with O nL-Iteration Complexity
Debin Fang1 and Qian Yu2
1
2
School of Economics and Management, Wuhan University, Wuhan 430072, China
School of Economics, Wuhan University of Technology, Wuhan 430070, China
Correspondence should be addressed to Qian Yu, yuqian365@126.com
Received 18 September 2011; Revised 23 November 2011; Accepted 24 November 2011
Academic Editor: Chong Lin
Copyright q 2011 D. Fang and Q. Yu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper proposes an improved predictor-corrector interior-point algorithm for the linear
complementarity problem LCP based on the Mizuno-Todd-Ye algorithm. The modified corrector
steps in our algorithm cannot only draw the iteration point back to a narrower neighborhood of the
center path but also reduce the duality gap. It implies that the improved algorithm can converge
faster than√the MTY algorithm. The iteration complexity of the improved algorithm is proved to
obtain O nL which is similar to the classical Mizuno-Todd-Ye algorithm. Finally, the numerical
experiments show that our algorithm improved the performance of the classical MTY algorithm.
1. Introduction
Since Karmarkar published the first paper on interior point method 1 in 1984, the interior
point methodologies have yielded rich theories and algorithms in the fields of linear
programming LP, quadratic programming QP, and linear complementarity problems
LCP. Among these interior point methods, predictor-corrector interior-point methods play
a special role due to their best polynomial complexity and superlinear convergence.
In 1993, Mizuno et al. 2 proposed the classical representative of predictor-corrector
√
method for linear programming. The Mizuno-Todd-Ye MTY algorithm has O nLiteration complexity which is the best iteration complexity obtained so far for all the
interior-point method 3, 4. Moreover, Ye et al. 5 proved the duality gap of classical
MTY algorithm converges to zero quadratically, which dedicated that MTY algorithm has
superlinear convergence. The classical MTY algorithm is the first algorithm for LP has
both polynomial complexity and superlinear convergence. So the classical MTY algorithm
therefore was considered as the most efficient interior point methods for LP.
2
Journal of Applied Mathematics
Afterwards, the MTY algorithm was extended to LCP 6, 7 for its excellent
performance. In recent papers, Potra 8, 9 presented several predictor-corrector methods for
linear complementarity problems acting in a wide neighborhood of the central path, and Stoer
et al. 10 proposed a predictor-corrector algorithm with arbitrarily high order of convergence
to degenerate sufficient linear complementarity problems.
Subsequently, Potra proposed a generalization of the MTY algorithm for infeasible
starting points for monotone LCP 11. Based on the so-called “fast step-safe step” strategy,
Wright proposed an infeasible interior point method with polynomial and quadratic
√
convergence for nondegenerate LCP 12. And Ai and Zhang presented an O nL-iteration
primal-dual path-following method for monotone LCP based on wide neighborhoods and
large updates 13. These workings improved the MTY algorithm in various aspects. In
this paper, the MTY algorithm for LCP will be improved by modifying the corrector of the
algorithm to obtain a larger iteration reduce factor than the original, which can guarantee a
faster convergence.
The typical iterations of the MTY algorithm operate between two neighborhoods of
the central path, ρ1/4 and ρ1/2. Before starting the MTY algorithm, an arbitrary initial
point ω ∈ ρ1/4 will be given. And then, the predictor produces a point ω ∈ ρ1/2 by
carefully choosing a steplength along the affine-scaling direction
at ω and reduces the primal
√
√
dual gap by a factor of at least 1 − χ/
n where χ 1/ 4 8 ≈ 0.5946. In the corrector steps,
the corrector produces a point ω
∈ ρ1/4 by taking a unit steplength along the centering
direction at ω. It is shown that the corrector put the iteration points back to ρ1/4 which
is the narrower neighborhood of the central path and maintains the same duality gap. Thus
√
n and holds
predictor-corrector reduces the duality gap μk by a factor of at least 1 − χ/
the iteration points in the neighborhood ρ1/4. It follows that the corresponding iterative
√
procedure has O nL-iteration complexity.
