Figures
Abstract
Conjugate Gradient (CG) methods are widely used for solving large-scale nonlinear systems of equations arising in various real-life applications due to their efficiency in employing vector operations. However, the global convergence analysis of CG methods remains a significant challenge. In response, this study proposes scaled versions of CG parameters based on the renowned Barzilai-Borwein approach for solving convex-constrained monotone nonlinear equations. The proposed algorithms enforce a sufficient descent property independent of the accuracy of the line search procedure and ensure global convergence under appropriate assumptions. Numerical experiments demonstrate the efficiency of the proposed methods in solving large-scale nonlinear systems, including their applicability to accurately solving the inverse kinematic problem of a 3DOF robotic manipulator, where the objective is to minimize the error in achieving a desired trajectory configuration.
Citation: Ibrahim SM, Muhammad L, Yunus RB, Waziri MY, Kamaruddin SbA, Sambas A, et al. (2025) The global convergence of some self-scaling conjugate gradient methods for monotone nonlinear equations with application to 3DOF arm robot model. PLoS ONE 20(1): e0317318. https://doi.org/10.1371/journal.pone.0317318
Editor: Carlos Alberto Cruz-Villar, CINVESTAV IPN: Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional, MEXICO
Received: October 5, 2024; Accepted: December 24, 2024; Published: January 24, 2025
Copyright: © 2025 Ibrahim et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All data are available in the manuscript.
Funding: This research was supported by the Ministry of Higher Education (MoHE) of Malaysia through the Fundamental Research Grant Scheme (FRGS/1/2021/STG06/UUM/02/4) with S/O code 20123. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Conjugate gradient (CG) methods are among the efficient family of optimization algorithms, particularly suited for large-scale nonlinear problems [1–3]. These methods are often defined to solve problems of the form:
(1)
where
is a continuous and monotone function, and
is nonempty, closed, and convex. The monotonicity of F implies that
(2)
The general CG algorithm is designed to solve the constrained problem (1) via the following iterative formula;
(3)
where αk > 0 is the step length which is obtained using a suitable line search strategy and dk is the search direction generated by:
(4)
From the search direction, the coefficient βk is a scalar termed as CG parameter and Fk = F(xk) defines the function value of F at xk. The choices of βk influence the of any CG method and thus, many researchers have studied effective choices of βk (see; [4–24]).
Among the modifications available in the literature, the Riavie-Mamat-Ismail-Leong (RMIL) [25] is considered one of the effective CG coefficients with a restart feature. Rivaie et al. [25] constructed the RMIL formula as follows:
(5)
The author further presented the modification of RMIL called MRMIL [26] with a formula defined as follows:
(6)
and proved the global convergence of the method under suitable assumptions. Results from numerical experiments demonstrate the efficiency of the MRMIL method on a set of unconstrained optimization problems. Motivated by the nice convergence features of the RMIL methods, several researchers have developed other versions of the formula (see; [27–32]).
Previous studies have shown that advanced optimization techniques can significantly enhance the performance of robotic systems. For instance, Kim et al., [33] introduced a class of Newton-type algorithms for optimizing robot motions that consider dynamics. This allows for the formulation of the exact gradient and Hessian of a dynamics-involved objective function, leading to efficient second-order Newton-type optimization algorithms for optimal robot motion generation. In the context of robotic control, Yoshikawa [34] highlighted the importance of accurate dynamic modeling and optimization for achieving precise and efficient robot movements. Lee et al., [35] describe Newton and quasi-Newton minimization procedures for dynamics-based robot movement generation by modelling the considered robot as rigid multi-body systems containing redundant actuators and sensors, active and passive joints, and multiple closed loops. Although it is theoretically possible to derive the equations of motion for such systems analytically, their complexity renders classical optimization methods impractical for anything but the simplest systems. Specifically, numerical approximation of the gradient and Hessian frequently results in ill-conditioning and poor convergence.
Recently, Yahaya et al., [36, 37] constructed some quasi-Newton schemes to meet the requirements of the weak secant equation. These methods are further extended to solve the 3DOF planar robot arm manipulator. All these methods are based on Newton and Quasi-Newton updates. Studies have shown that the Newton method faces many challenges when employed to solve large-scale problems. This is because it is nearly impossible to compute the Jacobian matrix exactly in each iteration. On the other hand, the quasi-Newton methods can inherit some appropriate theoretical and numerical properties from the Newton method since they approximate the Jacobian matrix [38]. However, its matrix operations can also be computationally expensive when solving large-scale problems, thus, the quasi-Newton approaches are also not as effective in handling these kinds of issues.
