Global Online Journal of Academic Research (GOJAR), Vol. 4, No. 1 February 2025


Global Online Journal of Academic Research (GOJAR), Vol. 4, No. 1 February 2025 

 

 

26 
 

Implementation and Comparative Analysis of AMGT 

Method in Maple 24: Convergence Performance in 

Optimization Problems 

 

Mark Laisin & Rosemary U. Adigwe 

 

 

 

Abstract 

This study investigates the utilization of the Accelerated Modified Gradient 

Technique (AMGT) in Maple 24 for optimization problem-solving. The research 

focuses on enhancing and evaluating the convergence speed of AMGT in 

comparison to the Gradient Descent (GD) and Conjugate Gradient (CG) methods. 

The analysis covers a range of function types, such as convex functions (e.g., 

production cost, energy consumption, and transportation cost functions) and a non-

convex function. Findings showcase the effectiveness and versatility of AMGT, 

shedding light on its utility for addressing practical optimization challenges. 

Keywords: AMGT method, Maple 24, convergence performance, optimization 

problems, comparative analysis 

 

 

1. Introduction 

Optimization techniques are crucial for solving complex problems in various 

fields, such as engineering, economics, machine learning, and operations 

research. They are essential for decision-making processes, from designing 

efficient systems to finding optimal financial strategies (Boyd & 

Vandenberghe, 2004). Gradient-based methods like Gradient Descent (GD) 

and Conjugate Gradient (CG) are widely used due to their computational 

efficiency and well-understood theoretical properties (Nocedal & Wright, 

2006). These methods iteratively improve solutions by using gradients to 

guide the search for the optimal solution, making them effective for 

continuous optimization problems. 

However, in some cases, basic gradient-based methods face limitations in 

terms of convergence speed and stability, especially in large or complex 



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problem spaces. As optimization problems become high-dimensional or non-

convex, traditional methods like GD and CG may struggle, leading to slower 

convergence rates and potential failure to escape local minima (Bertsekas, 

1999). Accelerated Gradient Methods, including the Accelerated Modified 

Gradient Technique (AMGT), have been proposed to address these 

challenges and offer improvements in convergence speed and stability 

(Nesterov, 1983).  

The AMGT combines the theoretical robustness of traditional gradient 

methods with acceleration strategies that optimize the search for a minimum. 

It adapts the gradient direction dynamically to enhance the rate of 

convergence, a feature that can be critical when dealing with large-scale 

optimization tasks or when high precision is required (Dauphin et al., 2015). 

Despite its potential, there is a limited comparative analysis of AMGT 

concerning traditional methods such as GD and CG in contemporary 

optimization applications, particularly in symbolic and numerical 

computation environments. 

This paper investigates the implementation of AMGT in Maple 24, a versatile 

software for mathematical modeling, simulation, and optimization 

(Maplesoft, 2023). The objective is to compare the convergence performance 

of AMGT with GD and CG across various optimization problems, including 

convex and non-convex scenarios, to assess their relative effectiveness in 

different contexts.  

II. Methods and Materials 

Problem Formulation Optimization problems are expressed as minimizing a 

cost function. The study considers the following types of functions: 

 Production cost function  𝐶(𝑥, 𝑦) = 𝑎𝑥2 + 𝑏𝑦2 + 𝑐𝑥𝑦 + 𝑑 where 

𝐶(𝑥, 𝑦) is the cost of producing x units of labour and y units of 

capital, and a, b, c, and d are constants. 

 Energy consumptionfunction  𝐸(𝑇, 𝐻) = 𝛼𝑇2 + 𝛽𝐻2 + 𝛾𝑇𝐻 + 𝜔 

where 𝐸(𝑇, 𝐻) is energy consumption as a function of temperature 

and humidity while  𝛼, 𝛽, 𝛾 𝑎𝑛𝑑 𝜔 are constants. 

 Transportation cost function  𝑇(𝑄, 𝐷) = 𝛼𝑄2 + 𝛽𝐷2 + 𝛾𝑄𝐷 + 𝜔 

where 𝑇(𝑄, 𝐷) is the transportation cost as a function of quantity (Q) 

of goods to be transported and distance to be covered in transportation 

(D).   𝛼, 𝛽, 𝛾 𝑎𝑛𝑑 𝜔 are constants. 