The modified algorithm in this paper has only one corrector step in neighborhood of
the central path ρ1/2 and one predictor step in a narrow neighborhood ρ1/4 same as
the MTY algorithm. Moreover, the modified iteration direction of corrector step can make
a reduction for duality gap. This improvement attained a larger iteration reduce factor and
results in a faster convergence than the classical MTY algorithm.
Section 2 describes the procedure of our predictor-corrector algorithm. Section 3 gives
the convergence analysis of our algorithm. It shows that the improved algorithm can generate
a sequence {xk , yk } in the neighborhood ρ1/4 from an arbitrary initial point x0 , y0 ∈
√
ρ1/4 and converge to optimal solution xk , yk with O nL-iteration complexity. Finally,
the performances of our algorithm are evaluated by numerical experiments in Section 4.
2. Description of the Algorithm
In this paper, LCP is to find a vector pair x, y ∈ Rn × Rn such that y Mx h, x, y ≥ 0, 0
and xT y 0, where h ∈ Rn and M is a n × n positive semidefinite matrix.
Denote the set of all feasible points of LCP by S {x, y | Mx−yh 0, x ≥ 0, y ≥ 0}
and the solution set of LCP by S∗ {x∗ , y∗ | x∗ , y∗ ∈ S, x∗ T y∗ 0}. The relative interior
of S, S {x, y | x, y ∈ S, x > 0, y > 0} is called the set of strictly feasible points, or
the set of interior points. We assume S / Φ in this paper. It is known that if S is nonempty,
then for any parameter τ > 0 the nonlinear system Mx − y h 0, xy τe has a unique
positive solution 14. The set of all such solutions defines the central path C of LCP. So the
assumption is sufficient to establish the existence of the central path and the existence of a
solution to LCP.
Journal of Applied Mathematics
3
The mean value of xT y is denoted as μ xT y/n throughout this paper. Then ρβ {x, y ∈ S | Xy − μe ≤ βμ, μ xT y/n} is considered as the neighborhood of central path,
where β > 0, e denotes the vector of ones, X diagx and · express the Euclidean norm.
The improved predictor-corrector interior-point algorithm IPCIP is as follows.
2.1. Main Steps of IPCIP Algorithm
Step 0. Choose an initial pair of interior point x0 , y0 with x0 , y0 ∈ ρ1/4, and set the
accuracy parameter ε > 0. Let k 0.
T
Step 1. If xk yk /n ≤ ε, then stop.
Step 2. Compute the predictor direction Δxp , Δyp by solving the system 2.1,
MΔxp − Δyp 0,
Y k Δxp X k Δyp 2 k
μ e − X k yk .
3
2.1
√ Step 3. Set xk , yk xk , yk Qk Δxp , Δyp where Qk min{1/8 n, μk /8ΔX p Δyp }.
k yk − μk e ≤ 1/2μk , where μk 1 − Qk μk holds for every
We will show later that X
xk , yk .
√
Step 4. Set r 1 − 1/4 2n, and compute the corrector direction Δxc , Δyc by solving the
system 2.2,
MΔxc − Δyc 0,
k Δyc rμk e − X
k yk .
Y k Δxc X
2.2
Step 5. Set xk1 , yk1 xk , yk Δxc , Δyc , update k k 1, and return to Step 1.
3. Convergence and Complexity Analysis
Lemma 3.1. If b, c ∈ Rn and b c h, bT c ≥ 0, B diagb, then
√
i Bc ≤ 2/4h2 ;
ii bT c ≤ 1/2h2 ;
iii b2 ≤ h2 .
Proof. The proof of i can be seen in Lemma 1 of 2.
From h2 b c2 b2 c2 2bT c and bT c ≥ 0, we can obtain bT c ≤ 1/2h2
and b2 ≤ h2 . Thus inequalities ii and iii hold. This completes the proof.
Lemma 3.2. If the point xk , yk was generated by the algorithm, then one has μ
k xk T yk /n ≥ μk ,
where μk 1 − Qk μk .