The CG methods appear to be an appropriate alternative for handling large-scale problems since their algorithms solely employ vector operations. Betts [39] in their work on trajectory optimization, demonstrated the efficiency of CG methods in solving large-scale problems in aerospace applications. Similarly, Nocedal and Wright [40] provided a comprehensive overview of optimization methods, including CG, emphasizing their applicability to a wide range of engineering problems. However, as previously stated, the global convergence analysis of CG methods is typically not easy to achieve. Hence, motivated by the above discussion and by exploring the advantages of the RMIL conjugate gradient procedure, the Quasi-Newton direction, the Quasi-Newton equation, and the famous Barzilai-Borwein strategy, the study proposed a family of scaled CG parameters for a monotone nonlinear system of equations. The main features of the new algorithms are as follows:
- Introducing two self-scaling matrix-free CG algorithms based on the famous Barzilai-Borwein strategy.
- Show that the proposed methods enforce a sufficient descent property irrespective of the line search procedure.
- Prove the global convergence of the new methods using monotonicity condition and Lipschitz continuous of the function F.
- Demonstrate practical implementation of the proposed methods for trajectory planning and real-time control in the 3DOF arm robot. This will highlight the benefits of self-scaling CG algorithms for complex robotic systems.
- Comparing the efficiency and robustness of existing CG algorithms in terms of residual Error and computational efficiency.
The study’s remaining sections are organized as follows. The suggested algorithms utilizing the scaling strategy will be covered in the following Section, and Section 3 presents the method’s global convergence under appropriate assumptions. Section 4 will present the numerical experiments, and Section 5 will include the conclusion.
2 A scaled RMIL and MRMIL CG algorithms
This section will present scale RMIL and MRMIL CG algorithms for monotone nonlinear equations. The RMIL and MRMIL CG parameters are given by
(7)
and MRMIL [26] with the following formula:
(8)
The scaling strategy has been an effective procedure for enhancing the theoretical performances of the CG algorithms, see [41–44]. In line with these developments, we proposed a scale RMIL and MRMIL CG parameters as
(9)
and
(10)
such that |γk| ≤ 1 and |λk| ≤ 1. Gilbert and Nocedal [45] defined a new CG formula and show that any CG method with parameter
in which |γk| ≤ 1, satisfies the SDC [46]. The primary contribution of this study is determining the value of the scalars γk and λk at each iteration.
2.1 Computing the scaling parameter based on Barzilai-Borwein approach
This section constructs the optimal scaled choice parameter γk that incorporates some important features of the famous Barzilai-Borwein (BB) [47] procedure including the spectral parameters whose formulas are defined as as
(11)
Now, considering Perry’s point of view [48], the scaled RMIL and MRMIL directions can be obtained as
(12)
where Ak+1 is the search direction matrix defined by
(13)
By taking advantage of the BB approach [47], the study constructed the formula for γk as the solution to the minimization problem of the form:
(14)
with ‖.‖F representing the Frobenius matrix norm and
(15)
where the 0 < λmin < λmax < ∞. If we assigned M = Ak+1 − λI and using the relation
, then
(16)
Differentiating this with respect to γk and equating it to zero will produce:
(17)
solving for γk, we obtained the following solution for (14) as
(18)
We suggested the following revised form of (18), satisfying
(19)
In a similar approach we can determine the scaling parameter in (10) using (8) as follows;
Let us define
(20)
where the search direction matrix is define by
The study also constructed the formula for λk as the solution to the minimization problem of the form:
(21)
with ‖.‖F representing the Frobenius matrix norm and
(22)
where the 0 < γmin < γmax < ∞
Differentiating with respect to λk and settting the result equal to zero we obtained
Solving for λk we obtained the following MRMIL scaling parameter
(25)
and suggested the following revised form of (25), satisfying
:
(26)
Additionally, to guarantee that the sufficient descent condition is fulfilled regardless of the line search method, the study defined the scaled RMIL and MRMIL CG directions by
(27)
and
(28)
where
(29)
Algorithm 1: The scaled RMIL projection-based CG algorithm (SRMILCG)
Step 1: Input and σ ∈ (0, 1), μ ∈ (0, 1), ν > 0, κ > 0, 0 < γmin < γmax and by setting k = 0 and computing d0 = −F0.
