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

Abstract

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(√𝑛𝐿)-iteration 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.

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, 𝜌(1/4) and 𝜌(1/2). Before starting the MTY algorithm, an arbitrary initial point 𝜔∈𝜌(1/4) will be given. And then, the predictor produces a point 𝜔∈𝜌(1/2) 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/√⁴8≈0.5946. In the corrector steps, the corrector produces a point 𝜔∈𝜌(1/4) by taking a unit steplength along the centering direction at 𝜔. It is shown that the corrector put the iteration points back to 𝜌(1/4) which is the narrower neighborhood of the central path and maintains the same duality gap. Thus predictor-corrector reduces the duality gap 𝜇^𝑘 by a factor of at least (1−𝜒/√𝑛) and holds the iteration points in the neighborhood 𝜌(1/4). 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 𝜌(1/2) and one predictor step in a narrow neighborhood 𝜌(1/4) 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 {(𝑥^𝑘,𝑦^𝑘)} in the neighborhood 𝜌(1/4) from an arbitrary initial point (𝑥⁰,𝑦⁰)∈𝜌(1/4) and converge to optimal solution (𝑥^𝑘,𝑦^𝑘) 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)∈𝑅^𝑛×𝑅^𝑛 such that 𝑦=𝑀𝑥+ℎ, (𝑥,𝑦)≥(0,0) and 𝑥^𝑇𝑦=0, where ℎ∈𝑅^𝑛 and 𝑀 is a 𝑛×𝑛 positive semidefinite matrix.

Denote the set of all feasible points of LCP by 𝑆={(𝑥,𝑦)∣𝑀𝑥−𝑦+ℎ=0,𝑥≥0,𝑦≥0} and the solution set of LCP by 𝑆^∗={(𝑥^∗,𝑦^∗)∣(𝑥^∗,𝑦^∗)∈𝑆,𝑥^∗𝑇𝑦^∗=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.

The mean value of 𝑥^𝑇𝑦 is denoted as 𝜇=𝑥^𝑇𝑦/𝑛 throughout this paper. Then 𝜌(𝛽)={(𝑥,𝑦)∈𝑆⁺∣‖𝑋𝑦−𝜇𝑒‖≤𝛽𝜇,𝜇=𝑥^𝑇𝑦/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.

2.1. Main Steps of IPCIP Algorithm

Step 0. Choose an initial pair of interior point (𝑥⁰,𝑦⁰) with (𝑥⁰,𝑦⁰)∈𝜌(1/4), and set the accuracy parameter 𝜀>0. Let 𝑘=0.

Step 1. If (𝑥^𝑘)^𝑇𝑦^𝑘/𝑛≤𝜀, then stop.

Step 2. Compute the predictor direction (Δ𝑥^𝑝,Δ𝑦^𝑝) by solving the system (2.1), 𝑀Δ𝑥^𝑝−Δ𝑦^𝑝𝑌=0,_𝑘Δ𝑥^𝑝+𝑋^𝑘Δ𝑦^𝑝=23𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘. (2.1)

Step 3. Set (𝑥^𝑘,𝑦^𝑘)=(𝑥^𝑘,𝑦^𝑘)+𝑄^𝑘(Δ𝑥^𝑝,Δ𝑦^𝑝) where 𝑄^𝑘=min{1/8√𝑛,𝜇^𝑘/8‖Δ𝑋^𝑝Δ𝑦^𝑝‖}. (We will show later that ‖𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒‖≤(1/2)𝜇^𝑘, where 𝜇^𝑘=(1−𝑄^𝑘)𝜇^𝑘 holds for every (𝑥^𝑘,𝑦^𝑘).)

Step 4. Set 𝑟=1−1/4√2𝑛, and compute the corrector direction (Δ𝑥^𝑐,Δ𝑦^𝑐) by solving the system (2.2), 𝑀Δ𝑥^𝑐−Δ𝑦^𝑐𝑌=0,_𝑘Δ𝑥^𝑐+𝑋^𝑘Δ𝑦^𝑐=𝑟𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘. (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.

