# A New Hybrid Conjugate Gradient Projection Method for Solving Nonlinear Monotone Equations

Authors: Minglei Fang; Min Wang; Defeng Ding
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, we propose a new modified hybrid conjugate gradient projection method with a new scale parameterφk for solving large-scale nonlinear monotone equations. The proposed method includes two major features: projection techniques and sufficient descent property independent of line search technique. Global convergence of the proposed method is proved under some suitable assumptions. Finally, numerical results illustrating the robustness of the suggested strategy and its comparisons are shown.


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## Body

## 1. Introduction

The following system of constrained nonlinear monotone equations is considered:(1)Ht=0,t∈C,where C⊆ℝn is a nonempty closed convex set and H:C⟶ℝn is continuous and monotone function. The monotone property of H means that(2)Ht1−Ht2Tt1−t2≥0,∀t1,t2∈ℝn.In many contemporary domains, some problems can be turned into nonlinear monotonic equation problems, such as variational inequality problems [1], image restoration problems [2], signal reconstruction problems [3], financial forecasting problems [4, 5], and optimal power flow control problems in power [6]. It is very necessary to solve nonlinear monotone equation.At present, there are many ways to solve problem (1), such as the Newtons method [7], Levenberg–Marquardt method [8], and various variants of the method. Although iterative approaches are recognized for their simplicity and good convergence, each iteration of these methods requires a substantial amount of space to compute and store the Jacobian or comparable to the Jacobian, which is not conducive to solving large-scale nonlinear monotone equations. In order to efficiently solve problem (1) and avoid solving a linear system of equations at each iteration, the emergence of the derivative-free method is necessary [9–11].In recent years, a number of researchers have proposed the conjugate gradient method in conjunction with the projection method [12] for solving large-scale monotone nonlinear equations. Sabi’u et al. [13] proposed the Hager–Zhang conjugate gradient method for nonlinear monotone equations using singular value analysis and proposed two adaptive parameter choices: the first by minimizing the Frobenius condition number of the search direction matrix; the other is achieved by minimizing the difference between the maximum and singular values, combined with the projection technique, using some appropriate assumptions to demonstrate that the method satisfies the global convergence, see [14, 15]. Yin et al. [16] proposed a hybrid three-term conjugate gradient projection method, whose search direction is close to the search direction generated by the memoryless BFGS method and possesses a descending characteristic independent of the line search technique as well as a trust region characteristic. Using the adaptive line search technique, the global convergence of the method is established under certain mild conditions, and numerical experiments demonstrate that the method inherits the beneficial properties of the three-term conjugate gradient method and the hybrid conjugate gradient method [17, 18]. Zhou et al. [19] proposed a novel hybrid PRPFR conjugate gradient method with sufficient descent and trust region properties. The method can be considered as a convex combination of the PRP method and the FR method, while using the hyperplane projection technique. In accordance with the acceleration step size, global convergence is attained with the help of some suitable assumptions. Experiments with numbers demonstrate that the PRPFR method is more competitive for solving nonlinear equations and image restoration problems [20, 21].In this paper, we propose a new hybrid conjugate gradient projection method to solve (1). Interestingly, the new search direction has better theoretical properties, that is, it automatically satisfies the sufficient descent condition and the trust region property. The rest of the paper is organized as follows. In Section 2, we detail the motivation for this paper and propose a new algorithm. In Section 3, we demonstrate that the search direction has sufficient descent property and trust region property, and we obtain global convergence under a few moderate conditions. In Section 4, we solve six large-scale nonlinear equations numerically to demonstrate the effectiveness of the proposed method. In Section 5, we draw the conclusion.


## 2. Motivation and Algorithm

This section quickly reviews the conjugate gradient method’s general formula, followed by a new hybrid conjugate gradient approach based on adaptive line search. Finally, we combine the suggested method with the projection technique to solve unconstrained optimization problems.Due to its ease of use and storage, the conjugate gradient approach has been effectively applied to the following unconstrained optimization problems:(3)mint∈ℝnht,where h:ℝn⟶ℝ is a continuously differentiable function.Generally, these methods generate a sequence of iterates recurrently by(4)tk+1=tk+αkdk,k=1,2,…,where αk>0 (it is the step size calculated by performing some suitably precise or imprecise line search) and dk is the search direction defined by(5)dk=−Hk,k=0,−Hk+βkdk−1,k≥1.The termβk is a scalar known as the conjugate gradient parameter. Different choices for the conjugate gradient update βk lead to different conjugate gradient methods. There are some famous conjugate gradient methods such as the FR method [22], the PRP method [23], the DY method [24], the LS method [25], the HS method [26], the CD method [27], and so on. The parameters we mentioned are as follows:(6)βkFR=Hk2Hk−12,βkPRP=HkTyk−1Hk−12,βkDY=Hk2dk−1Tyk−1,βkLS=HkTyk−1−Hk−1Tdk−1,βkHS=HkTyk−1dk−1Tyk−1,βkCD=Hk2−Hk−1Tdk−1,where yk−1=Hk−Hk−1 and the symbol ⋅ stands for the Euclidean norm.An important class of conjugate gradient methods, hybrid conjugate gradient methods, has been proposed by many scholars, in order to obtain a conjugate