Step 2: If ||Fk|| ≤ ϵ, stop; if not, move on to Step 3.
Step 3: Obtain the dk via (27) or (28), with yk = F(wk) − Fk + σsk and sk = vk − xk.
Step 4: Set wk = xk + αkdk and calculate αk = max{νμi i = 1, 2, 3…} to satisfy
(30)
Step 5 If wk ∈ ψ and ||F(wk)|| = 0 stop, otherwise
(31)
where P is the projection operator, and
(32)
Step 6: Proceed to step 2 after setting that k = k + 1.
3 Convergence analysis
In this section, we will show the global convergence of the scaled RMIL algorithm, assuming that F is monotonic and Lipschitz continuous.
Lemma 3.1 The line search in (30) is well-defined for all k ≥ 0.
Proof: Assume there exists k0 ≥ 0 under the condition that for i = 0, 1, 2, …
(33)
Let i → ∞ in (33), then,
(34)
Since dk defined by (27) or (28) satisfy the SDC, it follows that:
(35)
Hence, Eqs (34) and (35) lead to a contradiction, thereby completing the proof.
Lemma 3.2 [44] If the sequence {xk} and {uk} are given by the SRMILCG method, then, for certain M > 0, it implied:
(36)
proof: From line search (30), we get:
(37)
Let x* ∈ Rn satisfying F(x*) = 0, by the monotonicity of F, it implies:
(38)
(39)
Utilizing (37) and (38) to get:
(40)
By using the Cauchy Schwartz inequality in (40) and the definition of zk, we obtain:
(41)
hence, we get:
(42)
Thus, the sequence {||xk − x*||} is decreasing and therefore convergent. Next, using (45) and the Lipschitz continuity of F, it follows that:
(43)
By Cauchy Schwartz inequality, monotonicity of F, and (30), it follows that:
(44)
which gives
(45)
We can observe that from (45) the sequence {wk} is bounded. Again, from (41), it follows that:
(46)
This implies
(47)
which completes the proof.
Theorem 3.1 The proposed algorithm 1 demonstrates convergence, indicating that
(48)
Proof: Suppose that (49) does not hold true, i.e., for r > 0,
(49)
Also from Lipschitz continuity of F it can be deduce that where L is the taken as the Lipschitz constant. Thus,
(50)
(51)
(52)
Now, let therefore from (52) we have
(53)
Hence, for
the boundedness of the proposed direction is achieved. Similarly, (28) can also be shown to be bounded. From the famous Cauchy-Schwartz inequality and sufficient condition, it follows that:
(54)
From (54) we get
(55)
Thus, From (47) and (55) we obtain
(56)
Suppose there exists not satisfying the search strategy (30), that is
(57)
Obviously, the sequence {xk} is bounded and has an infinite index set J1 and an accumulation point satisfying
It also implies boundedness for
. Consequently, an infinite index set J2 ⊂ J1 with an accumulation point
exists, and
is satisfied by it.
After that, if we use the limit in (57), then,
(58)
once more putting a limit on (54)
(59)
The last two inequalities are contradictory, which completes the proof.
4 Numerical experiments
This section demonstrates the numerical efficiency of the to proposed algorithms through comparison to two existing algorithms, namely Algorithm 2.1a [49] and Rana [50]. The comparisons are based on the number of iterations (NI), the number of function evaluations (NF), and the CPU time. The parameter values for the proposed algorithms are set as ρ = 0.9, σ = 10−4, while the parameter values for Algorithm 2.1a are adopted from [49] and the parameters for Rana method are also adopted from [50]. Each method is implemented in MATLAB R2023a and executed on a PC equipped with an Infinix laptop featuring an Intel Core i7 processor, 32.0 GB of RAM, and a high-performance configuration. The termination criteria for the four algorithms are set as ‖Fk‖ ≤ 10−5.