4
Journal of Applied Mathematics
Proof. From Step 2 of our algorithm and M is a n × n positive semidefinite matrix, we have
Δxp T Δyp Δxp T MΔxp ≥ 0. Hence
xk
k
μ
xk Qk Δxp
x
k T
≥
yk
n
T
xk
T
n
yk Q
T n
k
yk QK Δyp
xk
T
2
Δyp Δxp T yk Qk Δxp T Δyp
n
yk
T Qk
2
n · μk − x k y k
n
3
3.1
2
μk Q k μk − Q k μk
3
≥ 1 − Q k μk
μk .
This completes the proof.
Lemma 3.3. If xk , yk ∈ ρ1/4, then the point xk , yk generated by the algorithm satisfies xk >
k yk − μk e ≤ 1/2μk , and xk , yk ∈ ρ1/2.
0, yk > 0, X
Proof. Let us define xQ, yQ xk QΔxp , yk QΔyp , where
⎫
⎧
⎬
⎨ 1 k
μ
.
0 ≤ Q ≤ Qk min
√ , ⎩8 n
8ΔX p Δyp ⎭
3.2
√ √
Note that 0 ≤ Q ≤ Qk min{1/8 n, μk /8ΔX p Δyp }, then we have 2/3 nQ ≤
√
1/12 from Q ≤ 1/8 n and Q2 ΔX p Δyp ≤ 1/8μk from Q ≤ μk /8ΔX p Δzp .
Furthermore X k yk − μk e ≤ 1/4μk holds since xk , yk ∈ ρ1/4. Therefore we have
XQyQ − 1 − Qμk e X k yk Q X p Δyp ΔX p yp Q2 ΔX p Δyp − 1 − Qμk e
k k
2 k
k k
2
p
p
k X
μ
y
Q
e
−
X
y
ΔX
Δy
−
−
Qμ
e
Q
1
3
2√
≤ 1 − QX k yk − μk e nQμk Q2 ΔX p Δyp 3
Journal of Applied Mathematics
5
1
1
1
≤ 1 − Q μk μk μk
4
12
8
1
1
1
1
1 − Qμk
4 1 − Q 12 8
5
1
≤
1 − Qμk
4 21
≤
1
1 − Qμk .
2
3.3
So xi Qyi Q ≥ 1/21 − Qμk > 0 i 1, 2, . . . , n for any Q 0 ≤ Q ≤ QK .
Since xQ, yQ xk QΔxp , yk QΔyp is continuous with respect to Q, xk > 0
k
and y > 0, it is easily seen that xk xQk > 0, yk yQk > 0.
k , Y Qk Y k , 1 − Qk μk μk , the above argument implies
Note that XQk X
k yk − μk e ≤ 1/2μk .
X
It’s well known that the inequality Xz − xT z/ne ≤ Xz − ce holds for arbitrary
vector x, z and arbitrary constant c.
k yk − μ
k yk − μk e ≤ 1/2μk ≤ 1/2
So we have X
k e ≤ X
μ from the proof above
k
k
and Lemma 3.2, thus x , y ∈ ρ1/2.
This completes the proof.
Remark 3.4. So Lemma 3.3 dedicates that the predictors of our algorithm operate in a wide
neighborhood of the central path ρ1/2.
k Y k k yk −μk e ≤ 1/2μk then X
Lemma 3.5. If X
−1/2
2
k yk ≤ 21/41−r2 nμk
rμk e − X
Proof.
k k −1/2 k
2
X
Y
k yk ≤
rμ
e
−
X
≤
1
min xk yk
1≤i≤n i i
1
2
k
k yk rμ e − X
2
k k
k
k X
y
−
μ
e
−
rμ
e
1
k
1 − 1/2μ
2 2 2 k k
≤ k X
y − μk e 1 − rμk e
μ
1
2
1 − r n μk .
≤2
4
3.4
This completes the proof.
Lemma 3.6. If n > 2 and the point xk1 , yk1 was generated by the algorithm, then μk1 ≤ 1 −
Qk μk always hold.
6
Journal of Applied Mathematics
k 1/2 Y k −1/2 then
Proof. Denote D X
−1/2 k yk
k Y k
rμk e − X
D−1 Δxc DΔyc X
−1
D Δx
c
T DΔy
c
c T
c
c T
3.5
c
Δx Δy Δx MΔx ≥ 0.