Non-Convex Function: A non-convex function in mathematics refers to a 

function where the line segment joining any two points on the graph of the 



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function is not entirely above or on the graph. More formally, a function f(x) 

is non-convex if, for some 𝑥1, 𝑥2 ∈ 𝑅𝑛 and for some 𝜆 ∈ [0,1], we have: 

𝑓(𝜆𝑥1 + (1 − 𝜆)𝑥2) > 𝜆𝑓(𝑥1) + (1 − 𝜆)𝑓(𝑥2) 

which means that the epigraph (the set of points above the graph) is not 

convex. 

A convex function satisfies the reverse inequality: 

𝑓(𝜆𝑥1 + (1 − 𝜆)𝑥2) ≤ 𝜆𝑓(𝑥1) + (1 − 𝜆)𝑓(𝑥2) 

A non-convex function is simply one that does not satisfy this condition in 

general. E.g. the convex function  𝑓(𝑥, 𝑦) = 2𝑥𝑦 + 𝑦 − 𝑥2 − 2𝑦2. With the 

three methods using 𝑥0 = 2 and 𝑦0 = 2,   while AMGT uses 𝑚0(𝑥, 𝑦) =

(1000,1000), 𝛼 = 50,  𝛽 = 0.13,  𝛾 = 0.1, is a non-convex function. 

Algorithms 

 Gradient Descent (GD): Iterative optimization algorithm using the 

gradient to update parameters. 

 Conjugate Gradient (CG): Enhances GD by using conjugate directions 

for faster convergence in quadratic problems. 

 Accelerated Modified Gradient Technique (AMGT): Combines 

momentum-based acceleration with gradient optimization for 

enhanced performance for the optimization of nonlinear unconstrained 

optimization. 

However, this is a gradient descent step length algorithm that is a 

modification of some of Adam’s algorithms. Our motivation for this method 

is the need to combine the benefits of adaptive learning rates and momentum 

while introducing a gradient technique mechanism to escape local minima. 

Update Rules for the proposed AMGT: 

a. Define the convex function to be optimized 

b. Initialize 𝑥0, 𝛼0, τ, β , γ, 𝑚0, tolerance and number of iterations 

c. Compute the gradient of the given convex function to be optimized: 

∇𝑓(𝑥𝑘) =
∂f

∂𝑥𝑘
 

d. Next compute the adaptive learning rate: 𝛼𝑘 = 𝛼𝑘−1/(1 + γ/𝜏) 

e. Compute the momentum: 𝑚𝑘 = β ∗ 𝑚𝑘−1 + (1 − 𝛽) ∗ ∇𝑓(𝑥𝑘) 



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f. Apply gradient technique on the momentum: �̂�𝑘 = 𝑚𝑘 ∗ (1 − γ ∗
|∇𝑓(𝑥𝑘)|

|𝑚𝑘|
) 

g. Update parameters or variable values of the convex function:  𝑥𝑘 =

𝑥𝑘−1 − 𝛼𝑘 ∗ �̂�𝑘 

h. Continue steps b to g till the specified number of iterations. 

Convergence Criteria: 

AMGT converges when the following conditions are met: 

 Parameter Convergence: The updates to the parameters (𝑥𝑘) become 

negligible, i.e., |𝑥𝑘 − 𝑥𝑘−1| < tolerance. 

 Objective Function Convergence: The change in the objective 

function value becomes negligible, i.e., |𝑓(𝑥𝑘) − 𝑓(𝑥𝑘−1)|<tolerance. 

 Maximum Iterations: The algorithm reaches the specified number of 

iterations. 

Inputs or initial parameters: 

 𝛼0this is the initial learning rate 

 τ: this is the learning rate decay rate 

 β: this is the momentum coefficient 

 γ: this is the gradient technique coefficient 

Rational behind each of the input parameters: 

o The Adaptive learning rate (𝛼𝑘) helps the method to converge faster. 

o The Momentum (𝑚𝑘) helps to escape local minima. 

o While the Gradient technique (𝛾) prevents overshooting. 

Implementation in Maple 24: The AMGT method is implemented using 

Maple 24's symbolic and numerical capabilities. Scripts for GD and CG are 

also developed for comparison. The functions are defined, and step sizes are 

chosen dynamically for each method to ensure fair benchmarking. 

Performance Metrics: 

 Convergence Rate: Number of iterations required to reach a 

predefined tolerance. 