3. Convergence and Complexity Analysis

Lemma 3.1. If 𝑏,𝑐∈𝑅^𝑛 and 𝑏+𝑐=ℎ,𝑏^𝑇𝑐≥0,𝐵=diag(𝑏), then(i)‖𝐵𝑐‖≤(√2/4)‖ℎ‖²; (ii)𝑏^𝑇𝑐≤(1/2)‖ℎ‖²; (iii)‖𝑏‖²≤‖ℎ‖².

Proof. The proof of (i) can be seen in Lemma  1 of [2].
From ‖ℎ‖²=‖𝑏+𝑐‖²=‖𝑏‖²+‖𝑐‖²+2𝑏^𝑇𝑐 and 𝑏^𝑇𝑐≥0, we can obtain 𝑏^𝑇𝑐≤(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 𝜇^𝑘=(𝑥^𝑘)^𝑇𝑦^𝑘/𝑛≥𝜇^𝑘, where 𝜇^𝑘=(1−𝑄^𝑘)𝜇^𝑘.

Proof. From Step 2 of our algorithm and 𝑀 is a 𝑛×𝑛 positive semidefinite matrix, we have (Δ𝑥^𝑝)^𝑇Δ𝑦^𝑝=(Δ𝑥^𝑝)^𝑇𝑀Δ𝑥^𝑝≥0. Hence 𝜇^𝑘=𝑥^𝑘𝑇𝑦^𝑘𝑛=𝑥^𝑘+𝑄^𝑘Δ𝑥^𝑝𝑇𝑦^𝑘+𝑄^𝐾Δ𝑦^𝑝𝑛=𝑥^𝑘𝑇𝑦^𝑘+𝑄^𝑘𝑥^𝑘𝑇Δ𝑦^𝑝+Δ𝑥^𝑝𝑇𝑦^𝑘+𝑄^𝑘2Δ𝑥^𝑝𝑇Δ𝑦^𝑝𝑛≥𝑥^𝑘𝑇𝑦^𝑘𝑛+𝑄^𝑘𝑛2𝑛⋅3𝜇^𝑘−𝑥^𝑘𝑇𝑦^𝑘=𝜇^𝑘+23𝑄^𝑘𝜇^𝑘−𝑄^𝑘𝜇^𝑘≥1−𝑄^𝑘𝜇^𝑘=𝜇^𝑘. (3.1)
This completes the proof.

Lemma 3.3. If (𝑥^𝑘,𝑦^𝑘)∈𝜌(1/4), then the point (𝑥^𝑘,𝑦^𝑘) generated by the algorithm satisfies 𝑥^𝑘>0,𝑦^𝑘>0, ‖𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒‖≤(1/2)𝜇^𝑘, and (𝑥^𝑘,𝑦^𝑘)∈𝜌(1/2).