From Lemmas 3.1 and 3.5 we have
c T
−1/2 2
1
k
X
k yk
k yk ≤ 1 1 − r2 n μk
rμ
e
−
X
2
4
k1 T k1
y
x
n
T k
k
y Δyc
x Δxc
n
k T k k T
x y x Δyc Δxc T yk Δxc T Δyc
n
Δx Δyc ≤
∴ μk1
3.6
nrμk Δxc T Δyc
n
1
k
2
≤ rμ 1 − r μk .
4n
√
Note that r 1 − 1/4 2n, we have
√
1
1
4 2n − 9
1
1
r
1 − r2 1 − √ 1−
.
4n
32n
4 2n 4n 32n
3.7
It is obvious that r 1/4n 1 − r2 < 1 when n > 2, so μk1 ≤ μk 1 − Qk μk .
This completes the proof.
Remark 3.7. From Lemmas 3.2 and 3.6, we can obtain immediately that μk1 ≤ μk ≤ μ
k . That
k
k1
means the corrector reduces μ
to μ . Because the corrector in the classical MTY algorithm
just maintains the same duality gap, the improvement of the corrector in our algorithm will
make a faster reduction of μk than MTY algorithm.
Then the polynomial convergence of the algorithm could be established.
Theorem 3.8. Suppose that the sequence {xk1 , yk1 } generated by the algorithm, then one has
xk1 , yk1 ∈ ρ1/4.
Proof. Let us define that xk1 α xk αΔxc , yk1 α yk αΔyc , and denote
μ
k1
k1 T k1
x α y α
.
α n
3.8
Journal of Applied Mathematics
7
As discussed in Lemma 3.6, we also have D−1 Δxc T DΔyc Δxc T Δyc Δxc T MΔxc ≥
0.
√
From Lemmas 3.1 and 3.5 and r 1 − 1/4 2n, then
√ 2
k k −1/2 k
ΔX c Δyc ≤ 2 X
Y
k yk rμ
e
−
X
4 √ 2 1
2
2
1 − r n μk
≤
4
4
√
√
2
1
2
2 2n1 − r μk
4
2
√
2 1
1
μk
≤
4
2
8
7 k
μ ,
32
k1 T k1
x α y α
k1
μ α n
T k
k
y αΔyc
x αΔxc
n
T
k T k
x y α xk Δyc Δxc T yk α2 Δxc T Δyc
n
k T k
T
x y α xk Δyc Δxc T yk
≥
n
T k T k
x y α nrμk − xk yk
n
k
μ
k α rμk − μ
≤
3.9
1 − α
μk αrμk
≥ 1 − 1 − rαμk .
So we can write μk1 α ≥ 1 − 1 − rαμk , that is, μk ≤ μk1 α/1 − 1 − rα.
Hence
xk αΔxc T yk αΔyc k1
k1
k1
k
c
k
c
αΔX
y αΔy −
e
X αy α − μ αe X
n
k yk α X
k Δyc ΔX c yk α2 ΔX c Δyc
X
−
xk
T
T
yk α xk Δyc Δxc T yk α2 Δxc T Δyc
n
e
8
Journal of Applied Mathematics
k yk α2 ΔX c Δyc
k yk α rμk e − X
X
−
xk
T
T yk α nrμk − xk yk α2 Δxc T Δyc
n
⎛
1 − α⎝X y −
k k
xk
T
yk
n
⎞
e
e⎠ α
2
Δxc T Δyc
ΔX Δy −
e .
n
c
c
3.10
Since the inequality Xz − xT z/ne ≤ Xz − ce holds for arbitrary vector x, z and
arbitrary constant c, thus
k T k c T
c x y Δy
Δx
k1
k1
k1
k
k
2
c
c
ΔX Δy −
e
e
X αy α − μ αe ≤ 1 − α
α X y −
n
n
k k
≤ 1 − αX
y − μk e α2 ΔX c Δyc 1
7
≤ 1 − α μk α2 μk
2
32
≤
1/21 − α 7/32α2 k1
μ α.
