A PATH-FOLLOWING INTERIOR-POINT ALGORITHM FOR MONOTONE LCP BASED ON A MODIFIED NEWTON SEARCH DIRECTION

. In this paper, we propose a short-step feasible full-Newton step path-following interior-point algorithm (IPA) for monotone linear complementarity problems (LCPs). The proposed IPA uses the technique of algebraic equivalent transformation (AET) induced by an univariate function to transform the centering equations which defines the central-path. By applying Newton’s method to the modified system of the central-path of LCP, a new Newton search direction is obtained. Under new appropriate defaults of the threshold 𝜏 which defines the size of the neighborhood of the central-path and of 𝜃 which determines the decrease in the barrier parameter, we prove that the IPA is well-defined and converges locally quadratically to a solution of the monotone LCPs. Moreover, we derive its iteration bound, namely, 𝒪 (︀ √ 𝑛 log 𝑛𝜖 )︀ which coincides with the best-known iteration bound for such algorithms. Finally, some numerical results are presented to show its efficiency.


Introduction
After the seminal work of Karmarkar [25] for linear optimization (LO), interior-point methods revitalized as an active area of research in mathematical programming.Among them the class of path-following primaldual IPAs deserved much more attention due to their polynomial complexity and their practical efficiency (see e.g., [2,3,7,22,23,30,34]). Determining a search direction plays an important role in IPAs.In the last decade, several types of search directions have been proposed.Some of them are based on the strategy of so-called self-regular and kernel barrier functions (see e.g., [4,5,8,9,33]).Meanwhile, others are based on the strategy of the AET technique applied to the centering equation of the system which characterizes the central-path.In 2003, Darvay [13], uses the AET technique based on the univariate function () = √  for LO.By means of this function, a new type of Newton direction is obtained and the best iteration bound for feasible short-step IPAs is derived.This method was extended successfully to convex quadratic optimization (CQO) and monotone LCP by Achache (see e.g., [1][2][3]), semidefinite optimization (SDO) and second-order cone optimization (SCOP) by Wang and Bai [35,36].Besides, Kheirfam and Haghighi [28] investigated the AETs based on the function () = et al. [24] presented a generalized direction in interior-point methods for monotone LCP.Their approach is based on AET induced by the class of smooth concave univariate functions.By utilizing the AET technique based on the logarithmic function () = log , Pan et al. [32] presented an infeasible IPA to solve LO.Furthermore, Darvay and Takács [15] developed a new IPA for LO based on a new modified search direction induced by an asymptotic barrier kernel function.The best polynomial complexity is provided.Recently, Darvay et al. [16], proposed an IPA for LO where their search direction is based on AET introduced by the new univariate function () =  − √ .Later, Darvay et al. [17] generalized this algorithm for sufficient LCP.Also we mention that Fischer proposed a damped Newton-type IPA for monotone LCP [20].Here he reformulated the LCP as an equivalent nonlinear system of equations based on the so-called NCP-functions.Under some conditions, the super-linear convergence of this algorithm is established.
For more details about the AETs technique we direct the reader to the papers (see e.g., [14,18,26,37]) and the references therein.
Based on the iteration bound obtained by [29], we notice that if  becomes very large then  gets very small.Consequently, the rate (1 − ) which determines the decrease in the barrier parameter converges to one.This leads to a slow convergence and even to a divergence of their algorithm.So it clear that taking a member of   () =   2 with a large value of  leads to bad numerical results.In this paper, in order to improve the numerical results of these algorithms, we reconsider the analysis of their IPAs designed for LO to monotone LCP where a non parametric univariate function, namely,   () =  5 2 is suggested.Therefore, similar to LO, we use the AET technique introduced by this function to nonlinear equations of the system which defines the central-path of monotone LCPs, a modified nonlinear equations is obtained.The application of Newton method to the latter, a modified search direction is offered.The proposed IPA uses full-Newton steps for tracing approximately the central-path.Unlike LO case, the presence of non orthogonality of scaled directions in LCPs, a different analysis is stated.Further, under new appropriate choices of defaults  and , we prove that the algorithm is well-defined and converges locally quadratically to a solution of monotone LCP (these results are based on the useful Lemma 4.4).Moreover, the currently best known iteration bound for the algorithm with short-update method, namely,  (︀√  log   )︀ is obtained.This complexity is analogue to those achieved by many authors (see e.g.[1-3, 14, 18, 26, 27, 37]).Some numerical results are presented to evaluate our proposed algorithm.In addition, to accelerate the speed of convergence of our original algorithm, some relaxations are imported on the selection of the default .Finally, we compare the performances of our algorithm with a previously Fischer type IPAs on a set of monotone LCPs.
The outline of the paper is as follows.In Section 2, preliminaries notions and the problem description are presented.In Section 3, the modified search directions based on AETs for the centering equation is discussed.Moreover, the generic feasible short-update full-Newton step IPA for monotone LCPs is presented.In Section 4, the analysis of the algorithm is given.Further, its iteration bound is obtained.In Section 5, some numerical results are reported.Also for the performances of our algorithm, some modifications are suggested.In the last section we present a general conclusion of the work carried out, some remarks, as well as some perspectives and suggestions for future work.

