A MODIFIED PERRY-TYPE DERIVATIVE-FREE PROJECTION METHOD FOR SOLVING LARGE-SCALE NONLINEAR MONOTONE EQUATIONS

In this paper, a new modified Perry-type derivative-free projection method for solving largescale nonlinear monotone equations is presented. The method is developed by combining a modified Perry’s conjugate gradient method with the hyperplane projection technique. Global convergence and numerical results of the proposed method are established. Preliminary numerical results show that the proposed method is promising and efficient compared to some existing methods in the literature. Mathematics Subject Classification. 65K10, 90C06, 90C30, 65K05. Received March 19, 2021. Accepted July 31, 2021.


Introduction
In this paper, we use a modified Perry-type derivative-free projection method to solve the nonlinear monotone equations F (x) = 0, x ∈ Ω, (1.1) where F : R n → R n is continuous and monotone, and Ω ⊆ R n is a nonempty closed convex set. By monotonicity, we mean that (F (x) − F (y)) T (x − y) ≥ 0, ∀x, y ∈ R n .
If Ω = R n then (1.1) is a general system of nonlinear equations problem, and when Ω ⊆ R n is a nonempty closed convex set then (1.1) is said to be constrained. Nonlinear monotone equations have many practical applications such as in chemical equilibrium systems [20] and the economic equilibrium problems [9]. Also, some monotone variational inequality problems can also be converted into nonlinear monotone equations by means of fixed point mappings or normal maps if the underlying function satisfies some coercive conditions [34]. Conjugate gradient-based projection methods are among the most famous and efficient methods for solving (1.1) and, thus, have recently received a lot of attention. This is due to their simplicity, global convergence properties and low memory requirements, which make them suitable for solving large-scale equations [8,16,26]. They are iterative methods, that is, given x k , the next iterate x k+1 is obtained as where z k = x k + α k d k , α k > 0 is a step length and with β k being a conjugate gradient parameter such that where F k = F (x k ) and · is the Euclidean norm. If Ω is a closed convex subset of R n then the projection operator P Ω [·] is a map from R n onto Ω, that is, This operator is non-expansive, that is, for any x, y ∈ R n , Conjugate gradient-based projection methods are obtained by combining conjugate gradient methods [3,5,6,30,32] with the projection technique proposed by Solodov and Svaiter [25]. Thus, they differ according to how the direction is obtained, or, more specifically, in how the parameter β k is constructed. Recently, by employing Perry's conjugate gradient parameter [22], in which or the Dai-Liao conjugate gradient parameter [5], in which where y k−1 = F k − F k−1 and s k−1 = x k − x k−1 , a number of modified forms of the Perry and Dai-Liao methods for nonlinear systems of equations have been proposed [1,4,7,27,28]. These Perry and Dai-Liao methods are based on a quasi-Newton aspect and have been considered to be among the most effective in the context of unconstrained optimization. Note that when t = 1, the Dai-Liao method reduces to the Perry method. In Dai et al. [7], a modified Perry method is combined with the hyperplane technique [25] to give a derivativefree method for solving large-scale nonlinear monotone equations. And in Waziri et al. [28], two enhanced Dai-Liao methods are presented based on two modified spectral coefficients and a revised form of the extended secant condition in [29]. All these methods were shown to be very competitive when compared to some of the existing methods for solving large-scale nonlinear monotone equations.
Abubakar et al., in [2], constructed three-term conjugate gradient projection methods by proposing the direction as In order to satisfy (1.3), they derived the parameter λ k using three different approaches, thus proposing three different conjugate gradient projection methods with and they named the resulting algorithms as M3TCD1, M3TCD2 and M3TCD3, respectively.
Another recently suggested three term derivative-free conjugate gradient-based projection method, which the authors named PDY, is that by Liu and Feng [16]. In this method, the authors proposed a three term derivative-free projection method based on the Dai-Yuan (DY) conjugate gradient parameter This method was shown to have nice convergence properties. And Yan et al., in [31], proposed two derivative-free projection methods based on the three term Hestenes-Stiefel (HS) conjugate gradient method of Zhang et al. [32]. They showed that their methods were also efficient for solving large-scale nonlinear systems of monotone equations. Based on a modified line search, an extension of the scaled conjugate gradient (SCG) method of [3] and the projection technique, Ou and Li [21] proposed a derivative-free SCG-type projection method for nonlinear monotone equations with convex constraints. They showed further that when F in (1.1) is a strongly monotone mapping, then the sequence {x k } generated by their method R-linearly converges to x * ∈ Ω * , where Ω * is the solution set of (1.1). Other gradient-based projection methods for large-scale nonlinear monotone equations can be found in [11-15, 18, 23, 26, 33, 35]. Motivated by the works of [19,30], we propose a descent Perry-type derivative-free projection method for solving large-scale nonlinear monotone equations. This method is presented in the next section. In Section 3, we prove the global convergence of the proposed method followed by the convergence rate in Section 4. Numerical results follow in Section 5. Finally, conclusion is presented in Section 6.