1 − 1 − rα
3.11
Let us define fα 1/21 − α 7/32α2 /1 − 1 − rα, then f α −1/2r 7/16α − 7/32α2 1 − r/1 − 1 − rα2 .
When n > 2, we have r ≥ 7/8, this results in f α ≤ 0 and fα decreases
monotonously, when 0 ≤ α ≤ 1. Because fα ≤ f0 1/2 holds for every α ∈ 0, 1 thus
1
k1
X αyk1 α − μk1 αe ≤ μk1 α.
2
3.12
Since μk > ε > 0 otherwise the algorithm will be terminated, it follows that
Xik1 αyik1 α ≥
1 k1
μ α
2
1
1 − 1 − rαμk
2
1
1 − 1 − rα 1 − Qk μk > 0.
2
≥
3.13
We have proved that xk xQk > 0, yk yQk > 0 in Lemma 3.3. So we have
xk1 0 xk > 0,
yk1 0 yk > 0,
when α 0.
3.14
Journal of Applied Mathematics
9
Note that xk1 α and yk1 α are continuous with respect to α. It implies that xk1 α >
0 and yk1 α > 0 hold for any α ∈ 0, 1.
Let α 1, then we have xk1 > 0 and yk1 > 0.
Furthermore, when α 1, fα 7/32r thus X k1 yk1 − μk1 e ≤ 7/32rμk1 ≤
1/4μk1 .
Consider that we can obtain Mxk1 − yk1 h 0 directly from the steps in our
algorithm, so xk1 , yk1 ∈ ρ1/4 holds for every xk1 , yk1 generated by the algorithm.
This completes the proof.
√
Theorem 3.9. The iteration complexity of the algorithm is O nL.
Proof. Let D X k 1/2 Y k −1/2 then
D
D
−1
−1
p
p
k
Δx D Δy X Y
Δx
p
T D Δy
p
k
−1/2 2
p T
3
p
k
k k
μ e−X y
,
3.15
p T
p
Δx Δy Δx MΔy ≥ 0.
From Lemma 3.1, we have
−1
ΔX p Δyp ΔX p D Δyp D
√
2
−1/2 2
2 k k
k
k k ≤
· X y
μ e−X y 4 3
⎞2
√ ⎛ n 2 −1 −1
2 ⎝
4
2
xik
yik
· μk
≤
− μk xik yik ⎠ .
4
3
3
i1
3.16
From X k yk − μk e ≤ 1/4μk , we have xik yik ≥ 3/4μk , then xik −1 yik −1 ≤ 4/3μk ,
√
n 2 4 4 k 4 k
·
· · μ − μ nμk
4 i1 9 3
3
√
2 7
· nμk
4 27
√
7 2 k
nμ .
108
ΔX p Δyp ≤
3.17
√
√ p
p
k
Hence μ /8ΔX Δy ≥ 27/14 2n, and we can obtain min{1/8 n, μk /8ΔX p Δyp }
√
≥ 27/14 2n.
√
So it implies that μk1 ≤ 1 − Qk μk ≤ 1 − 27/14 2nμk for each k. This means that
√
μk will decrease at least by a constant factor of 1 − 27/14 2n at each iteration, which
√
guarantees a O nL-iteration complexity for the algorithm.
This completes the proof.
10
Journal of Applied Mathematics
√
Remark 3.10. Note that the reduce factor in the MTY algorithm
is 1 − χ/
n where χ √
√
√
1/ 4 8 ≈ 0.5946, but the reduce factor in our algorithm is 1 − 27/14 2n 1 − λ/ n
√
where λ 27/14 2 ≈ 1.168, which is larger than χ.
It implies that our algorithm will
converge faster than the MTY algorithm, although both have the same iteration complexity.
4. Examples and Numerical Results
Finally, the numerical experiments were carried out to evaluate the performance and practical
efficiency of the algorithm,
Test Problem 1
Find a vector pair x, y ∈ R3 × R3 such that y M1 x h1 , x, y ≥ 0, 0, xT y 0, where
h1 ∈ R3 and M1 is a 3 × 3 positive semidefinite matrix,
⎛
2 1 3
⎞
⎞
−1
⎜ ⎟
⎟
h1 ⎜
⎝ 0 ⎠.