The benchmark problems considered for the experiments are presented below with dimensions of n = 1000;5, 000;10, 000;50, 000;100, 000, and five different initial points: x1 = (1, 1, 1,…, 1)T, x2 = (0.6, 0.6,…, 0.6)T, x3 = (0.5, 0.5,…, 0.5)T, x4(0.4, 0.4,…, 0.4)T, x5(0.1, 0.1,…, 0.1)T.
Problem 4.1 ([7]). Exponential Function:
Problem 4.2 ([8]). Logarithmic Function:
Problem 4.3 ([9]). Modified Semismooth function:
Problem 4.4 ([7]). Tridiagonal exponential function:
Problem 4.5 ([8]). Nonsmooth Function
Problem 4.6 ([11]). Modified exponential Function:
The computational results obtained from the experiments are presented in the Tables 1–6 below.
Results from Tables 1–6 show that the proposed SRMIL and SMRMIL algorithms exhibit nearly identical values for Iter, Fval, and CPU time across the tested problems. Moreover, the findings indicate that the four algorithms applied to Problems 4.1–4.6 are largely insensitive to variations in initial points and problem dimensions.
Additionally, the study utilize the performance profiles tool introduced by Dolan and Moré [51], which provide valuable insights into the efficiency and robustness of the algorithms. The performance profiles are cumulative distribution functions that measure the proportion of problems solved by an algorithm within a given performance ratio threshold. Particularly, for each method, the performance ratio is computed as the metric value (e.g., Iter, Fval, or CPU time) attained by the method on a given function divided by the best metric value attained by any method for that problem. A smaller ratio indicates better performance as illustrated in the following figures.
Figs 1–3 illustrates these profiles for the three evaluation criteria:
- Fig 1 represents the performance profile with respect to the number of iterations, highlighting how fast an algorithm converges to a solution.
- In Fig 2, the performance profile plot the curve for the number of function evaluations, assessing the algorithms’ efficiency in terms of computational resources utilized during the iteration process.
- Plot from Fig 3 illustrates the performance profile based on CPU time, showcasing the computational efficiency of the each algorithm.
In all three figures, SRMIL and SMRMIL which exhibit nearly identical values, consistently attain higher cumulative probabilities at various performance thresholds, indicating their superior competitiveness and efficiency compared to Algorithm 2.1a and Rana. This dominance across all performance criteria demonstrates the efficiency and practical advantage of the proposed methods.
5 Application
Robotic systems are increasingly becoming an integral part of modern technology, finding applications in industries ranging from manufacturing to healthcare. The three degrees of freedom (3DOF) arm robot model represents a class of manipulators characterized by its versatility and utility in complex tasks consisting of three rotational joints, each providing a degree of freedom [52]. This configuration allows the robot to perform an extensive array of movements and tasks, making it highly adaptable to various operational environments [53–55]. The kinematic equations of the 3DOF arm robot describe the position and orientation of the end-effector as functions of the joint angles while the dynamic equations govern the torques and forces required to achieve desired movements [56].
In this study, we are more interested in the discrete kinematic model equation with three degrees of freedom define below which also represents the planar three-joint kinematic model:
(60)
where r(⋅) denotes the kinematic transformation and positioning of a robot’s endpoint or any component relative to active adjustments in its joints, represented by
. The link is denoted by li for i = 1, 2, 3 represents the length of the corresponding link. In general, in the context of robotic motion, r(θ) represents the position and orientation of the robot end effector, while in this specific study case it only represents the x − y cartesian robot position. Effective control and trajectory planning for such robots necessitate advanced optimization techniques to handle the non-linearity and high dimensionality of the system dynamics [53–55].
Consider to denote the vector representing the desired path at a given time instant tk. We propose the following least-squares model, which will be evaluated at each time segment tk within the interval [0, tf]. The optimization problem is formulated as follows:
(61)
where
denotes the target path at the moment tk of a Lissajous curve. Furthermore, various Lissajous curves in 3DOF have been employed, for example, see [37, 57].
To simulate the results, the pseudo-code below was used to obtain the solution of the inverse kinematic problem.