gradient method with better performance than the classical one. For example, Djordjević [28] combined the LS method and FR method by convex combination form to get conjugate gradient parameters:(7)βhyb=1−φkβkLS+φkβkFR,using the good practical performance of the LS method and the strong convergence of the FR method.Soon after, Zhou et al. [19] gave a variant of the PRP method and FR method:(8)βkMPRP=HkTyk−1maxμdkyk−1,Hk−12,βkMFR=Hk2maxμdkHk,Hk−12,where μ>0. Similarly, define a new parameter(9)βkPRPFR=1−φkβkMPRP+φkβkMFR.In [29], based on the MMWU method [30] and RMAR method [31], Fanar and Ghada proposed a new hybrid conjugate gradient method (HFG) as follows:(10)βkHFG=1−φkHk2dk−12+φkHk2−Hk/dk−1HkTdk−1dk−12.This technique is a convex mixture of two types of conjugate gradient algorithms, and the search direction very easily satisfied sufficient descent, besides conforming to Newton direction under appropriate conditions. The global convergence of the proposed method can be established when a strong Wolfe line search is performed. Shockingly, this hybrid method not only performs better than the classical conjugate gradient method but also outperforms some complex conjugate gradient methods in many problems.Inspired by above, a new convex combination is proposed as follows:(11)βkWF=1−φkβk1+φkβk2,(12)βk1=HkTwk−1maxHk−12,μdk−1wk−1,βk2=Hk2μdk−12+Hk2,where wk−1=yk−1+Hk−1sk−1,sk−1=rk−1−tk−1,μ>1, and the choice of the parameter φk satisfies the conjugate condition in each iteration: dkTyk−1=0.Clearly,(13)dk=−Hk+1−φkβk1dk−1+φkβk2dk−1.Multiplying from the right both sides of the transposed equation byyk−1, we get(14)0=−HkTyk−1+1−φkβk1dk−1Tyk−1+φkβk2dk−1Tyk−1,and with some mathematical calculation, we obtain(15)φk=−HkTyk−1+βk1dk−1Tyk−1βk1dk−1Tyk−1−βk2dk−1Tyk−1.Ifφk is outside the interval [0, 1], to maintain the convex combination in (11), it can be fixed by(16)φk=0,if−HkTyk−1+βk1dk−1Tyk−1βk1dk−1Tyk−1−βk2dk−1Tyk−1≤0,1,if−HkTyk−1+βk1dk−1Tyk−1βk1dk−1Tyk−1−βk2dk−1Tyk−1≥1,−HkTyk−1+βk1dk−1Tyk−1βk1dk−1Tyk−1−βk2dk−1Tyk−1,else.For line search, the key is to obtain the step size at the lowest cost. Therefore, the most widely used line search proposed by Solodov and Svaiter [12] was used, for which the step size is computed as αk=maxaρi:i=0,1,… such that(17)−Htk+αkdkTdk≥σαkHtk+αkdkdk2.Another commonly used technique is the line search technique proposed by Zhang and Zhou [32], and the step size αk=maxaρi:i=0,1,… satisfied the following inequality:(18)−Htk+αkdkTdk≥σαkdk2.Recently, an adaptive line search approach that takes into consideration a disturbance component was presented by Liu et al. [33] and Guo and Wan [34], i.e., the step size αk=maxaρi:i=0,1,… satisfied the following inequality:(19)−Htk+αkdkTdk≥σαkγkdk2,where γk=Htk+αkdk/maxHtk+αkdk,ν and ν≥1. If Htk+αkdk≥ν, then γk=Htk+αkdk/Htk+αkdk=1; otherwise, if Htk+αkdk<ν, then γk=Htk+αkdk/ν<1.To establish the algorithm, we introduce the definition of projection operator. LetC be a nonempty closed convex set of ℝn; then,(20)PCt=argmint−y|y∈C,∀t∈ℝn.Also, it satisfies the nonexpansive property(21)PCt−PCy≤t−y,∀t,y∈ℝn.We now go into great depth about our algorithm, based on the abovementioned preliminaries.


## 3. Convergence Analysis

In order to show the global convergence of Algorithm1, the following assumptions need to be established:(H 3.1) The solution set of problem (1) is nonempty.(H 3.2) The mappingH is Lipschitz continuous on ℝn, i.e., there exists a constant L>0 such that(22)Ht1−Ht2≤Lt1−t2,∀t1,t2∈ℝn.Algorithm 1:
Step 0. Select an initial point t0∈C and the parameters ε>0, σ,ρ∈0,1,m∈0,2,a>0,μ>1,ν≥1. Let k=0.Step 1. Compute Htk, if Htk≤ε, stop. Otherwise, go to Step 2.Step 2. Compute dk by (5)–(12), and choose αk that satisfies inequality (19).Step  3. Set rk=tk+αkdk. If Hrk≤ε, stop and let tk+1=rk. Otherwise, compute the next iterate tk+1 by tk+1=PCtk−m⋅ξkHrk,whereξk=HrkTtk−rk/Hrk2.Step 4. Set k:=k+1, and go to Step 1.The following lemma demonstrates that the direction of the search satisfies sufficient descent condition and possesses trust region property independent on any line search.Lemma 1.
For allk≥0, if the search direction dk is generated by Algorithm 1, then the following two properties can be satisfied:(23)HkTdk≤−1−1μHk2,(24)dk≤21+1μHk.Proof.
Fork=0, it is easy to know that H0Td0=−H02 and (23) and (24) hold.
Fork≥1, it can be broken down into three cases for discussion:(i)
Firstly, ifφk=0, then βkWF=βk1=HkTyk−1/maxHk−12,μdk−1yk−1.IfHk−12≤μdk−1wk−1, then βkWF=HkTwk−1/μdk−1wk−1, and if Hk−12>μdk−1wk−1, then βkWF=HkTwk−1/Hk−12<HkTwk−1/μdk−1wk−1. It follows that βkWF=βk1≤HkTwk−1/μdk−1wk−1.Multiplying both sides of (5) by HkT, from (11) and (12), we get(25)HkTdk=−Hk2+βk1HkTdk−1≤−Hk2+HkTwk−1μdk−1wk−1HkTdk−1≤−Hk2+Hkwk−1μdk−1wk−1Hkdk−1=−1−1μHk2.According to (5), it is easy to obtain that(26)dk=−Hk+βk1dk−1≤−Hk+βk1dk−1≤Hk+HkTwk−1μdk−1wk−1dk−1=1+1μHk≤21+1μHk.(ii)
Secondly, ifφk=1, then βkWF=βk2=Hk2/μdk−12+Hk2.As in the case of the first case, we get(27)HkTdk=−Hk2+βk2HkTdk−1≤−Hk2+Hk2μdk−12+Hk2Hkdk−1≤−Hk2+Hk2μHkdk−1Hkdk−1=−1−1μHk2,(28)dk=−Hk+βk2dk−1≤−Hk+βk2dk−1≤Hk+Hk2μHkdk−1dk−1=1+1μHk≤21+1μHk.(iii)
Finally, for0<φk<1 there exist a1, a2 in which 0<a1≤φk≤a2<1, and we can simply rewrite formula (5):(29)dk=−Hk+βkWFdk−1=−Hk+1−φkβk1+φkβk2dk−1=−φkHk+1−φkHk+1−φkβk1+φkβk2dk−1=φk−Hk+βk2dk+1−φk−Hk+βk1dk.Then, we can get the relationship amongdk,dk1, and dk2:(30)dk=φkdk2+1−φkdk1.From (26) and (28), we have(31)dk≤a2dk2+1−a1dk1≤a21+1μHk+1−a11+1μHk≤1−a1+a21+1μHk≤21+1μHk.Obviously, the descending property can be obtained by combining the step relationship (30) with (25) and (27), and we get(32)HkTdk=φkHkTdk2+1−φkHkTdk1≤a1HkTdk2+1−a2HkTdk1≤a11μ−1Hk2+1−a21μ−1Hk2=−1−a2+a11−1μHk2≤−1−1μHk2.
The proof is completed.
The following lemma shows that Algorithm1 can get a step size and stop within finite steps, which shows that this line search is reasonable.Lemma 2.
If assumptions (H3.1) and (H3.2) hold, then Algorithm1 stops with a finite number of iterations.Proof.
Suppose that the assumptions are invalid or that Algorithm1 is not terminated. In this case, there exists a constant ε0>0 such that(33)Hk≥ε0,∀k∈N∪0.
