An Improved Predictor-Corrector Interior-Point Algorithm for Linear Complementarity Problems with O √ nL-Iteration Complexity

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.


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.
The mean value of x T y is denoted as μ x T y/n throughout this paper.Then ρ β { x, y ∈ S | Xy − μe ≤ βμ, μ x T y/n} is considered as the neighborhood of central path, where β > 0, e denotes the vector of ones, X diag x and • express the Euclidean norm.
The improved predictor-corrector interior-point algorithm IPCIP is as follows.

Main Steps of IPCIP Algorithm
Step 0. Choose an initial pair of interior point x 0 , y 0 with x 0 , y 0 ∈ ρ 1/4 , and set the accuracy parameter ε > 0. Let k 0.
Step 1.If x k T y k /n ≤ ε, then stop.
Step 2. Compute the predictor direction Δx p , Δy p by solving the system 2.1 ,

2.1
Step We will show later that , and compute the corrector direction Δx c , Δy c by solving the system 2.2 ,

2.2
Step 5. Set x k 1 , y k 1 x k , y k Δx c , Δy c , update k k 1, and return to Step 1.

Convergence and Complexity Analysis
Proof.The proof of i can be seen in Lemma 1 of 2 .From h 2 b c 2 b 2 c 2 2b T c and b T c ≥ 0, we can obtain b T c ≤ 1/2 h 2 and b 2 ≤ h 2 .Thus inequalities ii and iii hold.This completes the proof.Lemma 3.2.If the point x k , y k was generated by the algorithm, then one has Proof.From Step 2 of our algorithm and M is a n × n positive semidefinite matrix, we have Δx p T Δy p Δx p T MΔx p ≥ 0. Hence

3.1
This completes the proof.
Proof.Let us define x Q , y Q x k QΔx p , y k QΔy p , where It's well known that the inequality Xz − x T z/n e ≤ Xz − ce holds for arbitrary vector x, z and arbitrary constant c.
So we have 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 ρ 1/2 . Proof.

3.4
This completes the proof.
Lemma 3.6.If n > 2 and the point x k 1 , y k 1 was generated by the algorithm, then

3.5
From Lemmas 3.1 and 3.5 we have

3.6
Note that r 1 − 1/4 √ 2n, we have Remark 3.7.From Lemmas 3.2 and 3.6, we can obtain immediately that μ k 1 ≤ μ k ≤ μ k .That means the corrector reduces μ k to μ k 1 .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 μ k than MTY algorithm.
Then the polynomial convergence of the algorithm could be established.
Theorem 3.8.Suppose that the sequence { x k 1 , y k 1 } generated by the algorithm, then one has Proof.Let us define that x k 1 α x k αΔx c , y k 1 α y k αΔy c , and denote As discussed in Lemma 3.6, we also have From Lemmas 3.1 and 3.5 and r 1 − 1/4 √ 2n, then 3.9 So we can write

3.10
Since the inequality Xz − x T z/n e ≤ Xz − ce holds for arbitrary vector x, z and arbitrary constant c, thus

3.12
Since μ k > ε > 0 otherwise the algorithm will be terminated , it follows that

3.13
We have proved that x k x Q k > 0, y k y Q k > 0 in Lemma 3.3.So we have x k 1 0 x k > 0, y k 1 0 y k > 0, when α 0.

3.14
Journal of Applied Mathematics 9 Note that x k 1 α and y k 1 α are continuous with respect to α.It implies that x k 1 α > 0 and y k 1 α > 0 hold for any α ∈ 0, 1 .
Let α 1, then we have x k 1 > 0 and y k 1 > 0. Furthermore, when α Consider that we can obtain Mx k 1 − y k 1 h 0 directly from the steps in our algorithm, so x k 1 , y k 1 ∈ ρ 1/4 holds for every x k 1 , y k 1 generated by the algorithm.
This completes the proof.
Theorem 3.9.The iteration complexity of the algorithm is

3.15
From Lemma 3.1, we have

3.16
From X k y k − μ k e ≤ 1/4 μ k , we have

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 ∈ R 3 × R 3 such that y M 1 x h 1 , x, y ≥ 0, 0 , x T y 0, where h 1 ∈ R 3 and M 1 is a 3 × 3 positive semidefinite matrix, Test Problem 2 Find a vector pair x, y ∈ R 5 × R 5 such that y M 2 x h 2 , x, y ≥ 0, 0 , x T y 0, where h 2 ∈ R 5 and M 2 is a 5 × 5 positive semidefinite matrix,

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 ∈ R n × R n such that y M 3 x h 3 , x, y ≥ 0, 0 , x T y 0, where h 3 ∈ R n and M 3 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, 0, 0, 0, 0, 0 T 0, 0, 0, 0, 0 T Run time s 0.071335 0.037116 0, 0, 0, . .., 0, 0 T 0, 0, 0, . .., 0, 0 T y k 1, 1, 1, . .., 1, 1 T 1, 1, 1, . .., 1, 1 T Run time s 0.217375 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, μ k , the approximate solutions, and computing times are shown in Tables 1, 2, and 3.

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.
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.
Remark 3.10.Note that the reduce factor in the MTY algorithm is 1 − χ/ It implies that our algorithm will converge faster than the MTY algorithm, although both have the same iteration complexity.

Table 1 :
Computational results of test problem 1.

Table 2 :
Computational results of test problem 2.

Table 3 :
Computational results of test problem 3.