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 Computation Time: Time taken to converge. 

 Accuracy: Proximity of the solution to the true minimum. 

Mathematical Properties of Non-Convex Functions 

 Local Minima and Maxima: A non-convex function can have multiple 

local minima and maxima, unlike convex functions, which have at 

most one global minimum. 

 Optimization Challenges: Non-convex optimization is harder than 

convex optimization because local search algorithms like gradient 

descent may converge to local minima instead of the global minimum. 

(Bertsekas, 1999; Boyd & Vandenberghe, 2004) 

 Non-convexity in Higher Dimensions: Non-convexity is not limited to 

functions of one variable. A function f(x) defined on a higher-

dimensional space 𝑅𝑛 can be non-convex if it does not satisfy the 

convexity condition for all pairs of points in its domain (Bertsekas, 

1999; Boyd & Vandenberghe, 2004). 

III Results 

Theorems 3.1. Convergence of AMGT 

Let 𝑓(𝑥, 𝑦) be a continuously differentiable function, and let (𝑥𝑘, 𝑦𝑘) be the 

sequence generated by the Adaptive Momentum Gradient Technique 

(AGMT). Assume that  

 𝑓(𝑥, 𝑦) convex 

 The learning rate 𝑎𝑘 satisfies ∑ 𝑎𝑘 = ∞∞
𝑘=1  and ∑ 𝑎𝑘

2 < ∞∞
𝑘=1 . 

 The momentum coefficient 𝛽  satisfies 0 < 𝛽 < 1 

 The gradient threshold coefficient 𝛾 satisfies 0 < 𝛾 < 1 

Then the sequence (𝑥𝑘, 𝑦𝑘) converges to a stationary point of 𝑓(𝑥, 𝑦) 

Proof. 

Using the convexity of 𝑓(𝑥, 𝑦) and the update rule of AMGT, we can 

establish the above theorem.  

By the convexity 0f 𝑓(𝑥, 𝑦) we have the convex inequality: 

𝑓(𝑥𝑘+1, 𝑦𝑘+1) ≤ 𝑓(𝑥𝑘, 𝑦𝑘) + 〈∇𝑓(𝑥𝑘, 𝑦𝑘), (𝑥𝑘+1 − 𝑥𝑘, 𝑦𝑘+1 − 𝑦𝑘)〉 

 



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The essence of the convex inequality is that the value of the function at the 

new point (𝑥𝑘+1, 𝑦𝑘+1) is always less than the value of the function at the 

current point (𝑥𝑘, 𝑦𝑘) plus a linear approximation based on the gradient. 

In the proposed AMGT method, the update involves a combination of the 

current gradient and previous momentum. Thus, we can express the 

parameters as  

𝑥𝑘+1 = 𝑥𝑘 − 𝛼𝑘�̂�𝑘 and 𝑦𝑘+1 = 𝑦𝑘 − 𝛼𝑘�̂�𝑘 

Where �̂�𝑘 is the momentum term after gradient thresholding. 

Applying the update of AMGT on the convex inequality, we have; 

𝑓(𝑥𝑘+1, 𝑦𝑘+1) ≤ 𝑓(𝑥𝑘, 𝑦𝑘) –𝑎𝑘〈∇𝑓(𝑥𝑘 , 𝑦𝑘), �̂�𝑘〉 + 𝑎𝑘
2‖∇𝑓(𝑥𝑘 , 𝑦𝑘)‖2 

The second term in the above inequality 𝑎𝑘〈∇𝑓(𝑥𝑘, 𝑦𝑘), �̂�𝑘〉 is the gradient 

update using the momentum �̂�𝑘. If �̂�𝑘points in the same direction as 

∇𝑓(𝑥𝑘, 𝑦𝑘), the term will be negative which helps reduce the function value 

(minimization). 