Proof. Let us define (𝑥(𝑄),𝑦(𝑄))=(𝑥^𝑘+𝑄Δ𝑥^𝑝,𝑦^𝑘+𝑄Δ𝑦^𝑝), where 0≤𝑄≤𝑄^𝑘1=min8√𝑛,𝜇^𝑘8Δ𝑋^𝑝Δ𝑦^𝑝. (3.2)
Note that 0≤𝑄≤𝑄^𝑘=min{1/8√𝑛,𝜇^𝑘/8‖Δ𝑋^𝑝Δ𝑦^𝑝‖}, then we have (2/3)√𝑛𝑄≤1/12 from 𝑄≤1/8√𝑛 and 𝑄²‖Δ𝑋^𝑝Δ𝑦^𝑝‖≤(1/8)𝜇^𝑘 from 𝑄≤𝜇^𝑘/8‖Δ𝑋^𝑝Δ𝑧^𝑝‖.
Furthermore ‖𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒‖≤(1/4)𝜇^𝑘 holds since (𝑥^𝑘,𝑦^𝑘)∈𝜌(1/4). Therefore we have 𝑋(𝑄)𝑦(𝑄)−(1−𝑄)𝜇^𝑘𝑒=𝑋^𝑘𝑦^𝑘+𝑄𝑋^𝑝Δ𝑦^𝑝+Δ𝑋^𝑝𝑦^𝑝+𝑄²Δ𝑋^𝑝Δ𝑦^𝑝−(1−𝑄)𝜇^𝑘𝑒=𝑋^𝑘𝑦^𝑘2+𝑄3𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘+𝑄²Δ𝑋^𝑝Δ𝑦^𝑝−(1−𝑄)𝜇^𝑘𝑒≤(1−𝑄)𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒+23√𝑛𝑄𝜇^𝑘+𝑄²Δ𝑋^𝑝Δ𝑦^𝑝≤()11−𝑄4𝜇^𝑘+1𝜇12^𝑘+18𝜇^𝑘=14+111−𝑄+1128(1−𝑄)𝜇^𝑘≤14+5()𝜇211−𝑄^𝑘≤12(1−𝑄)𝜇^𝑘. (3.3)
So 𝑥_𝑖(𝑄)𝑦_𝑖(𝑄)≥(1/2)(1−𝑄)𝜇^𝑘>0(𝑖=1,2,…,𝑛) for any 𝑄(0≤𝑄≤𝑄^𝐾).
Since (𝑥(𝑄),𝑦(𝑄))=(𝑥^𝑘+𝑄Δ𝑥^𝑝,𝑦^𝑘+𝑄Δ𝑦^𝑝) is continuous with respect to 𝑄,𝑥^𝑘>0 and 𝑦^𝑘>0, it is easily seen that 𝑥^𝑘=𝑥(𝑄^𝑘)>0, 𝑦^𝑘=𝑦(𝑄^𝑘)>0.
Note that 𝑋(𝑄^𝑘)=𝑋^𝑘,𝑌(𝑄^𝑘)=𝑌^𝑘,(1−𝑄^𝑘)𝜇^𝑘=𝜇^𝑘, the above argument implies ‖𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒‖≤(1/2)𝜇^𝑘.
It’s well known that the inequality ‖𝑋𝑧−(𝑥^𝑇𝑧/𝑛)𝑒‖≤‖𝑋𝑧−𝑐𝑒‖ holds for arbitrary vector 𝑥,𝑧 and arbitrary constant 𝑐.
So we have ‖𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒‖≤‖𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒‖≤(1/2)𝜇^𝑘≤(1/2)𝜇 from the proof above and Lemma 3.2, thus (𝑥^𝑘,𝑦^𝑘)∈𝜌(1/2).
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 𝜌(1/2).

Lemma 3.5. If ‖𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒‖≤(1/2)𝜇^𝑘 then ‖(𝑋^𝑘𝑌^𝑘)^−1/2(𝑟𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘)‖²≤2(1/4+(1−𝑟)²𝑛)𝜇^𝑘

Proof. 𝑋^𝑘𝑌^𝑘−1/2𝑟𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘2≤1min_{1≤𝑖≤𝑛}𝑥^𝑘_𝑖𝑦^𝑘_𝑖𝑟𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘2≤1(1−1/2)𝜇^𝑘𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒+(1−𝑟)𝜇^𝑘𝑒²≤2𝜇^𝑘𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒²+(1−𝑟)𝜇^𝑘𝑒²1≤24+(1−𝑟)²𝑛𝜇^𝑘. (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 𝜇^𝑘+1≤(1−𝑄^𝑘)𝜇^𝑘 always hold.