Motivation and the algorithm
In this section, we describe the details of the proposed method. But first, we briefly review the work of Livieris and Pintelas [19] and that of Yao and Ning [30] which motivated this work.
Livieris and Pintelas, in [19], proposed a modified Perry's conjugate gradient method for the unconstrained optimization problem min{f (x) | x ∈ R n }, with f : R n → R being a continuously differentiable function bounded from below. The method generates iterations x k+1 = x k + α k d k using the direction is the gradient of f at x k , and The authors suggested h k be given as where t and r are positive constants, and y k−1 = g k − g k−1 and s k−1 = x k − x k−1 . The parameter β MP k was shown to satisfy the sufficient descent property where c > 0 is a positive constant and that the method is globally convergent. Yao and Ning [30], using a modified symmetric Perry matrix where t k is a positive parameter to be determined, defined their search direction as By minimizing the distance, in the Frobenius norm, between the above Perry matrix and the self-scaling memoryless BFGS matrix [24] they determined an optimal parameter t * k as Hence, using this optimal parameter t * k , the authors suggested an adaptive three term Perry's conjugate gradient method The method was shown to be effective numerically. Now, motivated by the above discussed modified Perry's conjugate gradient methods of [19,30], we propose a modified Perry-type derivative-free projection method for solving (1.1). By taking a careful look at the search direction presented in [19] and the β k parameter in [30], we propose our search direction as , and φ is a positive constant and κ ∈ (0, 1]. With d k given in (1.3) above, we determine the next iterate x k+1 using (1.2), where the step length α k = max{ρ i : i = 0, 1, 2, ...}, ρ ∈ (0, 1], is such that it satisfies We present our proposed method below.

3:
If F k ≤ , then stop. Otherwise, go to Step 4.

7:
Set k = k + 1 and go to Step 3. 8: end for

Global convergence
We now establish the global convergence of the presented Algorithm 1. Throughout the paper, we assume that F k = 0 for all k, otherwise a stationary point has been found. We also assume that the following assumption holds.
is Lipschitz continuous on R n , i.e. there exists a positive constant L such that Note that from the monotonicity of F , we have that This indicates that λ * is positive whenever s k−1 is not zero and hence λ k in (2.1) is well-defined.
Lemma 3.2. The search direction d k generated by Algorithm 1 satisfies the descent condition . Now, for k ≥ 1, we obtain from (2.1) and the relation λ k ≥ κ that Lemma 3.3. Suppose that Assumption 3.1 holds, and let the sequences {x k } and {z k } be generated by Algorithm 1. Then, for any x * ∈ Ω, it holds that Moreover, the sequences {x k } and {z k } are bounded, and Furthermore, it holds that For x * ∈ Ω, we obtain from (1.2) that . By monotonicity of F , we obtain From (3.4)-(3.6), we get that Thus, the sequence { x k − x * } is decreasing and convergent, and hence {x k } is bounded. From (3.4), it follows that which shows that {z k } is also bounded.
Furthermore, it follows from (3.7) that Therefore, since F is continuous, by the Lipschitz condition we get that where ϕ is a positive constant.
Proof. From (3.1) and Cauchy-Schwarz inequality , we obtain And from (2.1), we have From the definitions of u k−1 , w k−1 and s k−1 in (2.1), we get that there exist positive constants and such that Notice that for F k−1 = 0 and s k−1 = 0, we get Hence, there is a constant ω > 0 such that It then follows that From (3.10) we get Lemma 3.5. Let {x k } and {z k } be generated by Algorithm 1. Then Proof. From the line search procedure (2.2), if α k = 1, then α k = ρ −1 α k does not satisfy (2.2). This means that where z k = x k + α k d k . This together with (2.2), (3.1) and (3.9) imply that Thus α k ≥ κρ (L + σγ)ϕ 2 > 0, which gives the desired result. Proof. We assume that (3.12) does not hold, that is, ∃ η > 0 such that F k ≥ η, ∀ k ≥ 0. It then follows from (3.9) that On the other hand, (3.11) implies that α k > 0, ∀k ≥ 0, which contradicts (3.13). Thus, (3.12) holds.