Algorithm 2: Solution of 3DOF robotic arm model using (SRMILCG)
Step 1: Inputs: Initialize parameters t0, , tf, g, and Kmax
Step 2: For k = 1 to Kmax do
tk = k*g;
Step 3: Evaluate Lissajous curves ;
Step 4: Compute using SRMILCG
stated in Algorithm 1;
Step5: Set ;
Step 6: Output: θnew
Remark 5.1. When applying the CG methods to the specific case of the 3DOF robotic arm, the continuity of the model naturally follows from the well-defined kinematic equations of the robotic system. While monotonicity is not explicitly proven for the specific 3DOF case, the embedding of the kinematic equations into the optimization problem aligns well with the assumptions of the CG framework.
To solve the model and then simulate the results, the following parameters are used in the implementation:
- At the initial time instant t0 = 0, the joint angular vector is
,
- The link has a length of li (where i = 1, 2, 3),
- The anticipated total task duration is tf = 10 seconds.
The algorithms used in this experiment were developed using MATLAB R2022a, and the computations were performed on an Intel (R) CORE(TM) i7-3537U processor with a clock speed of 2.00 GHz and 8 GB of RAM. The effectiveness of the new formulae SRMIL and SMRMIL is demonstrated by comparing their results with those of other existing algorithms that share similar characteristics. For this comparison in numerical experiments, we consider the following methods:
- The Rivaie- Mustafa-Ismail-Leong (RMIL) CG method by [25] with the following parameter:
- The modified RMIL method (MRMIL) by [26] given as:
- The CG_Descent method by Hager and Zhang [58], defined by the following formula:
.
Problem 5.1. Consider the following Lissajous curve presented in [59, 60] defined by:
(62)
The numerical results are illustrated in the following figures. In particular, Fig 4(a) depicts the robot trajectories generated by the Algorithm 2. Fig 4(b) effectively illustrates the successful synthesis of robot trajectories for the task given. Moreover, Fig 5(a) and 5(b) display the residual error for each algorithm used in the experiment.
After analyzing the Figures that show the output of solving Eq (60) with the corresponding Lissajous curve (62) using the proposed and compared algorithms, Fig 4(a) shows the robot’s end effector model precisely following the target path. The success of creating robot trajectories for the assignment is aptly illustrated in Fig 4(b). The residual error rates are shown in Fig 5(a) and 5(b), respectively. SRMIL and SMRMIL have the lowest error at about 10−6, followed by CG_Descent at 10−5, MRMIL at 10−4, and RMIL at 10−3. The algorithms exhibit good performance in this example, as evidenced by the impressively low residual error rates of SRMIL and SMRMIl, which confirms their efficacy in finding the accuracy on the solution of the inverse kinematic problem.
Problem 5.2. Consider the following Lissajous curve given in [61] defined by:
(63)
The simulation results for problem 2 are presented in the following figures. Specifically, Fig 6(a) shows the robot trajectories generated by Algorithm 2. Fig 6(b) effectively illustrates the successful synthesis of robot trajectories for the given task. Additionally, Fig 7(a) and 7(b) display the residual error for each algorithm used in the experiment.
The plots in Fig 6 depict the output data from the model’s solution. Specifically, Fig 6(b) illustrates how the robot trajectories completed the task, with the end-effector following a precise path. Fig 7 shows the residual error norm rates, revealing that SMRMIL has the lowest error rate at approximately 10−6, followed by SRMIL, CG_Descent, MRMIL and RMIL, with error rates of around 10−5, 10−4, 10−4, and 10−3, respectively.
Problem 5.3. The end-effector is guided to follow a Lissajous curve [57], represented as
(64)
The simulation results for problem 3 are depicted in the following figures. Fig 8(a) displays the robot trajectories produced by Algorithm 2. Fig 8(b) effectively demonstrates the successful synthesis of robot trajectories for the specified task. Moreover, Fig 9(a) and 9(b) show the residual error for each algorithm utilized in the experiment.
The plots in Fig 8 showcase the output data derived from the model’s solution. Specifically, Fig 8(b) demonstrates the robot trajectories completing the task, with the end-effector precisely following its intended path. Fig 9 presents the residual error norm rates, indicating that the SMRMIL algorithm achieves the lowest error rate at approximately 10−7. This is followed by the SRMIL, CG_Descent, MRMIL, and RMIL algorithms, which have error rates around 10−6, 10−5, 10−5, and 10−3, respectively. These results highlight the superior accuracy of the SMRMIL algorithm in minimizing residual error.