Now since �̂�𝑘 is calculated using both the gradient and momentum, when we 

apply the gradient threshold, we have: 

�̂�𝑘 ≈ ∇𝑓(𝑥𝑘, 𝑦𝑘) (for large gradients) 

Therefore, we can simplify the function update to: 

𝑓(𝑥𝑘+1, 𝑦𝑘+1) ≤ 𝑓(𝑥𝑘, 𝑦𝑘)-𝑎𝑘‖∇𝑓(𝑥𝑘 , 𝑦𝑘)‖2 + 𝑎𝑘
2‖∇𝑓(𝑥𝑘, 𝑦𝑘)‖2 

Next, by using the assumptions of 𝑎𝑘 and 𝛽 we can simplify the right-hand 

side to obtain the: 

𝑓(𝑥𝑘+1, 𝑦𝑘+1) ≤ 𝑓(𝑥𝑘, 𝑦𝑘)-
𝑎𝑘

2
‖∇𝑓(𝑥𝑘 , 𝑦𝑘)‖2 

To prove the convergence of the AMGT, when we sum both sides of the 

inequality over all the iterations 𝑘 

𝑓(𝑥𝑘+1, 𝑦𝑘+1) ≤ 𝑓(𝑥𝑘, 𝑦𝑘)-
𝑎𝑘

2
‖∇𝑓(𝑥𝑘 , 𝑦𝑘)‖2 

∑ 𝑓(𝑥𝑘+1, 𝑦𝑘+1)

𝑚

𝑘=1

− 𝑓(𝑥𝑘, 𝑦𝑘) ≤ ∑ −
𝑎𝑘

2
‖∇𝑓(𝑥𝑘, 𝑦𝑘)‖2

𝑚

𝑘=1

 

Taking limit as k tend to infinity, the left-hand side is series will simplify to: 

𝑓(𝑥∞, 𝑦∞) − 𝑓(𝑥0, 𝑦𝑜) 

Where (𝑥0, 𝑦𝑜) is the point where the sequence converges. 



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Thus, we now have  

𝑓(𝑥𝑚, 𝑦𝑚) − 𝑓(𝑥0, 𝑦𝑜) ≤ −
1

2
∑ 𝑎𝑘‖∇𝑓(𝑥𝑘, 𝑦𝑘)‖2

𝑚

𝑘=1

) 

Therefore, since the sum 𝑎𝑘 is infinite, but the of 𝛼𝑘
2 is finite  ∑ 𝛼𝑘

2𝑚
𝑘=1 < ∞, 

the implication of the above is that the gradient term ‖∇𝑓(𝑥𝑘, 𝑦𝑘)‖2must 

converges to 0 as 𝑘 approaches infinity, thereby proving the convergence to a 

stationary point. 

Thus, the sequence (𝑥𝑘, 𝑦𝑘) generated by the proposed AMGT converges to a 

stationary point of 𝑓(𝑥𝑘, 𝑦𝑘) proving that AMGT is a convergent 

optimization method for a convex function. 

IV Numerical Application 

Application 1: We shall now implement the AMGT method in Maple 24 and 

compare the convergence rate with that of the Gradient Descent method and 

the Conjugate Gradient method, using the convex function  𝑓(𝑥, 𝑦) = 𝑥2 −

2𝑥𝑦 + 2𝑦2 + 2𝑥 − 4𝑦 + 5 . 

With the three methods using 𝑥0 = 2 and 𝑦0 = 2,    

while AMGT uses 𝑚0(𝑥, 𝑦) = (1000,1000), 𝜏 = 50,  𝛽 = 0.13,  𝛾 = 0.1, 

we have the following results; 

Table 4.1: Comparing the convergence of AMGT, GD and CG 

Methods Gradient 

Descent 

Conjugate 

Gradient 

AMGT 

Iteration f(x,y)at 

𝛼 = 0.1 

f(x,y)at 

𝛼 = 0.1 

f(x,y)at 

𝛼0 = 0.1 

f(x,y) at 

𝛼0 = 0.2 

f(x,y)at 

𝛼0 = 0.3 

f(x,y) at 

𝛼0 = 0.4 

1 4.627200 4.640000 3.734479 5.926594 11.576348 20.68373853 

10 3.326152 3.161294 3.005368 3.059090 3.049195 14.79960452 

20 3.056349 3.012675 3.001100 3.002134 3.000265 28.59793180 

30 3.009737 3.000996 3.000224 3.000077 3.000001 58.61401015 

40 3.001682 3.000078 3.000046 3.000003 3.000000 123.8269287 

50 3.000291 3.000006 3.000009 3.000000 3.000000 265.508434 

100 3.000000 3.000000 3.000000 3.000000 3.000000 12709.93742 

 

The results show that the new AMGT is convergent and converges faster with 

proper selection of the momentum coefficient, the gradient threshold, and the 

learning rate decay rate, which are absent in the gradient descent and 

conjugate gradient methods. 

