Improving the Giant-Armadillo Optimization Method

Abstract

Global optimization is widely adopted presently in a variety of practical and scientific problems. In this context, a group of widely used techniques are evolutionary techniques. A relatively new evolutionary technique in this direction is that of Giant-Armadillo Optimization, which is based on the hunting strategy of giant armadillos. In this paper, modifications to this technique are proposed, such as the periodic application of a local minimization method as well as the use of modern termination techniques based on statistical observations. The proposed modifications have been tested on a wide series of test functions available from the relevant literature and compared against other evolutionary methods.

1. Introduction

Global optimization aims to discover the global minimum of an optimization problem by searching in the domain range of the problem. Typically, a global optimization method aims to discover the global minimum of a continuous function $f:S\to R,S\subset R^n$ and hence the global optimization problem is formulated as follows:

$$x^{*}=\arg \underset{x\in S}{min}f\left(x\right).$$

(1)

The set S is defined as:

$$S=\left[a_1,b_1\right]\otimes \left[a_2,b_2\right]\otimes \dots \left[a_n,b_n\right]$$

The vectors a and b stand for the left and right bounds, respectively, for the point x. A review of the optimization procedure can be found in the paper by Rothlauf [1]. Global optimization refers to techniques that seek the optimal solution to a problem, mainly using traditional mathematical methods, for example, methods that try to locate either maxima or minima [2,3,4]. Every optimization problem contains its decision variables, a possible series of constraints, and the definition of the objective function [5]. Every optimization method targets the discovery of appropriate values for the decision variables to minimize the objective function. The optimization methods are commonly divided into deterministic and stochastic approaches [6]. The techniques used in most cases for the first category are the interval methods [7,8]. In interval methods, the set S is divided through several iterations into subareas that may contain the global minimum using some criteria. On the other hand, stochastic optimization methods are used in most cases because they can be programmed faster than deterministic ones, and they do not depend on any previously defined information about the objective function. Such techniques may include Controlled Random Search methods [9,10,11], Simulated Annealing methods [12,13], Clustering methods [14,15,16], etc. Systematic reviews of stochastic methods can be located in the paper by Pardalos et al. [17] or in the paper by Fouskakis et al. [18]. Also, due to the widespread use of parallel computing techniques, a series of methods have been presented that exploit such architectures [19,20].

A group of stochastic programming techniques that have been proposed to tackle optimization problems are evolutionary techniques. These techniques are biologically inspired, heuristic, and population-based [21,22]. Some techniques that belong to evolutionary techniques are, for example, Ant Colony Optimization methods [23,24], Genetic Algorithms [25,26,27], Particle Swarm Optimization (PSO) methods [28,29], Differential Evolution techniques [30,31], evolutionary strategies [32,33], evolutionary programming [34], genetic programming [35], etc. These methods have found success in a series of practical problems from many fields, for example biology [36,37], physics [38,39], chemistry [40,41], agriculture [42,43], and economics [44,45].

Metaheuristic algorithms, from their appearance in the early 1970s to the late 1990s, have seen significant developments. Metaheuristic algorithms have gained much attention in solving difficult optimization problems and are paradigms of computational intelligence [46,47,48]. Metaheuristic algorithms are grouped into four categories based on their behavior: evolutionary algorithms, algorithms based on considerations derived from physics, algorithms based on swarms, and human-based algorithms [49].

Recently, Alsayyed et al. [50] introduced a new bio-inspired algorithm that belongs to the group of metaheuristic algorithms. This new algorithm is called Giant-Armadillo Optimization (GAO) and aims to replicate the behavior of giant armadillos in the real world [51]. The new algorithm is based on the giant armadillo’s hunting strategy of heading toward prey and digging termite mounds.

Owaid et al. presented a method [52] concerning the decision-making process in organizational and technical systems management problems, which also uses giant-armadillo agents. The article presents a method for maximizing decision-making capacity in organizational and technical systems using artificial intelligence. The research is based on giant-armadillo agents that are trained with the help of artificial neural networks [53,54], and in addition, a genetic algorithm is used to select the best one.

The GAO optimizer can also be considered to be a method based on Swarm Intelligence [55]. Some of the reasons why methods based on Swarm Intelligence are used in optimization problems are their robustness, scalability, and flexibility. With the help of simple rules, simple reactive agents such as fish and birds exchange information with the basic purpose of finding an optimal solution [56,57]. This article focuses on enhancing the effectiveness and the speed of the GAO algorithm by proposing some modifications and, more specifically:

• The application of termination rules, which are based on asymptotic considerations and are defined in the recent bibliography. This addition will achieve early termination of the method and will not waste computational time on iterations that do not yield a better estimate of the global minimum of the objective function.
• A periodic application of a local search procedure. Using local optimization, the local minima of the objective function will be found more efficiently, which will also lead to a faster discovery of the global minimum.

The current method was applied to a series of objective problems found in the literature of global optimization, and it is compared against an implemented Genetic Algorithm and a variant of the PSO technique.

The rest of this paper is divided into the following sections: in Section 2, the proposed modifications are fully described. In Section 3, the benchmark functions are listed, accompanied by the experimental results, and finally, in Section 4, some conclusions and guidelines for future work are provided.

2. The Proposed Method

The GAO algorithm is based on processes inspired by nature and initially generates a population of candidate solutions that are possible solutions to the objective problem. The GAO algorithm aims to evolve the population of solutions through iterative steps. The algorithm has two major phases: the exploration phase, where the candidate solutions are updated with a process that mimics the attack of armadillos on termite mounds, and the exploitation phase, where the solutions are updated similarly to digging in termite mounds. The basic steps of the GAO algorithm are presented below:

1.