Suppose there existsk′, so that line search (19) does not hold. Then, for all m∈N∪0, let αk′m=aρm, and we have(34)−Hrk′Tdk′<σαk′mdk′2Hrk′maxHrk′,ν≤σαk′mdk′2,where rk′=tk′+αk′mdk′. It can be obtained by assumption (H3.2) and formula (23) that(35)1−1μHk′2≤−Hk′Tdk′≤Hrk′−Htk′Tdk′−Hrk′Tdk′<Lαk′mdk′2+σαk′mdk′2=ρ−1αk′mL+σdk′2.
Using (24), we get(36)αk′m>1−1/μHk′2ρ−1L+σdk′2>μ−12ρ−1L+σμ+1>0.
This is contradictory to the definition ofαk′m. Thus, line search (19) can attain a positive step size in finite steps and we complete the proof.
The following outcomes are required in order to establish that Algorithm1 has achieved global convergence.Lemma 3.
If assumptions (H3.1) and (H3.2) hold, lettk and rk be the sequences generated by Algorithm 1, and t∗ is the solution of problem (1); for all t∗∈S, the sequence tk−t∗ converges. Then, we have(37)limk⟶∞αkdk=0.Proof.
First, according to the definition ofrk and line search, we have(38)HrkTtk−rk=−αkHrkTdk≥σαk2dk2Htk+αkdkmaxHtk+αkdk,ν=σtk−rk2Htk+αkdkmaxHtk+αkdk,ν.
From (2) and t∗∈S, the following relation holds:(39)HrkTtk−t∗=HrkTtk−rk+HrkTrk−t∗≥HrkTtk−rk+Ht∗Trk−t∗=HrkTtk−rk≥σtk−rk2.
Then, combining the relation above with equations in Algorithm1 and (21), we come to the conclusion that(40)tk+1−t∗2=PCtk−ξkHrk−PCt∗2≤tk−ξkHrk−t∗=tk−t∗2−2ξkHrkTtk−t∗+ξk2Hrk2≤tk−t∗2−2ξkHrkTtk−rk+ξk2Hrk2=tk−t∗2−HrkTtk−rk2HrkT2≤tk−t∗2−σ2tk−rk4Hrk2Htk+αkdkmaxHtk+αkdk,ν2.
(40) shows the relation 0≤tk+1−t∗≤tk−t∗, which means that the sequence tk−t∗ is decreasing and bounded. Thus, the sequence tk−t∗ is convergent, and the sequence tk is bounded. Combined with assumption (H3.2), we have(41)Htk=Htk−Ht∗≤Ltk−t∗≤Lt0−t∗.
Therefore,Htk has an upper bound. Through Lemma 1 and the continuity of H, it is obtained that dk is bounded; furthermore, sequence rk is bounded. This shows that there is a constant M>0 such that(42)Hrk≤M,∀k≥0.
LetHtk+αkdk/maxHtk+αkdk,ν=c1, where 0<c1≤1.
This together with (40) implies(43)∑k=0∞c12σ2Mtk−rk4≤∑k=0∞tk−t∗2−tk+1−t∗2=t0−t∗2−limk⟶∞tk+1−t∗2<∞.
Sincerk=tk+αkdk, the above formula implies that limk⟶∞αkdk=limk⟶∞tk−rk=0.
On the basis of the lemma above, we show that the algorithm is globally convergent under certain conditions.Theorem 1.
If assumptions (H3.1) and (H3.2) hold, let sequencedk and tk be generated by Algorithm 1. Then, we have(44)limk⟶∞infHk=0.Proof.
It can be proved by contradiction. Assuming that formula (44) does not hold, then there is a constant η>0 such that Hk≥η,∀k≥0. According to HkTdk=ξHkdk, where −1≤ξ≤1, and formula (23), we get(45)−Hkdk≤HkTdk≤−1−1μHk2.
Moreover,(46)dk≥1−1μHk2≥1−1μη.
Combining the above formula with Lemma3, it is easy to show that(47)limk⟶∞αk=0.
We know that the sequencestk and dk are bounded by Lemma 3 and formula (24), so there is a cluster point t˜ and an infinite index set N1 such that limk⟶∞tk=t˜, and there is a clustering point d˜ and an infinite index set N2 such that limk⟶∞dk=d˜. There exists a α^k=ρ−1αk in line search (19) such that(48)−Htk+α^kdkTdk<σα^kdk2Htk+α^kdkmaxHtk+α^kdk,ν≤σα^kdk2.
Taking the limit of both sides of the above equation, for allk∈N2, it turns out that(49)Ht˜Td˜>0.
Taking the limit of both sides of (19), we get(50)Ht˜Td˜≤0.
Obviously, the above two formulas are contradictory. Therefore, the assumption does not hold, that is,limk⟶∞infHk=0.


## 4. Numerical Results

In this section, in order to ensure the effectiveness of the proposed algorithm (Algorithm1), specific numerical experiments are given below. We denote the proposed Algorithm 1 by WF, and compare with some existing algorithms. The conjugate gradient method in [35, 36, 37, 38] are denoted by JKL, EMDY, PDY, and HG, respectively. These are derivative-free methods, where JKL and EMDY methods are the state-of-the-art methods. Among them, the JKL method, PDY method, and HG method use the line search of formula (17), and the EMDY method uses the line search of formula (18). All codes used were written in Matlab R2015a and run on PC with 4 GB of RAM and Windows 10 operating system.For different methods, we choose the optimal parameters:WF:ρ=0.5,σ=0.0001,a=1,m=1.5,mu=3,ν=1.25.JKL:ρ=0.55,σ=0.0001,a=1,m=1.55,c=0.6.EMDY:ρ=0.8,σ=0.0001,a=1,m=1.2.PDY:ρ=0.5,σ=0.01,a=1,ξ=0.1,c=1.HG:ρ=0.4,σ=0.0001,a=1,γ=1.We choose five methods whose termination condition isHk≤1×10−6 or NG+NI>10000.The test questions are as follows:Problem 1 [11]: set hit=eti−2, for i=1,2,…,n and C=ℝ+n.Problem 2 [34]: set nonsmooth functionhit=2ti−sinti, for i=1,2,…,n and C=ℝ+n.Problem 3 [11]: set logarithmic function hit=lnti+1−ti/n, for i=1,2,…,n and C=ℝ+n.Problem 4 [33]: set(51)h1t=2t1+sint1−1,hit=2t1+2ti−1+sinti−1,fori=2,3,…,n−1,hnt=2tn+sintn−1,C=ℝ+n.Problem 5 [33]: set(52)h1t=t1−ecosht1+t2,hit=ti−ecoshti−1+ti+ti+1,fori=2,3,…,n−1,hnt=tn−ecoshtn−1+tn,whereh=1/n+1 and C=ℝ+n.Problem 6 [33]: set hit=eti2+3sinticosti−1, for i=1,2,…,n and C=ℝ+n.Problems 1–6 use six initial points:t1=0.5,0.5,…,0.5T,t2=1,1,…,1T,t3=1.5,1.5,…1.5T,t4=2,2,…,2T,t5=1/21,1/22,…,1/2nT,t6=1/n,2/n,…,1T. In addition, choose 5000,10000, 150000,20000, and 30000 as the dimension of the problem.In addition, we compare the efficacy of the five different approaches by utilizing the performance characteristics that Dolan and Moré [39] have provided. Assuming we have ns solvers and np test problems, compare the performance of solving s∈S on the problem p∈P with the best performance of any solver on the problem. This method is used to compare and measure how well solver set S works on test set P. The comparison between different solvers is based on(53)rp,s=tp,smintp,s:s∈S.In order to obtain the performance curve of each solvers, ρsτ is defined as the probability of solver s, that is, the performance ratio of rp,s is within the factor τ∈R of the best possible ratio. Its performance is superior to that of the following distribution functions:(54)ρsτ=1npsizep∈P:log2rp,s≤τ,where size A is the number of items in set A and τ≥0. In the event that the solver s is unable to find a solution to the problem p, we will adjust the ratio rp,s to a value that is sufficiently high.Tables1–6 display all numerical findings derived from the five approaches. In the tables, “Dim” represents the dimension of the problem, “Initial” represents the initial point, “—” indicates that the method failed to converge inside the iteration termination condition, “NI” refers to the number of iterations, “NF” refers to the number of function evaluation, “CPU” denotes the CPU time, and “H∗” refers to the final value of Hk when the program is stopped. In problem 3, the MEDY method outperforms the WF method at some initial points; however, in problem 6, the MEDY method cannot find the corresponding solution for initial point t6. In problem 4, the PDY method is superior to the WF method in the number of iterations. It can also be observed that in most of the remaining problems, the FW method has better performance than other methods in terms of the number of iterations, the number of function evaluation and CPU time.Table 1
Numerical results on problem 1.