 

 



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Fig 4.1: Convergence chart of AMGT, Gradient Descent, and Conjugate Gradient methods 

 

Fig 4.2: Chart for convergence at different learning rates 

 
Fig 4.3: AMGT optimization path 

 

 
Fig 4.4: 3D plot of the convex function 

 

0

1

2

3

4

5

1 2 3 4 5 6 7

Convergence Chart 

GRADIENT DESCENT CONJUGATE GRADIENT AMGT

0

5

10

15

1 2 3 4 5 6 7 8

Chart for convergence at 
different learning rates

α=0.1 α=0.2 α=0.3



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Application 2: Let us consider the production cost function  𝐶(𝑥, 𝑦) = 𝑎𝑥2 +

𝑏𝑦2 + 𝑐𝑥𝑦 + 𝑑 where 𝐶(𝑥, 𝑦) is the cost of producing x units of labour and y 

units of capital and a, b, c, and d are constants. This function is convex if a>0 

and b>0. Supposed a=2, b=4, c=5 and d=9  then;  

𝐶(𝑥, 𝑦) = 2𝑥2 + 4𝑦2 + 5𝑥𝑦 + 9 and with the three methods using 𝑥0 = 2 

and 𝑦0 = 2, while AMGT uses 𝑚0(𝑥, 𝑦) = (1000,1000), 𝜏 = 50,  𝛽 = 0.13,  

𝛾 = 0.1, we have the following results: 

 

Table 4.2: Production cost function   

Methods Gradient 

Descent 

Conjugate 

Gradient 

AMGT 

Iteration C(x,y)at 

𝛼 = 0.1 

C(x,y)at 

𝛼 = 0.1 

C(x,y)at 

𝛼0 = 0.1 

C(x,y) at 

𝛼0 = 0.2 

C(x,y)at 

𝛼0 = 0.3 

1 9.740000 9.920000 35.214992 293.4537351 827.7162300 

10 9.000489 9.013961 9.002196 10524.22355 3.893467833*10^9 

20 9.000091 9.001823 9.000617 574721.6499 1.019715961*10^17 

30 9.000017 9.000238 9.000174 31411103.41 

 

2.670679931*10^24 

40 9.000003 9.000031 9.000049 1716782917 

 

6.994625548*10^31 

50 9.000001 9.000004 9.000014 93831291380 

 

1.831922500*10^39 

100 9.000000 9.000000 9.000000 4.576244859*10^19 2.257460882*10^76 

 

 

 

Fig 4.5: Convergence chart 

 

 

 

0

10

20

30

40

1 2 3 4 5 6 7

Convergence Chart

GRADIENT DESCENT CONJUGATE GRADIENT AMGT



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Fig 4.6: AMGT optimization path                  Fig 4.7: Objective function for production in 3D 

Application 3: Let us consider the energy consumption function  𝐸(𝑇, 𝐻) =

𝛼𝑇2 + 𝛽𝐻2 + 𝛾𝑇𝐻 + 𝜔 where 𝐸(𝑇, 𝐻) is energy consumption as a function 

of temperature and humidity while  𝛼, 𝛽, 𝛾 𝑎𝑛𝑑 𝜔 are constants. This function 

is convex if 𝛼>0 and 𝛽>0. Supposed𝛼 =5, 𝛽 =5, 𝛾 =7 and 𝜔 =10  then; 

𝐸(𝑇, 𝐻) = 5𝑇2 + 5𝐻2 + 7𝑇𝐻 + 10 and with the three methods using 𝑥0 = 2 

and 𝑦0 = 2,   while AMGT uses 𝑚0(𝑥, 𝑦) = (1000,1000), 𝜏 = 50,  𝛽 =

0.13,  𝛾 = 0.1, we have the following results: 