−2
⎛
⎟
⎜
⎟
M1 ⎜
⎝3 2 0⎠,
1 1 5
4.1
Test Problem 2
Find a vector pair x, y ∈ R5 × R5 such that y M2 x h2 , x, y ≥ 0, 0, xT y 0, where
h2 ∈ R5 and M2 is a 5 × 5 positive semidefinite matrix,
⎛
7
⎜
⎜2
⎜
⎜
M2 ⎜
⎜0
⎜
⎜0
⎝
8
0 0 2 0
⎞
⎛
⎟
8 3 5 9⎟
⎟
⎟
0 3 0 3⎟
⎟,
⎟
1 4 6 1⎟
⎠
0 0 2 5
−9.5
⎞
⎟
⎜
⎜−36.5⎟
⎟
⎜
⎟
⎜
⎜
h2 ⎜ −5 ⎟
⎟.
⎟
⎜
⎜ −14 ⎟
⎠
⎝
−18.5
4.2
Test Problem 3
This example is a general test problem used by Noor et al. 15. The problem is to find a vector
pair x, y ∈ Rn × Rn such that y M3 x h3 , x, y ≥ 0, 0, xT y 0, where h3 ∈ Rn and M3
is a n × n positive semidefinite matrix. To test the efficiency of our algorithm by large-scale
problem, we set the dimension of test problem 3 as 100, that is, n 100,
⎛
4 −2 0 · · · 0
⎟
0⎟
⎟
⎟
0⎟
⎟,
⎟
.. ⎟
.⎟
⎠
0 ··· 4
⎜
⎜1 4 −2 · · ·
⎜
⎜
⎜
M3 ⎜ 0 1 4 · · ·
⎜
⎜ .. .. .. ..
⎜. . . .
⎝
0 0
⎞
⎛ ⎞
1
⎜ ⎟
⎜1⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
h3 ⎜ ... ⎟.
⎜ ⎟
⎜ ⎟
⎜1⎟
⎝ ⎠
1
4.3
Journal of Applied Mathematics
11
Table 1: Computational results of test problem 1.
x0
y0
ε
Iterations
μk
xk
yk
Run time s
MTY
1, 1, 1T
5, 5, 5T
1e − 06
12
3.76e − 07
0.0, 0.0004, 0.3999T
0.2002, 0.0009, 9.29e − 07T
0.078220
IPCIP
1, 1, 1T
5, 5, 5T
1e − 06
11
5.73e − 07
0.0, 0.0009, 0.3998T
0.2004, 0.0018, 1.81e − 07T
0.049975
Table 2: Computational results of test problem 2.
x0
y0
ε
Iterations
μk
xk
yk
Run time s
MTY
1.5, 1.5, 1.5, 1.5, 1.5T
4, 4, 4, 4, 4T
1e − 06
10
2.61e − 08
0.8402, 1.3726, 0.0347, 1.8094, 1.6320T
0, 0, 0, 0, 0T
0.071335
IPCIP
1.5, 1.5, 1.5, 1.5, 1.5T
4, 4, 4, 4, 4T
1e − 06
9
2.18e − 07
0.8402, 1.3726, 0.0347, 1.8094, 1.6320T
0, 0, 0, 0, 0T
0.037116
Table 3: Computational results of test problem 3.
x0
y0
ε
Iterations
μk
xk
yk
Run time s
MTY
1, 1, 1, . . ., 1, 1T
3, 4, 4, . . ., 4, 6T
1e − 05
8
1.00e − 08
0, 0, 0, . . ., 0, 0T
1, 1, 1, . . ., 1, 1T
0.217375
IPCIP
1, 1, 1, . . ., 1, 1T
3, 4, 4, . . ., 4, 6T
1e − 05
7
1.92e − 08
0, 0, 0, . . ., 0, 0T
1, 1, 1, . . ., 1, 1T
0.177515
These test problems are solved by our IPCIP algorithm and the classical MTY
algorithm. The experiments are run on a PC 2.6 GHz CPU, 2 G DDR RAM using MATLAB 7.