Problem 5.4. The following describes how the end-effector is guided to follow a Lissajous curve:
(65)
The ensuing figures show the simulation results for problem 4. The robot trajectories generated by Algorithm 2 are shown in Fig 10(a). The synthesis of robot trajectories for the given job is successfully demonstrated in Fig 10(b). Additionally, the residual error for each method used in the experiment is displayed in Fig 11(a) and 11(b).
Fig 10 displays the plots of output data obtained from the solution of the model. In particular, Fig 10(b) illustrates the robot trajectories completing the task, with the end-effector precisely following its designed path. This indicates the robustness of the model in guiding the end-effector along the intended trajectory.
Furthermore, Fig 11 presents the residual error norm rates, providing a comparative analysis of the algorithms’ performance. The SRMIL algorithm achieves the lowest error rate at approximately 10−7, showcasing its superior accuracy. Following SRMIL, the SMRMIL, CG_Descent, MRMIL, and RMIL algorithms have error rates of around 10−6, 10−5, 10−5, and 10−3, respectively. These results highlight the enhanced precision of the SRMIL algorithm in minimizing residual error compared to the other methods.
Problem 5.5. As shown below, the end-effector is directed to follow a Lissajous curve:
(66)
The following figures present the simulation results for problem 5. Fig 12(a) illustrates the robot trajectories produced by Algorithm 2. Fig 12(b) demonstrates the successful synthesis of robot trajectories for the specified task. Furthermore, Fig 13 shows the residual errors for each method employed in the experiment.
Fig 13 illustrates the residual error norm rates, offering a comparative analysis of the algorithms’ performances. The SMRMIL algorithm demonstrates superior accuracy with the lowest error rate, approximately 10−6. It is followed by the SRMIL, CG_Descent, MRMIL, and RMIL algorithms, which exhibit error rates around 10−6, 10−5, 10−5, and 10−3, respectively. These findings underscore the enhanced precision of the SRMIL and SMRMIL algorithms in reducing residual error compared to the other methods. Furthermore, Table 7 presents the results of the problems, detailing the number of iterations, CPU time, and residual error. Regarding iteration counts, the SMRMIL method performs the best, achieving the fewest iterations, followed closely by SRMIL, CG_Descent, MRMIL, and RMIL, which have the highest count. Additionally, regarding CPU time, SMRMIL solves the problems in the shortest computational time, outperforming the SRMIL, CG_Descent, MRMIL, and RMIL methods.
Problem 5.6 Consider a trajectory that passes through a singularity or requires the end effector to transition between quadrants.
(67)
This trajectory does pass through points where both components become zero at tk = nπ (where n is an integer), which could indicate potential singularities in the robotic system depending on its configuration and the corresponding Jacobian.
The following figures showcase the simulation outcomes for Problem 5.6 Fig 14(a) depicts the robot trajectories generated by Algorithm 2, while Fig 14(b) highlights the successful synthesis of robot trajectories tailored to the specified task. Moreover, Fig 15 shows the residual errors for each method employed in the experiment.
6 Conclusion
In this study, we developed two self-scaling conjugate gradient (CG) parameters for solving nonmonotone nonlinear equations. The formulas were derived by integrating the renowned Barzilai-Borwein approach. The proposed algorithms ensure a sufficient descent property, independent of the accuracy of the line search procedure, and guarantee global convergence under appropriate assumptions. Furthermore, we numerically investigate the robustness and computational efficiency of the proposed methods by conducting experiments on benchmark test problems. Their performance is compared against existing methods. The results of these experiments demonstrate that the proposed methods outperform the alternatives based on the adopted comparison metrics. To validate the approach, the discrete kinematic equations of a 3DOF robotic arm were embedded within an optimization framework. This embedding minimized objective functions representing joint trajectory errors. Numerical simulations demonstrated that specific self-scaling CG algorithms exhibit superior convergence properties, making them particularly well-suited for solving inverse kinematics problems in robotic arms. By providing empirical validation and detailed analysis, we believe this study makes a significant contributions to advancing optimization techniques in robotics, paving the way for more reliable and efficient robotic systems.
Acknowledgments
The authors express their sincere gratitude to the anonymous reviewers for their insightful comments and constructive suggestions, which have significantly improved the quality of this manuscript.
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