DimInitialWFJKLEMDYHGPDYNF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗5000t14/9/0.028/7.24E − 1014/30/0.041/7.92E − 075/25/0.029/8.93E − 0811/23/0.017/8.66E − 0716/48/0.036/5.83E − 07t24/10/0.020/4.54E − 1215/32/0.044/4.02E − 076/31/0.030/2.24E − 0811/23/0.020/2.69E − 0718/55/0.039/4.18E − 07t34/10/0.023/5.15E − 0816/34/0.046/3.65E − 075/28/0.028/7.16E − 0811/24/0.019/3.29E − 0718/55/0.043/7.27E − 07t45/13/0.023/3.22E − 1115/33/0.044/8.64E − 076/33/0.031/8.18E − 0813/29/0.024/3.04E − 0718/56/0.042/7.83E − 07t512/30/0.045/7.39E − 0727/55/0.078/7.24E − 0710/50/0.050/2.22E − 0821/43/0.038/8.62E − 0718/54/0.038/8.69E − 07t614/36/0.052/5.88E − 0734/72/0.094/8.95E − 0713/68/0.067/2.50E − 0825/51/0.043/9.69E − 0719/57/0.038/3.92E − 0710000t14/9/0.021/1.02E − 0915/32/0.073/3.66E − 075/25/0.044/1.26E − 0712/25/0.035/2.45E − 0716/48/0.059/8.25E − 07t24/10/0.021/6.44E − 1215/32/0.076/5.69E − 076/31/0.055/3.17E − 0811/23/0.034/3.81E − 0718/55/0.068/5.91E − 07t34/10/0.021/7.28E − 0816/34/0.085/5.17E − 075/28/0.047/1.01E − 0711/24/0.035/4.65E − 0719/58/0.072/3.69E − 07t45/13/0.027/4.56E − 1116/35/0.083/3.99E − 076/33/0.059/1.16E − 0713/29/0.043/4.30E − 0719/59/0.071/3.98E − 07t512/30/0.062/6.81E − 0730/62/0.144/5.33E − 079/47/0.081/2.23E − 0722/45/0.064/9.08E − 0719/57/0.070/4.42E − 07t614/36/0.072/8.32E − 0736/76/0.175/6.30E − 0713/68/0.117/1.04E − 0726/53/0.080/2.50E − 0719/57/0.072/5.54E − 0715000t14/9/0.027/1.25E − 0915/32/0.111/4.48E − 075/25/0.065/1.55E − 0712/25/0.056/3.00E − 0717/51/0.093/3.63E − 07t24/10/0.029/7.86E − 1215/32/0.110/6.97E − 076/31/0.078/3.88E − 0811/23/0.050/4.66E − 0718/55/0.093/7.24E − 07t34/10/0.033/8.91E − 0816/34/0.114/6.33E − 075/28/0.070/1.24E − 0711/24/0.052/5.69E − 0719/58/0.105/4.53E − 07t45/13/0.039/5.59E − 1116/35/0.119/4.89E − 076/33/0.082/1.42E − 0713/29/0.057/5.27E − 0719/59/0.105/4.88E − 07t511/27/0.080/4.10E − 0730/62/0.217/6.58E − 078/41/0.096/3.44E − 0723/47/0.103/8.39E − 0719/57/0.102/5.41E − 07t615/38/0.111/1.37E − 0737/78/0.269/4.12E − 0713/68/0.170/1.96E − 0726/53/0.111/2.97E − 0719/57/0.100/6.78E − 0720000t14/9/0.036/1.45E − 0915/32/0.146/5.18E − 075/25/0.086/1.79E − 0712/25/0.070/3.46E − 0717/51/0.116/4.19E − 07t24/10/0.036/9.08E − 1215/32/0.140/8.05E − 076/31/0.103/4.48E − 0811/23/0.064/5.39E − 0718/55/0.126/8.36E − 07t34/10/0.040/1.03E − 0716/34/0.151/7.31E − 075/28/0.089/1.43E − 0711/24/0.062/6.57E − 0719/58/0.130/5.23E − 07t45/13/0.048/6.46E − 1116/35/0.156/5.65E − 076/33/0.110/1.64E − 0713/29/0.078/6.09E − 0719/59/0.131/5.64E − 07t510/25/0.100/4.78E − 0722/48/0.211/4.37E − 078/41/0.128/3.65E − 0724/49/0.135/7.96E − 0719/57/0.125/6.25E − 07t615/38/0.150/1.59E − 0736/77/0.341/2.62E − 0713/68/0.233/2.87E − 0726/53/0.143/3.38E − 0719/57/0.129/7.83E − 0730000t14/9/0.050/1.77E − 0915/32/0.208/6.34E − 075/25/0.123/2.19E − 0712/25/0.102/4.24E − 0717/51/0.172/5.13E − 07t24/10/0.055/1.11E − 1115/32/0.208/9.86E − 076/31/0.148/5.49E − 0811/23/0.091/6.60E − 0719/58/0.195/3.68E − 07t34/10/0.056/1.26E − 0716/34/0.222/8.95E − 075/28/0.134/1.75E − 0711/24/0.097/8.05E − 0717/53/0.175/4.09E − 07t45/13/0.073/7.90E − 1116/35/0.227/6.92E − 076/33/0.158/2.00E − 0713/29/0.117/7.45E − 0721/69/0.230/6.83E − 07t512/29/0.160/2.37E − 0726/56/0.358/7.75E − 078/41/0.200/2.88E − 0723/47/0.192/4.52E − 0719/57/0.186/7.66E − 07t615/38/0.218/1.94E − 0735/75/0.503/3.67E − 0713/68/0.326/4.59E − 0726/53/0.219/4.08E − 0719/57/0.195/9.59E − 07Table 2
Numerical results on problem 2.