Table 4.3: Energy consumption function 

Methods Gradient 

Descent 

Conjugate 

Gradient 

AMGT 

Iteration 𝐸(𝑇, 𝐻) at 

𝛼 = 0.1 

𝐸(𝑇, 𝐻) at 

𝛼 = 0.1 

𝐸(𝑇, 𝐻) at  

𝛼0 = 0.1 

𝐸(𝑇, 𝐻) at 

𝛼0 = 0.2 

1 14.998000 43.320000 129.59045 917.0755341 

10 10.000013 10.000000 10.043577 3.831893089*10^9 

20 10.000000 10.000000 10.000006 8.801297678*10^16 

30 10.00000 10.00000 10.000000 2.021529286*10^24 

40 10.000000 10.000000 10.000000 4.643156981*10^31 

50 10.000000 10.000000 10.000000 6.817314683*10^75 
 

 
Fig 4.8:  Convergence chart for GD, CG and AMGT 

0

50

100

150

1 2 3 4 5 6

Convergence Chart

Gradient Descent Conjugate Gradient AMGT



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Fig 4.9: AMGT optimization path for GD and CG     Fig 4.10: Objective function for energy cost 

 

Application 4: Let us consider transportation cost function  𝑇(𝑄, 𝐷) =

𝛼𝑄2 + 𝛽𝐷2 + 𝛾𝑄𝐷 + 𝜔 where 𝑇(𝑄, 𝐷) is the transportation cost as function 

of quantity (Q) of goods to be transported and distance to be covered in 

transportation (D).   𝛼, 𝛽, 𝛾 𝑎𝑛𝑑 𝜔 are constants. This function is convex if 

𝛼>0 and 𝛽>0 indicating that transportation cost increases at an increasing rate 

as either quantity or distance increases. Supposed𝛼 =4, 𝛽 =1, 𝛾 =-1 and 𝜔 =0  

then;  

𝑇(𝑄, 𝐷) = 4𝑄2 + 𝐷2 − 2𝑄𝐷 and with the three methods using 𝑥0 = 2 and 

𝑦0 = 2, while AMGT uses 𝑚0(𝑥, 𝑦) = (1000,1000), 𝜏 = 50,  𝛽 = 0.13,  

𝛾 = 0.1, we have the following results: 

Table 4.4: Transportation cost function 
Methods GD at 

𝛼 = 0.1 

CG at  

𝛼 = 0.1 

AMGT at 𝛼 = 0.1, 0.2, 𝑎𝑛𝑑 0.3 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑣𝑖𝑒𝑙𝑦 

Iteration 𝑇(𝑄, 𝐷) 𝑇(𝑄, 𝐷) 𝑇(𝑄, 𝐷) 𝑇(𝑄, 𝐷) 𝑇(𝑄, 𝐷) 

1 2.841600 3.360000 2.100208 31.996444 101.6887051 

10 0.135361 0.050166 0.011700 0.024588 185200.1144 

20 0.005533 0.000429 0.000578 0.000012 8.452832713*10^8 

30 0.000226 0.000004 0.000029 0.000000 3.858009564*10^12 

40 0.000009 0.000000 0.000001 0.000000 1.760857964*10^16 

50 0.000000 0.000000 0.000000 0.000000 8.036840592*10^19 

100 0.000000 0.000000 0.000000 0.000000 1.591807268*10^38 
 

 
Fig 4.11: Convergence path for transportation cost function 



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Fig 4.12 AMGT optimization path for GD and CG     Fig 4.13: Objective function for transportation cost 

Application 5: Let's consider the convex function 

𝑓(𝑥, 𝑦) = 2𝑥𝑦 + 𝑦 − 𝑥2 − 2𝑦2. 

With the three methods using 𝑥0 = 2 and 𝑦0 = 2,   while AMGT uses 

𝑚0(𝑥, 𝑦) = (1000,1000), 𝜏 = 50,  𝛽 = 0.13,  𝛾 = 0.1, we have the 

following results; 

Table 4.5: Non convex  

Methods Gradient Descent Conjugate 

Gradient 

AMGT 

Iteration f(x,y) at 

𝛼 = 0.1 

f(x,y) at 

𝛼 = 0.1 

f(x,y) at 

𝛼0 = 0.1 

1 -3.080000 -3.08 -0.084282334 

10 -4263.708589 -603182.468126262 -490.5670125 

20 -34952895.710000 -1110219359887.00 -2146227.90 

 

        

Fig 4.14: AMGT optimization path for GD and CG     Fig 4.15: Objective function: Nonconvex 

function 

From the diagram above, this is a non-convex function; thus, it doesn’t have a 

minimum. Instead, what it has is a maximum and therefore none of the 



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methods converges. To solve the above, we may have to minimize the 

negative of the above function which will then be convex. 