Proof. Denote 𝐷=(𝑋^𝑘)^1/2(𝑌^𝑘)^−1/2 then 𝐷⁻¹Δ𝑥^𝑐+𝐷Δ𝑦^𝑐=𝑋^𝑘𝑌^𝑘−1/2𝑟𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘𝐷⁻¹Δ𝑥^𝑐𝑇𝐷Δ𝑦^𝑐=Δ𝑥^𝑐𝑇Δ𝑦^𝑐=Δ𝑥^𝑐𝑇𝑀Δ𝑥^𝑐≥0. (3.5)
From Lemmas 3.1 and 3.5 we haveΔ𝑥^𝑐𝑇Δ𝑦^𝑐≤12𝑋^𝑘𝑦^𝑘−1/2𝑟𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘2≤14+(1−𝑟)²𝑛𝜇^𝑘∴𝜇^𝑘+1=𝑥^𝑘+1𝑇𝑦^𝑘+1𝑛=𝑥^𝑘+Δ𝑥^𝑐𝑇𝑦^𝑘+Δ𝑦^𝑐𝑛=𝑥^𝑘𝑇𝑦^𝑘+𝑥^𝑘𝑇Δ𝑦^𝑐+Δ𝑥^𝑐𝑇𝑦^𝑘+Δ𝑥^𝑐𝑇Δ𝑦^𝑐𝑛=𝑛𝑟𝜇^𝑘+Δ𝑥^𝑐𝑇Δ𝑦^𝑐𝑛≤𝑟𝜇^𝑘+1+()4𝑛1−𝑟²𝜇^𝑘. (3.6)
Note that 𝑟=1−1/4√2𝑛, we have 1𝑟++()4𝑛1−𝑟²1=1−4√+12𝑛+14𝑛4√32𝑛=1−2𝑛−9.32𝑛 (3.7)
It is obvious that (𝑟+1/4𝑛+(1−𝑟)²)<1 when 𝑛>2, so 𝜇^𝑘+1≤𝜇^𝑘=(1−𝑄^𝑘)𝜇^𝑘.
This completes the proof.

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 (𝑥^𝑘+1,𝑦^𝑘+1)∈𝜌(1/4).