DimInitialWFJKLEMDYHGPDYNF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗5000t12/4/0.004/02/4/0.004/04/10/0.009/1.12E − 0630/60/0.037/3.56E − 2018/37/0.022/3.79E − 07t22/4/0.004/02/4/0.004/01/3/0.002/031/62/0.037/7.49E − 2018/37/0.024/7.61E − 07t31/3/0.003/01/3/0.002/01/4/0.003/032/64/0.036/5.62E − 2119/39/0.027/3.63E − 07t41/3/0.003/01/3/0.002/01/4/0.003/033/66/0.038/9.36E − 2218/38/0.023/6.14E − 07t56/8/0.009/02/4/0.003/05/11/0.008/6.46E − 0826/53/0.024/9.52E − 0714/29/0.013/3.69E − 07t610/13/0.014/3.12E − 0717/19/0.024/1.97E − 076/13/0.014/7.93E − 0735/71/0.044/6.36E − 0718/37/0.022/4.42E − 0710000t12/4/0.007/02/4/0.006/04/10/0.016/1.59E − 0630/60/0.072/8.47E − 2018/37/0.041/5.36E − 07t22/4/0.006/02/4/0.007/01/3/0.004/036/73/0.074/8.79E − 0719/39/0.044/3.87E − 07t31/3/0.005/01/3/0.004/01/4/0.005/032/64/0.068/1.19E − 1918/38/0.039/4.76E − 07t41/3/0.004/01/3/0.005/01/4/0.005/031/62/0.069/2.81E − 1920/43/0.046/5.29E − 07t56/8/0.013/02/4/0.004/05/11/0.012/6.46E − 0826/53/0.037/9.52E − 0714/29/0.023/3.69E − 07t610/13/0.029/4.40E − 0717/19/0.047/2.78E − 077/15/0.019/4.49E − 0832/64/0.076/7.55E − 0718/37/0.044/6.25E − 0715000t12/4/0.010/02/4/0.008/04/10/0.024/1.95E − 0633/66/0.091/3.00E − 2018/37/0.061/6.56E − 07t22/4/0.010/02/4/0.011/01/3/0.006/030/60/0.097/1.26E − 1819/39/0.058/4.74E − 07t31/3/0.006/01/3/0.006/01/4/0.007/037/75/0.111/8.14E − 0720/43/0.066/5.57E − 07t41/3/0.007/01/3/0.007/01/4/0.008/031/62/0.103/4.21E − 1920/43/0.065/6.48E − 07t56/8/0.015/02/4/0.006/05/11/0.017/6.46E − 0826/53/0.048/9.52E − 0714/29/0.028/3.69E − 07t610/13/0.035/5.38E − 0717/19/0.063/3.40E − 077/15/0.025/5.49E − 0836/73/0.112/6.61E − 0718/37/0.053/7.65E − 0720000t12/4/0.011/02/4/0.013/04/10/0.029/2.25E − 0636/73/0.139/6.99E − 0718/37/0.087/7.58E − 07t22/4/0.012/02/4/0.013/01/3/0.008/037/74/0.212/2.04E − 2019/39/0.080/5.47E − 07t31/3/0.008/01/3/0.008/01/4/0.009/037/75/0.148/9.40E − 0720/43/0.084/6.43E − 07t41/3/0.008/01/3/0.009/01/4/0.010/030/60/0.121/2.84E − 1921/48/0.096/6.22E − 07t56/8/0.018/02/4/0.008/05/11/0.023/6.46E − 0826/53/0.065/9.52E − 0714/29/0.037/3.69E − 07t610/13/0.046/6.21E − 0717/19/0.075/3.92E − 077/15/0.034/6.34E − 0836/73/0.149/7.63E − 0718/37/0.076/8.84E − 0730000t12/4/0.017/02/4/0.017/04/10/0.041/2.75E − 0636/73/0.212/8.56E − 0718/37/0.127/9.28E − 07t22/4/0.018/02/4/0.019/01/3/0.012/037/75/0.218/9.13E − 0720/43/0.128/5.80E − 07t31/3/0.010/01/3/0.011/01/4/0.013/030/60/0.170/1.61E − 1820/43/0.122/7.88E − 07t41/3/0.011/01/3/0.011/01/4/0.016/038/77/0.224/7.43E − 0721/48/0.148/7.62E − 07t56/8/0.027/02/4/0.012/05/11/0.029/6.46E − 0826/53/0.092/9.52E − 0714/29/0.054/3.69E − 07t610/13/0.064/7.59E − 0717/19/0.119/4.81E − 077/15/0.048/7.77E − 0836/73/0.210/9.34E − 0719/39/0.109/3.89E − 07Table 3
Numerical results on problem 3.