Discussion 

The discussion emphasizes AMGT's rapid convergence and stability, making 

it well-suited for a variety of optimization problems. However, there is still 

room for further investigation into computational costs and parameter 

adjustments. The findings confirm that Maple 24 is effective in executing 

sophisticated optimization algorithms. 

Conclusion 

This study underscores AMGT's advantages in convergence rate and stability, 

making it a promising alternative to GD and CG. However, the convex 

functions showed the following results. 

 Production Cost Function: AMGT demonstrated faster convergence 

compared to GD and CG, with reduced iterations for similar accuracy 

levels. 

 Energy Consumption Function: AMGT showcased enhanced stability 

in scenarios with fluctuating gradients. 

 Transportation Cost Function: The performance gain of AMGT was 

significant in multi-variable optimization tasks. 

Non-Convex Function: The non-convex function presented challenges in 

local minima. AMGT outperformed GD but showed similar convergence to 

CG in escaping local minima. 

Future research will focus on testing the scalability of AMGT in high-

dimensional and real-time optimization scenarios. 

 

References 

Bertsekas, D. P. (1999). Nonlinear Programming (2nd ed.). Athena 

Scientific. 

Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge 

University Press. 

Dauphin, Y. N., Pascanu, R., Gulcehre, C., Cho, K., Ganguli, S., & Bengio, 

Y. (2015). Identifying and attacking the saddle point problem in high-

dimensional non-convex optimization. In Proceedings of the 34th 

International Conference on Machine Learning (Vol. 37, pp. 2967-



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2975). 

Diederikm, P. K and Jimmy, L. B.  (2015). Adam: A method for Stochastic 

Optimization. Published as a conference paper at ICLR, 2015. 

Kingma, D. P., & Ba, J. (2015). "Adam: A Method for Stochastic 

Optimization." arXiv preprint arXiv:1412.6980. 

MapleSoft. (2023). Maple 24 Documentation. [Online]. Available: 

https://www.maplesoft.com/documentation. 

Nesterov, Y. (1983). A method of solving a convex programming problem 

with convergence rate O(1/k^2). DokladyAkademiiNauk SSSR, 269(3), 

543–547. 

Nesterov, Y. (2004). Introductory Lectures on Convex Optimization: A Basic 

Course. Springer. 

Nocedal, J., & Wright, S. J. (2006). Numerical Optimization (2nd ed.). 

Springer. 

Noel J. (2023). Nesterovs Method for Convex Optimization. Society for 

Industrial and Applied Mathematics. Vol. 65, No. 2, pp. 539–562. 

Polyak, B. T. (1964). "Some Methods of Speeding up the Convergence of 

Iteration Methods."  

USSR Computational Mathematics and Mathematical Physics, 4(5), 1-17. 

Rahul, A. (2023). Complete Guide to the Adam Optimization Algorithm. 

https://builtin.com/machine-learning/adam-optimization. (Retrieved 

November 11, 2024) 

Satyam, T (2024)/Adagrad Optimizer Explained: How It Works, 

Implementation, & Comparisons | DataCamp 

www.gabormelli.com/RKB/Root_Mean_Square_Propagation_Algorithm_(R

MSprop) (Retrieved November 11, 2024). 

 

 

 

Author Information: Mark Laisin is a Professor of 

Applied Mathematics at Chukwuemeka Odumegwu 

Ojukwu University, Uli, Anambra State, Nigeria. Email: 

laisinmark@gmail.com 

 
 

 

 

 

Rosemary U. Adigwe is of the Department of 

Mathematics, Chukwuemeka Odumegwu Ojukwu 

University, Uli, Anambra State, Nigeria. Email: 

 



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40 
 

rosemaryadigwe14@gmail.com 
 

 

 

 
 

APA 
 

Laisin, M. & Adigwe, R. U.  (2025). Implementation and 

Comparative Analysis of AMGT Method in Maple 24: Convergence 

Performance in Optimization Problems. Global Online Journal of 

Academic Research (GOJAR), 4(2), 26-40. https://klamidas.com 

/gojar-v4n1-2025-02/.  
 

MLA 
 

Laisin, Mark and Adigwe, Rosemary U. “Implementation and 

Comparative Analysis of AMGT Method in Maple 24: Convergence 

Performance in Optimization Problems”. Global Online Journal of 

Academic Research (GOJAR), vol. 4, no. 1, 2025, pp. 26-40. 

https://klamidas.com /gojar-v4n1-2025-02/.