NCP seeks to identify a vector satisfying

represents a nonlinear function from onto itself. This study considers to be continuous and monotone with respect to . Additionally, it is assumed that the solution set for (1), represented by , is not empty.

Richard W. Cottle’s introduction of NCP marked in his Ph.D. thesis during the early 1960s. Since then, complementarity problems have captured the interest of researchers, leading to numerous publications that lay down the essential theoretical foundations of this field (see [3, 4]). A standard approach to addressing the NCP involves identifying a vector such that , where the operator . Here, denotes the normal cone to the nonnegative orthant.

A widely adopted strategy for tackling the NCP is the PPA method. This method starts with an arbitrary and 0 \end{document} ]]> , generates for solving :

with

Recently, numerous studies have focused on developing innovative interior point methods to address NCP. These methods share a common characteristic which enforce the new iterates to stay in the interior of . Auslender, et al. [5] have proposed a new type of proximal interior algorithms via replacing the quadratic function (3) by which could be defined as

Let \mu > 0 \end{document} ]]> be two predetermined constants, let

where is a -divergence function that respects these necessary aspects:

In [6], Auslender, et al. A specialized logarithmic-quadratic proximal (LQP) algorithm has been employed by leveraging by using in the definition of (with .

Later on, Noor and Bnouhachem [7] and [8], have proposed a new modified LQP method by using as a -divergence function (with

Let and , in our contribution, we used the function as proposed in [9] and [2], we get

Assume 0 \end{document} ]]> , the updated iteration of the problem (1) becomes the unique solution of the following set-valued equation:

where

It is evident to see that

and

Now, the problem (4) is equivalent to :

Solving the subproblem (7) exactly presents significant challenges in practice, often excluding practical applications. To mitigate this issue, it is advisable to pursue approximate solutions instead of exact ones. For this reason, we introduce such that :

and satisfies

In this paper, we proposed a prediction-correction method to solve (7) approximately. Numerical results are provided to substantiate the eficacy of the proposed method.

Key properties are essential for our subsequent analysis.

First, we denote

A fundamental characteristic of this projection mapping is :

From (10), one can readily confirm that :

Definition 2.1.The operator is said to be monotone, if

At the kth iteration, Using a three-step SRQP approach, compute the exact solution for the system of equations specified below:

We now introduce an SRQP approach for solving problem (1). For given 0 , D _ { 0 } = 0 \end{document} ]]> and , the suggested approach comprises three steps.

Step 1 : Find of (12), such that

and satisfies

Step 2: For 0 \end{document} ]]> . Compute

and

where

is defined by

where

Step 3: For Compute

and

where

The new updated is

where

Remark 3.1. (14) Leads to the conclusion that

Remark 3.2.Consider the case where ). If F is Lipschitz continuous within , with 0 \end{document} ]]> , i.e.,

satisfying , then the above inequalities (14) are satisfied.

This Lemma is essential in analyzing convergence and plays a pivotal role in this regard.

Lemma 3.3.if we let 0 \end{document} ]]> and , Let x be the positive solution of the following equation :

and where

then we have

Proof.[2] ⊓⊔

Lemma 3.4.[10]Using the definition of and , then

Proof.[2]

Lemma 3.5.[10]For any , we have

Proof.[2]

Theorem 3.6.[11]Let represent any solution of (1) . For given and 0 \end{document} ]]> , let and satisfy the condition (14) , then it holds

Proof.[2]

To guarantee that moves closer to the solution set compared to , we introduce the following definition:

Theorem 3.7.Let and be defined by (30), (21) and (24)respectively, then and 0 \end{document} ]]> , we have

where

Proof.

Using the definition of and , then (6) is proved.

The function evaluates the progress achieved during the kth iteration. A logical choice is to select a step length that maximizes this progress. It is important to note that represents a quadratic function of achieving its maximum at

and

In the following theorem, we demonstrate that both and maintain bounds strictly greater than zero. This result plays a pivotal role in establishing the proof of global convergence.

Theorem 3.8.Given and 0 \end{document} ]]> , let and satisfy the condition

(14) . Under these assumptions, we arrive at the following results :

and

Proof. If , since 0 \end{document} ]]> it follows from (14), (4), (21) and (28) that

from (29) and (38), we obtain

Otherwise, if , it follows that

and thus

To ensure that is closer to the solution set than . For this purpose, we define

Theorem 3.9.Let , then we have

where

Proof.[1].

Remark 3.10.By using Theorem 3 and Theorem 1 in the reference [1], we get

and

From the computational point of view, a relaxation factor is preferable in the new iteration. It follows from (39) and (7) that there is a constant 0 \end{document} ]]> such that

The following result can be proved by similar arguments as those in [1]. Hence the proof will be omitted.

Theorem 3.11. [1, 7] 0 \end{document} ]]> , then the sequence generated by the proposed method converges to some which is a solution of the NCP.

In numerical experiments, determining the value of the approximate solution is essential. In the specific scenario where

equation (13) can be rewritten as an equivalent system of nonlinear equation

hence

Then

The iterative procedure of the Newton method for addressing the specified problem can be outlined as follows:

The solution satisfies 0 \end{document} ]]>. To prevent non-positive values of during the iteration process, we implement the following approach

To test the suggested algorithm, we consider the

where

with representing the nonlinear component and denoting the linear component of

The linear component of the test problems is constructed in a manner similar to the approach outlined by Harker and Pang [4]. Specifically, the matrix M is formed as , with A being an matrix whose entries are randomly selected within the range . Additionally, B is a skew-symmetric matrix generated under the same conditions. The vector q is drawn from a uniform distribution within the interval (−500, 500). Regarding , which represents the nonlinear part of , its components are defined as where is a random variable within the range (0, 1). Problems of a similar nature have been previously explored in [12] and [13].

The iterations begin with and are terminated once the condition

is satisfied. All codes were implemented in Matlab, and the proposed method is compared with those presented in [14]. The test results for problem (44) are summarized in Table 4.1 and Table 4.2. Here, k represents the number of iterations, and refers to the count of mapping calculations for .

Comparison to the method in [14] using only the first and second step of the proposed method:

Comparison to the method in [1] using the three-step proposed method:

CompComparod in [1] using the three-steps proposed method:Comparison to the m

[1] using the three-steps proposed method:

Table 4.1 and Table 4.2 illustrate the enhanced eficiency of the proposed method. The numerical findings reveal that this method substantially decreases the iteration count and computational efort needed to compute the function . This paper introduces a novel category of proximal algorithms aimed at addressing nonlinear complementarity problems, utilizing a novel SRQP term and a two-stage conjugate gradient algorithm. This approach integrates two refined descent directions with two optimal step sizes to achieve multiplicative improvements in each iteration, considerably speeding up the convergence process.