DimInitialWFJKLEMDYHGPDYNF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗5000t11/2/0.002/01/2/0.002/02/3/0.004/04/5/0.005/1.42E − 0717/34/0.026/5.90E − 07t21/2/0.002/01/2/0.003/02/3/0.004/05/6/0.007/6.26E − 0918/36/0.030/7.24E − 07t34/5/0.008/4.30E − 072/3/0.005/02/3/0.005/05/6/0.008/4.05E − 0719/38/0.029/5.33E − 07t45/6/0.011/1.46E − 083/4/0.006/03/4/0.006/06/7/0.010/2.36E − 0919/38/0.030/9.08E − 07t512/13/0.016/5.73E − 0812/13/0.016/1.49E − 076/7/0.009/7.48E − 1318/28/0.019/9.83E − 0713/26/0.014/9.16E − 07t616/17/0.025/4.64E − 0717/18/0.029/5.45E − 076/7/0.011/022/34/0.029/8.49E − 0718/36/0.028/4.02E − 0710000t11/2/0.004/01/2/0.004/02/3/0.007/04/5/0.010/9.73E − 0817/34/0.049/8.32E − 07t21/2/0.003/01/2/0.003/02/3/0.007/05/6/0.011/3.62E − 0919/38/0.053/3.68E − 07t34/5/0.014/3.11E − 072/3/0.007/02/3/0.008/05/6/0.012/2.93E − 0719/38/0.056/7.52E − 07t45/6/0.015/8.98E − 093/4/0.012/03/4/0.013/06/7/0.015/1.24E − 0920/41/0.056/6.32E − 07t512/13/0.023/4.90E − 0812/13/0.027/1.61E − 076/7/0.014/1.46E − 0819/30/0.035/7.46E − 0713/26/0.022/9.15E − 07t616/17/0.045/7.36E − 0717/18/0.055/7.73E − 077/8/0.019/020/30/0.054/5.07E − 0718/36/0.055/5.67E − 0715000t11/2/0.005/01/2/0.006/02/3/0.011/04/5/0.014/7.97E − 0818/36/0.183/3.66E − 07t21/2/0.004/01/2/0.004/02/3/0.009/05/6/0.016/2.74E − 0919/38/0.082/4.50E − 07t34/5/0.019/2.66E − 072/3/0.011/02/3/0.011/05/6/0.017/2.50E − 0720/41/0.092/4.58E − 07t45/6/0.024/6.97E − 093/4/0.017/03/4/0.017/06/7/0.020/8.93E − 1021/44/0.084/6.56E − 07t512/13/0.034/4.61E − 0812/13/0.034/1.65E − 077/8/0.021/3.06E − 1220/32/0.050/6.46E − 0713/26/0.035/9.15E − 07t616/17/0.068/9.46E − 0717/18/0.079/9.49E − 077/8/0.029/020/30/0.071/4.83E − 0718/36/0.081/6.94E − 0720000t11/2/0.007/01/2/0.007/02/3/0.013/04/5/0.019/6.98E − 0818/36/0.106/4.23E − 07t21/2/0.005/01/2/0.005/02/3/0.014/05/6/0.020/2.28E − 0919/38/0.098/5.19E − 07t34/5/0.026/2.41E − 072/3/0.014/02/3/0.014/05/6/0.023/2.27E − 0721/44/0.112/4.50E − 07t45/6/0.030/5.88E − 093/4/0.022/03/4/0.022/06/7/0.026/7.23E − 1021/44/0.112/7.57E − 07t512/13/0.044/4.47E − 0812/13/0.051/1.67E − 077/8/0.027/3.85E − 1220/32/0.058/8.44E − 0713/26/0.043/9.15E − 07t617/18/0.092/8.96E − 0718/19/0.109/6.50E − 077/8/0.038/021/32/0.096/7.83E − 0718/36/0.089/8.02E − 0730000t11/2/0.009/01/2/0.010/02/3/0.020/04/5/0.026/5.88E − 0818/36/0.149/5.18E − 07t21/2/0.007/01/2/0.008/02/3/0.019/05/6/0.030/1.78E − 0919/39/0.150/9.43E − 07t34/5/0.036/2.16E − 072/3/0.021/02/3/0.021/05/6/0.037/2.02E − 0721/44/0.167/5.51E − 07t45/6/0.044/4.69E − 093/4/0.033/03/4/0.034/06/7/0.036/5.50E − 1021/44/0.174/9.26E − 07t512/13/0.064/4.32E − 0812/13/0.068/1.70E − 077/8/0.044/3.98E − 1222/34/0.085/2.26E − 0913/26/0.075/9.15E − 07t618/19/0.137/7.90E − 0818/19/0.152/7.96E − 077/8/0.058/6.51E − 0823/36/0.149/6.18E − 0718/36/0.145/9.81E − 07Table 4
Numerical results on problem 4.
DimInitialWFJKLEMDYHGPDYNF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗5000t127/92/0.090/8.52E − 0748/144/0.144/8.73E − 0735/300/0.220/5.76E − 0738/115/0.066/9.98E − 0724/101/0.062/5.04E − 07t227/93/0.092/5.73E − 0743/133/0.131/8.44E − 0735/301/0.221/8.76E − 0737/112/0.068/9.38E − 0725/105/0.054/5.69E − 07t330/101/0.100/6.30E − 0751/151/0.149/8.29E − 0747/391/0.302/4.77E − 0738/115/0.067/8.12E − 0723/95/0.051/8.61E − 07t432/107/0.102/9.94E − 0743/131/0.138/8.29E − 0743/361/0.279/4.89E − 0737/112/0.071/8.88E − 0728/118/0.063/3.54E − 07t529/98/0.089/8.87E − 0739/124/0.114/4.82E − 0750/416/0.311/5.70E − 0732/97/0.057/8.85E − 0722/92/0.061/4.92E − 07t627/92/0.085/7.19E − 0738/121/0.115/8.74E − 0776/629/0.485/6.69E − 0733/100/0.061/6.77E − 0723/96/0.049/8.90E − 0710000t125/86/0.157/7.71E − 0744/135/0.255/8.78E − 0733/284/0.415/6.82E − 0738/115/0.127/8.79E − 0721/87/0.104/6.60E − 07t228/95/0.160/8.78E − 0743/134/0.243/9.47E − 0731/269/0.371/7.53E − 0737/112/0.126/9.17E − 0723/96/0.093/9.35E − 07t337/122/0.212/9.40E − 0740/127/0.235/6.04E − 0736/307/0.442/5.32E − 0737/112/0.121/8.42E − 0724/100/0.103/6.65E − 07t435/116/0.206/9.76E − 0741/128/0.235/7.82E − 0737/314/0.443/5.27E − 0737/112/0.122/7.76E − 0724/100/0.100/8.62E − 07t527/92/0.166/9.26E − 0737/122/0.222/3.70E − 0731/269/0.376/6.02E − 0733/100/0.111/7.70E − 0723/96/0.098/8.08E − 07t625/86/0.218/7.42E − 0741/129/0.351/9.46E − 0731/268/0.561/7.68E − 0738/115/0.184/8.01E − 0722/92/0.136/7.26E − 0715000t128/95/0.237/9.71E − 0741/129/0.354/7.73E − 0733/285/0.590/6.80E − 0738/115/0.181/7.81E − 0725/105/0.156/7.75E − 07t231/104/0.261/9.76E − 0743/132/0.364/9.62E − 0733/283/0.587/5.68E − 0737/112/0.191/8.88E − 0723/95/0.140/8.67E − 07t335/117/0.303/5.68E − 0740/125/0.340/9.44E − 0734/291/0.597/5.58E − 0737/112/0.190/7.44E − 0725/106/0.150/9.66E − 07t427/92/0.241/5.07E − 0733/108/0.288/8.33E − 0737/316/0.657/6.70E − 0733/100/0.158/6.95E − 0722/91/0.131/8.12E − 07t527/92/0.234/9.12E − 0736/119/0.310/6.57E − 0735/301/0.616/5.44E − 0733/100/0.161/8.38E − 0723/96/0.143/5.30E − 07t629/99/0.433/6.08E − 0746/140/0.483/9.32E − 0746/388/1.064/8.32E − 0738/115/0.242/8.67E − 0723/97/0.174/9.81E − 0720000t129/99/0.329/6.96E − 0740/125/0.440/7.82E − 0735/301/0.826/6.31E − 0737/112/0.235/9.79E − 0723/95/0.178/7.88E − 07t235/117/0.385/4.74E − 0737/120/0.417/6.96E − 0731/268/0.740/5.95E − 0737/112/0.312/8.48E − 0724/102/0.208/7.98E − 07t336/119/0.393/9.75E − 0736/114/0.398/9.22E − 0732/275/0.744/5.80E − 0737/112/0.246/7.37E − 0730/130/0.252/3.72E − 07t425/86/0.281/7.24E − 0730/104/0.348/8.35E − 0735/300/0.821/7.13E − 0733/100/0.208/7.48E − 0723/96/0.183/4.52E − 07t527/92/0.311/8.88E − 0738/124/0.437/7.51E − 0738/324/0.901/5.22E − 0733/100/0.210/8.95E − 0723/96/0.175/6.19E − 07t624/83/0.328/9.44E − 0733/112/0.466/6.08E − 0733/284/0.958/7.53E − 0733/100/0.260/7.93E − 0723/96/0.219/3.89E − 0730000t129/98/0.495/8.30E − 0739/125/0.644/8.74E − 0732/276/1.124/9.77E − 0737/112/0.353/9.54E − 0721/87/0.255/6.96E − 07t236/120/0.585/6.17E − 0741/128/0.682/7.81E − 0746/381/1.554/5.96E − 0737/112/0.354/8.22E − 0722/91/0.254/5.56E − 07t339/128/0.630/6.93E − 0738/120/0.616/8.52E − 0739/332/1.336/6.66E − 0736/109/0.344/9.84E − 0723/97/0.268/9.14E − 07t438/126/0.623/5.47E − 0736/118/0.603/6.67E − 0739/332/1.369/6.36E − 0737/112/0.352/8.23E − 0728/121/0.326/7.65E − 07t524/83/0.392/7.69E − 0733/111/0.592/8.48E − 0732/276/1.123/7.89E − 0733/100/0.323/8.33E − 0722/91/0.259/7.99E − 07t627/92/0.470/8.56E − 0736/121/0.621/4.89E − 0741/347/1.422/4.98E − 0733/100/0.321/9.87E − 0723/96/0.267/7.77E − 07Table 5
Numerical results on problem 5.
DimInitialWFJKLEMDYHGPDYNF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗5000t13/5/0.009/6.07E − 0713/15/0.040/1.75E − 076/13/0.024/6.42E − 0737/75/0.086/9.71E − 0719/39/0.052/5.91E − 07t23/5/0.010/5.66E − 0713/15/0.039/1.94E − 076/13/0.026/4.97E − 0737/75/0.088/7.52E − 0719/39/0.048/4.58E − 07t33/5/0.009/4.70E − 0712/14/0.037/9.98E − 076/13/0.024/3.53E − 0736/73/0.098/8.88E − 0718/37/0.042/9.02E − 07t43/5/0.011/3.17E − 0712/14/0.037/7.20E − 076/13/0.024/2.08E − 0735/71/0.080/8.73E − 0718/37/0.045/5.32E − 07t53/5/0.009/5.09E − 0713/15/0.041/5.86E − 076/13/0.024/7.87E − 0738/77/0.090/7.14E − 0719/39/0.046/7.24E − 07t65/8/0.018/1.24E − 1017/19/0.053/7.48E − 076/13/0.025/6.48E − 0737/75/0.088/9.79E − 0719/39/0.046/5.96E − 0710000t13/5/0.018/1.52E − 0711/13/0.063/8.12E − 076/13/0.045/9.09E − 0738/77/0.176/8.24E − 0720/43/0.094/7.99E − 07t23/5/0.018/1.42E − 0711/13/0.063/8.84E − 076/13/0.048/7.04E − 0738/77/0.167/6.38E − 0719/39/0.084/6.47E − 07t33/5/0.018/1.17E − 0711/13/0.063/8.15E − 076/13/0.048/4.99E − 0737/75/0.175/7.54E − 0719/39/0.084/4.59E − 07t43/5/0.018/7.93E − 0810/12/0.058/8.49E − 076/13/0.047/2.94E − 0736/73/0.165/7.41E − 0718/37/0.086/7.52E − 07t53/5/0.017/1.27E − 0711/13/0.064/6.72E − 077/15/0.052/4.45E − 0839/79/0.185/6.06E − 0721/47/0.099/6.91E − 07t64/7/0.025/5.62E − 0714/16/0.098/6.89E − 076/13/0.047/9.16E − 0738/77/0.178/8.31E − 0720/43/0.091/8.06E − 0715000t13/5/0.026/6.75E − 0811/13/0.092/5.86E − 077/15/0.082/4.45E − 0839/79/0.265/6.05E − 0721/47/0.149/6.91E − 07t23/5/0.026/6.30E − 0811/13/0.093/5.32E − 076/13/0.069/8.62E − 0738/77/0.249/7.81E − 0720/43/0.136/7.58E − 07t33/5/0.025/5.22E − 0811/13/0.090/4.44E − 076/13/0.067/6.11E − 0737/75/0.254/9.23E − 0719/39/0.121/5.62E − 07t43/5/0.024/3.53E − 0810/12/0.088/8.90E − 076/13/0.068/3.60E − 0736/73/0.251/9.07E − 0718/37/0.119/9.21E − 07t53/5/0.023/5.66E − 0811/13/0.089/6.39E − 077/15/0.074/5.45E − 0839/79/0.261/7.42E − 0721/47/0.149/8.46E − 07t64/7/0.036/3.06E − 0715/17/0.123/8.73E − 077/15/0.079/4.49E − 0839/79/0.259/6.10E − 0721/47/0.143/6.97E − 0720000t13/5/0.033/3.79E − 0811/13/0.121/5.98E − 077/15/0.104/5.14E − 0839/79/0.337/6.99E − 0721/47/0.198/7.98E − 07t23/5/0.033/3.54E − 0811/13/0.122/4.89E − 076/13/0.093/9.95E − 0738/77/0.327/9.02E − 0720/43/0.187/8.76E − 07t33/5/0.032/2.94E − 0811/13/0.120/3.71E − 076/13/0.090/7.06E − 0738/77/0.364/6.40E − 0719/39/0.168/6.49E − 07t43/5/0.033/1.98E − 0811/13/0.118/2.36E − 076/13/0.090/4.16E − 0737/75/0.325/6.29E − 0719/39/0.166/3.83E − 07t53/5/0.032/3.18E − 0811/13/0.123/7.10E − 077/15/0.102/6.30E − 0839/79/0.338/8.56E − 0721/47/0.187/9.77E − 07t63/5/0.033/7.94E − 0713/15/0.141/5.08E − 077/15/0.106/5.18E − 0839/79/0.342/7.05E − 0721/47/0.194/8.04E − 0730000t13/5/0.048/1.69E − 0811/13/0.181/7.04E − 077/15/0.155/6.29E − 0839/79/0.501/8.56E − 0721/47/0.283/9.77E − 07t23/5/0.051/1.57E − 0811/13/0.175/5.50E − 077/15/0.153/4.88E − 0839/79/0.513/6.63E − 0721/47/0.278/7.57E − 07t33/5/0.045/1.31E − 0811/13/0.177/3.94E − 076/13/0.131/8.64E − 0738/77/0.497/7.84E − 0720/43/0.268/7.60E − 07t43/5/0.045/8.82E − 0911/13/0.181/2.36E − 076/13/0.129/5.10E − 0737/75/0.485/7.70E − 0719/39/0.254/4.69E − 07t53/5/0.042/1.41E − 0811/13/0.177/8.59E − 077/15/0.153/7.71E − 0840/81/0.520/6.29E − 0724/59/0.343/3.67E − 07t63/5/0.047/4.32E − 0713/15/0.212/8.94E − 077/15/0.156/6.35E − 0839/79/0.508/8.63E − 0721/47/0.284/9.85E − 07Table 6
Numerical results on problem 6.