Convergence Rate
From the above discussions, it is evident that the sequence {x k } converges to a solution of problem (1.1). Therefore, we always assume that x k converges to x * as k → ∞, where x * belongs to the solution set Ω * of problem (1.1). To determine the rate of convergence for the proposed algorithm, we also assume the following assumption holds.  Proof. Letx k := arg min{ x k −x :x ∈ Ω * }, which implies thatx k is the closest solution to x k , namely, Denoting x * byx k , it follows from (3.2) that This, together with (3.9) and (4.1), give that forx k ∈ Ω * , Taking κ 2 ≤ 1 σψ , we get that 1−σ 2 ψ 2 κ 4 α 4 k ∈ (0, 1) holds. This implies that the sequence {dist(x k , Ω * )} converges to 0 Q-linearly. Therefore, the whole sequence {x k } converges to x * R-linearly.

Numerical Experiments
In this section, we report some numerical results to test the efficacy of our proposed Algorithm 1, herein denoted as N M P CG. We compare it with other three term derivative-free projection methods that have recently been proposed in the literature. These are the three-term conjugate descent projection method of Abubakar et al. [2], denoted M 3T CD1, the derivative-free iterative method of Liu and Feng [16], denoted P DY , and the partially symmetrical derivative-free Liu-Storey projection method of Liu et al. [18], denoted sLS. All algorithms are coded in MATLAB R2019b and the methods are compared using number of iterations, number of function evaluations and CPU time taken for each method to reach the optimal value or termination. We test the algorithms on eight test problems, with various dimensions, using four different starting points  F 3 (x), ..., F n (x)) T , are as follows.
, for i = 2, 3, ..., n − 1 and Ω = R n + , In our experiments, the algorithms are stopped whenever the inequality F k ≤ = 10 −6 is satisfied, or the total number of iterations exceeds 1000. The N M P CG method is implemented with the parameters σ = 10 −4 , ρ = 0.5, φ = 10 −5 and κ = 10 −5 , while parameters for the algorithms sLS, P DY and M 3T CD1 are set as in respective papers.
The numerical results are reported in Tables 1 and 2, where IT ER refers to the number of iterations, F E stands for the number of function evaluations and CP U is the CPU time in seconds. We note here that all algorithms managed to solve all the eight test problems successfully. From Tables 1 and 2, we see that in terms of number of iterations, the N M P CG and M 3T CD1 methods are very competitive for most problems, and perform much better than the other methods. However, in terms of function evaluations, we see that the proposed N M P CG method is the best. Overall, the results indicate that the proposed N M P CG method performs better than the other competing methods, with the sLS method the weaker one among the four competing methods.    For a comprehensive performance of the methods, we use the performance profiles tool proposed by Dolan and Moré [10] to represent the results in Tables 1 and 2 in figures, based on the number of iterations, number of function evaluations, and CPU time. These are presented in Figures 1-3, respectively. With the Dolan and Moré performance profiles, the graph that is above the others for the most part is regarded as the best solver. We can see from Figure 1 that in terms of number of iterations, the N M P CG and M 3T CD1 methods are very competitive, and perform much better than both the P DY and sLS methods. Figure 2 shows clearly that in terms of function evaluations, the proposed N M P CG method outperforms all the other methods, followed by the M 3T CD1 method. Figure 3, for the CP U performance profiles, shows that the proposed N M P CG is equally efficient.

Conclusion
In this paper, we proposed a new modified derivative-free Perry's conjugate gradient-based projection method for solving systems of large-scale nonlinear monotone equations. Its global convergence and rate of convergence were established. The proposed algorithm was tested on some benchmark problems, with different starting points and dimensions, and the numerical results show that the method is efficient as compared to other methods from the literature. Future work includes extending the new method to solve other kinds of problems like robotic motion control, signal restoration and image deblurring.