Proof. Let us define that 𝑥^𝑘+1(𝛼)=𝑥^𝑘+𝛼Δ𝑥^𝑐,𝑦^𝑘+1(𝛼)=𝑦^𝑘+𝛼Δ𝑦^𝑐, and denote 𝜇^𝑘+1(𝛼)=𝑥^𝑘+1(𝛼)^𝑇𝑦^𝑘+1(𝛼)𝑛. (3.8) As discussed in Lemma 3.6, we also have (𝐷⁻¹Δ𝑥^𝑐)^𝑇(𝐷Δ𝑦^𝑐)=(Δ𝑥^𝑐)^𝑇Δ𝑦^𝑐=(Δ𝑥^𝑐)^𝑇𝑀Δ𝑥^𝑐≥0.
From Lemmas 3.1 and 3.5 and 𝑟=1−1/4√2𝑛, thenΔ𝑋^𝑐Δ𝑦^𝑐≤√24𝑋^𝑘𝑌^𝑘−1/2𝑟𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘2≤√24214+(1−𝑟)²𝑛𝜇^𝑘=14√22+2√2𝑛(1−𝑟)²𝜇^𝑘≤14√22+18𝜇^𝑘≤732𝜇^𝑘,𝜇^𝑘+1(𝛼)=𝑥^𝑘+1(𝛼)^𝑇𝑦^𝑘+1(𝛼)𝑛=𝑥^𝑘+𝛼Δ𝑥^𝑐𝑇𝑦^𝑘+𝛼Δ𝑦^𝑐𝑛=𝑥^𝑘𝑇𝑦^𝑘+𝛼𝑥^𝑘𝑇Δ𝑦^𝑐+Δ𝑥^𝑐𝑇𝑦^𝑘+𝛼²Δ𝑥^𝑐𝑇Δ𝑦^𝑐𝑛≥𝑥^𝑘𝑇𝑦^𝑘+𝛼𝑥^𝑘𝑇Δ𝑦^𝑐+Δ𝑥^𝑐𝑇𝑦^𝑘𝑛=𝑥^𝑘𝑇𝑦^𝑘+𝛼𝑛𝑟𝜇^𝑘−𝑥^𝑘𝑇𝑦^𝑘𝑛=𝜇^𝑘+𝛼𝑟𝜇^𝑘−𝜇^𝑘=(1−𝛼)𝜇^𝑘+𝛼𝑟𝜇^𝑘≥(1−(1−𝑟)𝛼)𝜇^𝑘. (3.9)
So we can write 𝜇^𝑘+1(𝛼)≥(1−(1−𝑟)𝛼)𝜇^𝑘, that is, 𝜇^𝑘≤𝜇^𝑘+1(𝛼)/(1−(1−𝑟)𝛼).
Hence 𝑋^𝑘+1(𝛼)𝑦^𝑘+1(𝛼)−𝜇^𝑘+1(𝛼)𝑒=𝑋^𝑘+𝛼Δ𝑋^𝑐𝑦^𝑘+𝛼Δ𝑦^𝑐−𝑥^𝑘+𝛼Δ𝑥^𝑐𝑇𝑦^𝑘+𝛼Δ𝑦^𝑐𝑛𝑒=𝑋^𝑘𝑦^𝑘+𝛼𝑋^𝑘Δ𝑦^𝑐+Δ𝑋^𝑐𝑦^𝑘+𝛼²Δ𝑋^𝑐Δ𝑦^𝑐−𝑥^𝑘𝑇𝑦^𝑘+𝛼𝑥^𝑘𝑇Δ𝑦^𝑐+Δ𝑥^𝑐𝑇𝑦^𝑘+𝛼²Δ𝑥^𝑐𝑇Δ𝑦^𝑐𝑛𝑒=𝑋^𝑘𝑦^𝑘+𝛼𝑟𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘+𝛼²Δ𝑋^𝑐Δ𝑦^𝑐−𝑥^𝑘𝑇𝑦^𝑘+𝛼𝑛𝑟𝜇^𝑘−𝑥^𝑘𝑇𝑦^𝑘+𝛼²Δ𝑥^𝑐𝑇Δ𝑦^𝑐𝑛𝑒=(1−𝛼)𝑋^𝑘𝑦^𝑘−𝑥^𝑘𝑇𝑦^𝑘𝑛𝑒+𝛼²Δ𝑋^𝑐Δ𝑦^𝑐−Δ𝑥^𝑐𝑇Δ𝑦^𝑐𝑛𝑒. (3.10)
Since the inequality ‖𝑋𝑧−(𝑥^𝑇𝑧/𝑛)𝑒‖≤‖𝑋𝑧−𝑐𝑒‖ holds for arbitrary vector 𝑥,𝑧 and arbitrary constant 𝑐, thus𝑋^𝑘+1(𝛼)𝑦^𝑘+1(𝛼)−𝜇^𝑘+1(𝛼)𝑒≤(1−𝛼)𝑋^𝑘𝑦^𝑘−𝑥^𝑘𝑇𝑦^𝑘𝑛𝑒+𝛼²Δ𝑋^𝑐Δ𝑦^𝑐−Δ𝑥^𝑐𝑇Δ𝑦^𝑐𝑛𝑒≤(1−𝛼)𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒+𝛼²Δ𝑋^𝑐Δ𝑦^𝑐≤()11−𝛼2𝜇^𝑘+𝛼²732𝜇^𝑘≤(1/2)(1−𝛼)+(7/32)𝛼²()𝛼𝜇1−1−𝑟^𝑘+1(𝛼). (3.11)
Let us define 𝑓(𝛼)=((1/2)(1−𝛼)+(7/32)𝛼²)/(1−(1−𝑟)𝛼), then 𝑓^′(𝛼)=(−(1/2)𝑟+(7/16)𝛼−(7/32)𝛼²(1−𝑟))/[1−(1−𝑟)𝛼]².
When 𝑛>2, we have 𝑟≥7/8, this results in 𝑓^′(𝛼)≤0 and 𝑓(𝛼) decreases monotonously, when 0≤𝛼≤1. Because 𝑓(𝛼)≤𝑓(0)=1/2 holds for every 𝛼∈[0,1] thus 𝑋^𝑘+1(𝛼)𝑦^𝑘+1(𝛼)−𝜇^𝑘+1(𝛼)𝑒≤12𝜇^𝑘+1(𝛼). (3.12)
Since 𝜇^𝑘>𝜀>0 (otherwise the algorithm will be terminated), it follows that 𝑋_𝑖^𝑘+1(𝛼)𝑦_𝑖^𝑘+1(𝛼)≥12𝜇^𝑘+1(𝛼)≥12(1−(1−𝑟)𝛼)𝜇^𝑘=12(1−(1−𝑟)𝛼)1−𝑄^𝑘𝜇^𝑘>0. (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,when𝛼=0. (3.14)
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=1, 𝑓(𝛼)=7/32𝑟 thus ‖𝑋^𝑘+1𝑦^𝑘+1−𝜇^𝑘+1𝑒‖≤(7/32𝑟)𝜇^𝑘+1≤(1/4)𝜇^𝑘+1.
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 𝑂(√𝑛𝐿).