DimInitialWFJKLEMDYHGPDYNF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗NF/NI/CPU/H∗5000t11/2/0.003/01/2/0.004/01/2/0.004/01/2/0.002/014/57/0.056/8.20E − 07t21/5/0.009/01/6/0.009/01/3/0.006/01/3/0.003/0.00E + 0013/53/0.050/8.96E − 07t31/2/0.004/01/2/0.004/01/2/0.004/01/2/0.002/015/62/0.060/5.10E − 07t41/3/0.006/01/8/0.014/01/3/0.006/014/46/0.048/2.18E − 0716/69/0.068/4.89E − 07t51/5/0.004/3.14E − 1610/41/0.034/5.99E − 073/14/0.014/8.24E − 1118/55/0.026/6.90E − 0711/45/0.021/8.47E − 07t617/61/0.093/2.18E − 0813/54/0.071/3.83E − 07—/—/—/—23/71/0.069/7.31E − 0716/66/0.061/6.73E − 0710000t11/2/0.006/01/2/0.007/01/2/0.007/01/2/0.004/015/61/0.112/2.97E − 07t21/5/0.016/01/6/0.019/01/3/0.010/01/3/0.006/0.00E + 0014/57/0.099/3.25E − 07t31/2/0.006/01/2/0.008/01/2/0.008/01/2/0.004/016/68/0.119/4.32E − 07t41/3/0.009/01/8/0.027/01/3/0.012/014/46/0.094/3.09E − 0716/69/0.123/6.91E − 07t51/5/0.005/3.14E − 1610/41/0.054/5.99E − 073/14/0.022/8.24E − 1118/55/0.046/6.90E − 0711/45/0.031/8.47E − 07t618/65/0.171/1.50E − 0713/54/0.140/5.42E − 07—/—/—/—24/74/0.123/2.28E − 0716/66/0.111/9.54E − 0715000t11/2/0.008/01/2/0.010/01/2/0.010/01/2/0.006/015/61/0.167/3.63E − 07t21/5/0.021/01/6/0.029/01/3/0.015/01/3/0.009/0.00E + 0015/63/0.163/8.73E − 07t31/2/0.009/01/2/0.011/01/2/0.010/01/2/0.007/016/68/0.186/5.29E − 07t41/3/0.016/01/8/0.037/01/3/0.017/014/46/0.138/3.78E − 0717/77/0.208/8.15E − 07t51/5/0.008/3.14E − 1610/41/0.076/5.99E − 073/14/0.025/8.24E − 1118/55/0.069/6.90E − 0711/45/0.047/8.47E − 07t618/65/0.259/1.72E − 0713/54/0.204/6.64E − 07—/—/—/—24/74/0.182/2.79E − 0717/70/0.1667/2.99E − 0720000t11/2/0.011/01/2/0.013/01/2/0.014/01/2/0.008/015/61/0.226/4.19E − 07t21/5/0.028/01/6/0.038/01/3/0.023/01/3/0.011/0.00E + 0016/67/0.230/2.58E − 07t31/2/0.013/01/2/0.015/01/2/0.015/01/2/0.009/016/68/0.235/6.11E − 07t41/3/0.019/01/8/0.050/01/3/0.020/014/46/0.180/4.37E − 0718/83/0.302/6.07E − 07t51/5/0.010/3.14E − 1610/41/0.102/5.99E − 073/14/0.031/8.24E − 1118/55/0.090/6.90E − 0711/45/0.068/8.47E − 07t618/65/0.331/1.74E − 0713/54/0.275/7.66E − 07—/—/—/—24/74/0.259/3.23E − 0717/70/0.2252/3.46E − 0730000t11/2/0.016/01/2/0.022/01/2/0.022/01/2/0.011/015/61/0.365/5.14E − 07t21/5/0.044/01/6/0.052/01/3/0.030/01/3/0.016/0.00E + 0016/67/0.341/3.16E − 07t31/2/0.018/01/2/0.022/01/2/0.020/01/2/0.013/016/71/0.372/7.58E − 07t41/3/0.027/01/8/0.076/01/3/0.033/014/46/0.264/5.35E − 0718/83/0.444/7.44E − 07t51/5/0.013/3.14E − 1610/41/0.144/5.99E − 073/14/0.043/8.24E − 1118/55/0.128/6.90E − 0711/45/0.091/8.47E − 07t618/65/0.481/5.58E − 0713/54/0.397/9.39E − 07—/—/—/—24/74/0.367/3.96E − 0717/70/0.3493/4.24E − 07As shown in Figures1, compared with JKL, EMDY, HG, and PDY methods, the proposed method achieves about 70%, 55%, and 43% wins in terms of the number of iterations, function evaluation, and CPU time, respectively. These demonstrate that when compared to the JKL and MEDY methods, the FW method is more robust in terms of the number of iterations, the number of function evaluation, and CPU time.Figure 1
(a) Performance profile on NG. (b) Performance profile on NI. (c) Performance profile on CPU.
(a)(b)(c)


## 5. Conclusion

In this paper, we suggested a way to solve nonlinear monotone equations without using derivatives. This method was a combination of the hybrid conjugate gradient method and the hyperplane projection method. An adaptive technique was used to get the search directiondk. We showed that the proposed method had global convergence under appropriate conditions. This technique worked well for solving large-scale monotone equations since it required a very small amount of memory. Preliminary numerical findings demonstrated the viability of our strategy. In the future, more experiments will be conducted to prove the performance of the proposed algorithm and we will try to extend the algorithm to practical large-scale nonlinear problems.



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*Source: 5540504-2022-11-14.xml*