Preliminaries and the problem statement
where  ∈ R × is positive semi-definite (PSD) and  ∈ R  .Throughout the paper, we assume that the interior-point condition (IPC) holds for monotone LCP (1), i.e., there exists a pair of vectors ( 0 ,  0 ) such that In this case the monotone LCP (1) has a solution.For more comprehension of LCP, we recommended the monograph of Cottle et al. [11].
The main idea of path-following IPAs is to replace the equation  = 0 in (1) by the parameterized equation  =  where  > 0. Hence, we obtain the system of equations: Under the IPC condition, Kojima et al. [30] shows that system (2) has a unique solution denoted by ((), ()) for each  > 0, which is called the −center of monotone LCP.The set of −centers is called the central-path of monotone LCP.If  goes to zero, then the limit of central-path exists and since the limit point satisfies the complementarity condition  = 0, the limit yields a solution of LCP.Applying Newton's method to system (2) for a given strictly feasible point (, ), i.e., the IPC holds, then the Newton direction (∆, ∆) at this point is the unique solution of the linear system of equations: The system (3) gives the classical Newton search direction for LCP [30,34,41].)︁ = () where  : (0, +∞) → R is a continuously differentiable function and invertible, i.e.,  −1 exists.Then, ( 2) is transformed to the following system:

The modified search directions for LCP
where  is applied coordinate-wisely.Applying Newton's method to system (4) for a given strictly feasible point (, ) yields the new Newton system: where  ′ denotes the derivative of .
Next, to facilitate the analysis of the algorithm, we introduce the following notations: From ( 3) and (6), system (5) can be written as follows: where and M :=  ,  := ().The system (7) determines a family of new scaled Newton search directions related to the function .Now, we consider the AET introduced by the function () =   2 with  = 5.This yields Moreover, system (5) becomes Next, according to (8), we define a norm-based proximity measure as follows: Clearly, () = 0 ⇔  =  ⇔  = .Therefore, the value of () can be considered as a measure of the distance between the given pair (, ) and the central-path.Furthermore, we define the  -neighborhood of the central-path as follows: where  is a threshold (default) and  > 0, is fixed.

The Algorithm
Now we are ready to describe the generic full-Newton step IPA for monotone LCP as follows.First, we use a suitable threshold value, with 0 <  < 5 and we suppose that an initial point ( 0 ,  0 ) ∈  (,  0 ) exists for certain  0 > 0 is known.The full-Newton step between successive iterates is defined as ( + ,  + ) = ( + ∆,  + ∆) where the Newton directions ∆ and ∆ are solutions for linear system (9).Then it updates the parameter  by the factor (1 − ) with 0 <  < 1, and target a new -center and so on.This procedure is repeated until the stopping criterion    ≤  is satisfied for a given accuracy parameter .
Therefore, the generic algorithm the full-Newton step IPA for monotone LCP is stated in Figure 1 as follows.

Analysis of the algorithm
In this section, we are going to show across our new selecting defaults of  and  , described in Figure 1, that Algorithm 3.1 is well-defined and converges locally quadratically to a solution of monotone LCP.Moreover, we prove that our algorithm solves the monotone LCP in polynomial time.We mention that due to the non orthogonality of the scaled direction, our analysis is quite different from those used in LO case.
Next, the following technical results are fundamental tools in our analysis.
Lemma 4.1.Let  > 0 and (  ,   ) be a solution of system (7) with  := (; ).Then one has and Proof.For the first part of (11) we have, The second part of it, follows trivially from the following equality For the first claim in (12), since and On the other hand, • Next, to prove the last claim of ( 12), we have Hence, 2 .This completes the proof.
In the next lemma, we show the feasibility of a full-Newton step under the condition  < 5 throughout the algorithm.
Next, we prove that the iterate across the proximity measure is locally quadratically convergent during the algorithm.
The next lemma shows the influence of a full-Newton step on the duality gap and gives an upper bound for it.
In the next theorem we analyze the effect of a full-Newton step on the proximity by updating the parameter  by a factor (1 − ).
Now, from the triangular inequality it follows that )︁ .A solution of the monotone LCP is denoted by  ⋆ .Finally, the "Iter" and "CPU" denote the number of iterations and the elapsed time in seconds, respectively.

Conclusion and remarks
In this paper, we have presented a feasible full-Newton step IPA for monotone LCPs.For this purpose, we have introduced a function  to the equation which characterizes the central-path of LCPs and we have applied Newton's method to offer new search directions.For () =  5 2 , we have proved its polynomial complexity under appropriate defaults of  and .The evaluation of Algorithm 3.1, across the obtained numerical results, showed its slow convergence and even for the version proposed by [29].We note also that Fischer's algorithm failed for monotone LCPs with size  > 100.However, the import ameliorations made on our algorithm, the new obtained numerical results on NETLIB collection of monotone LCP problems, are significantly improved.Finally, an interesting topic of research in the future is the extension of Algorithm 3.1 to the large class of  * ()-LCPs.Moreover, the development of an infeasible full-Newton step IPA for monotone LCP based on our AET remains a good subject of research.

Following [ 2 ,
13], the AET technique for computing a new search direction for IPAs is based on the transformation of the centrality equation  =  in (2) to the new equation  (︁

Problem 1 .
Consider the following monotone LCP where  ∈ R 5×5 and  ∈ R 5 are given by:

Table 2 .
Number of iterations and CPU time for Problems 1-3.

Table 3 .
Number of iterations and CPU time for NETLIB set problems.