Proof. Let 𝐷^′=(𝑋^𝑘)^1/2(𝑌^𝑘)^−1/2 then 𝐷^′−1Δ𝑥^𝑝+𝐷^′Δ𝑦^𝑝=𝑋^𝑘𝑌^𝑘−1/223𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘,𝐷^′−1Δ𝑥^𝑝𝑇𝐷^′Δ𝑦^𝑝=Δ𝑥^𝑝𝑇Δ𝑦^𝑝=Δ𝑥^𝑝𝑇𝑀Δ𝑦^𝑝≥0. (3.15)
From Lemma 3.1, we have Δ𝑋^𝑝Δ𝑦^𝑝=𝐷^′−1Δ𝑋^𝑝𝐷^′Δ𝑦^𝑝≤√24⋅𝑋^𝑘𝑦^𝑘−1/223𝜇^𝑘𝑒−𝑋^𝑘𝑦^𝑘2≤√24⋅^𝑛_𝑖=123𝜇^𝑘2𝑥^𝑘_𝑖⁻¹𝑦^𝑘_𝑖⁻¹−43𝜇^𝑘+𝑥^𝑘_𝑖𝑦^𝑘_𝑖². (3.16)
From ‖𝑋^𝑘𝑦^𝑘−𝜇^𝑘𝑒‖≤(1/4)𝜇^𝑘, we have 𝑥^𝑘_𝑖𝑦^𝑘_𝑖≥(3/4)𝜇^𝑘, then (𝑥^𝑘_𝑖)⁻¹(𝑦^𝑘_𝑖)⁻¹≤4/3𝜇^𝑘, Δ𝑋^𝑝Δ𝑦^𝑝≤√24⋅^𝑛_𝑖=149⋅43⋅𝜇^𝑘−43𝜇^𝑘+𝑛𝜇^𝑘=√24⋅727𝑛𝜇^𝑘=7√2108𝑛𝜇^𝑘. (3.17)
Hence 𝜇^𝑘/8‖Δ𝑋^𝑝Δ𝑦^𝑝‖≥27/14√2𝑛, and we can obtain min{1/8√𝑛,𝜇^𝑘/8‖Δ𝑋^𝑝Δ𝑦^𝑝‖}≥27/14√2𝑛.
So it implies that 𝜇^𝑘+1≤(1−𝑄^𝑘)𝜇^𝑘≤(1−27/14√2𝑛)𝜇^𝑘 for each 𝑘. This means that 𝜇^𝑘 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.

Remark 3.10. Note that the reduce factor in the MTY algorithm is (1−𝜒/√𝑛) where 𝜒=1/√⁴8≈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=𝑀₁𝑥+ℎ₁,(𝑥,𝑦)≥(0,0),𝑥^𝑇𝑦=0, where ℎ₁∈𝑅³ and 𝑀₁ is a 3×3 positive semidefinite matrix, 𝑀₁=213320115,ℎ₁=0.−1−2 (4.1)

Test Problem 2
Find a vector pair (𝑥,𝑦)∈𝑅⁵×𝑅⁵ such that 𝑦=𝑀₂𝑥+ℎ₂,(𝑥,𝑦)≥(0,0),𝑥^𝑇𝑦=0, where ℎ₂∈𝑅⁵ and M₂ is a 5×5 positive semidefinite matrix, 𝑀₂=7002028359003030146180025,ℎ₂=.−9.5−36.5−5−14−18.5 (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)∈𝑅^𝑛×𝑅^𝑛 such that 𝑦=𝑀₃𝑥+ℎ₃,(𝑥,𝑦)≥(0,0),𝑥^𝑇𝑦=0, where ℎ₃∈𝑅^𝑛 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, 𝑀₃=4−20⋯014−2⋯0014⋯0⋮⋮⋮⋮⋮000⋯4,ℎ₃=11⋮11. (4.3)

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.

Table 1 

Computational results of test problem 1.

Table 2 

Computational results of test problem 2.

Table 3 

Computational results of test problem 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.

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).