The results including the corresponding numbers of iterations, μk , the approximate solutions,
and computing times are shown in Tables 1, 2, and 3.
5. Conclusion
This paper modified the MTY algorithm for solving monotone LCPs to strengthen the
convergence results. Although the iteration complexity of the improved algorithm was
√
proved to be O nL which is similar to the classical MTY algorithm, but the reduce factor of
duality gap was enhanced to 1.168 and results in a faster convergence than the classical MTY
algorithm.
12
Journal of Applied Mathematics
The numerical results show that our IPCIP algorithm is more efficient than the MTY
algorithm. The number of iterations was decreased, and the computing times could be
reduced by nearly 20% to 50%. That indicated IPCIP algorithm has a better performance than
the MTY algorithm.
Acknowledgments
This paper was supported by the Humanities and Social Science Research Projects of
the Ministry of Education of China no. 11YJCZH221 and the National Natural Science
Foundation of China no. 71103135.
References
1 N. Karmarkar, “A new polynomial-time algorithm for linear programming,” Combinatorica, vol. 4, no.
4, pp. 373–395, 1984.
2 S. Mizuno, M. J. Todd, and Y. Ye, “On adaptive-step primal-dual interior-point algorithms for linear
programming,” Mathematics of Operations Research, vol. 18, no. 4, pp. 964–981, 1993.
3 T. Illés, M. Nagy, and T. Terlaky, “A polynomial path-following interior point algorithm for general
linear complementarity problems,” Journal of Global Optimization, vol. 47, no. 3, pp. 329–342, 2010.
4 Y. Q. Bai, G. Lesaja, and C. Roos, “A new class of polynomial interior-point algorithms for linear
complementary problems,” Pacific Journal of Optimization. An International Journal, vol. 4, no. 1, pp.
19–41, 2008.
√
5 Y. Ye, O. Güler, R. A. Tapia, and Y. Zhang, “A quadratically convergent O nL-iteration algorithm
for linear programming,” Mathematical Programming,
vol. 59, no. 2, pp. 151–162, 1993.
√
6 Y. Ye and K. Anstreicher, “On quadratic and O nL convergence of a predictor-corrector algorithm
for LCP,” Mathematical Programming, vol. 62, no. 3, pp. 537–551, 1993.
7 J. Ji, F. A. Potra, and S. Huang, “Predictor-corrector method for linear complementarity problems with
polynomial complexity and superlinear convergence,” Journal of Optimization Theory and Applications,
vol. 85, no. 1, pp. 187–199, 1995.
8 F. A. Potra, “The Mizuno-Todd-Ye algorithm in a larger neighborhood of the central path,” European
Journal of Operational Research, vol. 143, no. 2, pp. 257–267, 2002.
9 F. A. Potra, “A superlinearly convergent predictor-corrector
method for degenerate LCP in a wide
√
neighborhood of the central path with O nL-iteration complexity,” Mathematical Programming A,
vol. 100, no. 2, pp. 317–337, 2004.
10 J. Stoer, M. Wechs, and S. Mizuno, “High order infeasible-interior-point methods for solving sufficient
linear complementarity
√ problems,” Mathematics of Operations Research, vol. 23, no. 4, pp. 832–862, 1998.
11 F. A. Potra, “An O nL infeasible-interior-point algorithm for LCP with quadratic convergence,”
Annals of Operations Research, vol. 62, pp. 81–102, 1996.
12 S. J. Wright, “A path-following interior-point algorithm for linear and quadratic problems,” Annals of
Operations Research, vol. 62, pp.√103–130, 1996.
13 W. Ai and S. Zhang, “An O nL iteration primal-dual path-following method, based on wide
neighborhoods and large updates, for monotone LCP,” SIAM Journal on Optimization, vol. 16, no. 2,
pp. 400–417, 2005.
14 M. Kojima, N. Megiddo, T. Noma, and A. Yoshise, “A unified approach to interior point algorithms
for linear complementarity problems,” Lecture Notes in Computer Science, vol. 538, 1991.
15 M. A. Noor, Y. J. Wang, and N. Xiu, “Projection iterative schemes for general variational inequalities,”
Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 3, pp. 1–8, 2002.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014