Initialization step

• Define as $N_c$ as number of armadillos in the population.
• Define as $N_g$ the number of allowed iterations.
• Initialize randomly the $N_c$$g_i,i=1,\dots ,N_c$ armadillos in S.
• Set iter = 0.
• Set $p_l$ the local search rate.

2.

Evaluation step

• For $i=1,\dots ,N_c$ do Set $f_i=f\left(g_i\right)$.
• end for

3.

Computation step

• For  $i=1,\dots ,N_c$  do

  (a)

  Phase 1: Attack on termite mounds

  –

  Create a set that contains the termites as ${\mathrm{T}\mathrm{M}}_i=\left\{g_{k_i}:f_{k_i}<f_i\mathrm{a}\mathrm{n}\mathrm{d}{k}_i\ne i\right\}$

  –

  Select the termite mound ${\mathrm{S}\mathrm{T}\mathrm{M}}_i$ for armadillo i.

  –

  Create a new position $g_i^{P1}$ for the armadillo according to the formula: $g_{i,j}^{P1}=g_{i,j}+r_{i,j}\left({\mathrm{S}\mathrm{T}\mathrm{M}}_{i,j}-I_{i,j}{g}_{i,j}\right)$ where $r_{i,j}$ are random numbers in $[0,1]$ and $I_{i,j}$ are random numbers in $[1,2]$ and $j=1,\dots ,n$

  –

  Update the position of the armadillo i according to:

  $$g_i=\left\{\begin{array}{c}\begin{array}{cc}{g}_i^{P1},& f\left(g_i^{P1}\right)\le f_i\\ g_i& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathbf{e}\end{array}\hfill \end{array}\right.$$

  (b)

  Phase 2: Digging in termite mounds

  –

  Create a new trial position

  $$g_{i,j}^{\mathrm{P}2}=g_{i,j}+(1-2r_{i,j})\frac{b_j-a_j}{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}$$

  where $r_{i,j}$ are random numbers in $[0,1]$.

  –

  Update the position of the armadillo i according to:

  $$g_i=\left\{\begin{array}{c}\begin{array}{cc}{g}_i^{\mathrm{P}2},& f\left(g_i^{\mathrm{P}2}\right)\le f_i\\ g_i& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\hfill \end{array}\right.$$

  (c)

  Local search. Pick a random number $r\in [0,1]$. If $r\le p_l$, then a local optimization algorithm is applied to $g_i$. Some local search procedures found in the optimization literature are the BFGS method [58], the Steepest Descent method [59], the L-Bfgs method [60] for large-scaled optimization, etc. A BFGS modification proposed by Powell [61] was used in the current work as the local search optimizer. Using the local optimization technique ensures that the outcome of the global optimization method will be one of the local minima of the objective function. This ensures maximum accuracy in the end result.

• end for

4.

Termination check step

• Set iter = iter + 1.
• For the valid termination of the method, two termination rules that have recently appeared in the literature are proposed here, and they are based on stochastic considerations. The first stopping rule will be called DoubleBox in the conducted experiments, and it was introduced in the work of Tsoulos in 2008 [62]. The steps for this termination rule are as follows:

  (a)

  Define as ${\sigma }^{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}$ the variance of located global miniumum at iteration iter.

  (b)

  Terminate when $\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}\ge N_g\mathrm{O}\mathrm{R}{\sigma }^{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}}\le \frac{{\sigma }^{k_T}}{2}$

  where $k_T$ is the iteration where a new and better estimation for the global minimum was first found.
  The second termination rule was introduced in the work of Charilogis et al. [63] and will be called Similarity in the experiments. In the Similarity stopping rule, at every iteration k, the absolute difference between the current located global minimum $f_{min}^{\left(k\right)}$ and the previous best value $f_{min}^{(k+1)}$ is calculated:

  $$\left|f_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\left(k\right)}-f_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(k-1)}\right|$$
  If this difference is zero for a predefined number of consecutive generations $N_k$, the method terminates.
• If the termination criteria are not held, then go to step 3.

The steps of the proposed method are also outlined in Figure 1.

3. Experiments

This section will begin by detailing the functions that will be used in the experiments. These functions are widespread in the modern global optimization literature and have been used in many research works. Next, the experiments performed using the current method will be presented, and a comparison will be made with methods that are commonly used in the literature of global optimization.

3.1. Experimental Functions

The functions used in the conducted experiments can be found in the related literature [64,65]. The definitions for the functions are listed as follows.

• Bf1 function, defined as:

$$f\left(x\right)=x_1^2+2x_2^2-\frac{3}{10}cos\left(3\pi x_1\right)-\frac{4}{10}cos\left(4\pi x_2\right)+\frac{7}{10}$$

• Bf2 function, defined as:

  $$f\left(x\right)=x_1^2+2x_2^2-\frac{3}{10}cos\left(3\pi x_1\right)cos\left(4\pi x_2\right)+\frac{3}{10}$$
• Branin function, defined as: $f\left(x\right)={\left(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6\right)}^2+10\left(1-\frac{1}{8\pi }\right)cos\left(x_1\right)+10$.
• Camel function defined as:

  $$f\left(x\right)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4,x\in {[-5,5]}^2$$
• Easom defined as:

  $$f\left(x\right)=-cos\left(x_1\right)cos\left(x_2\right)exp\left({\left(x_2-\pi \right)}^2-{\left(x_1-\pi \right)}^2\right)$$
• Exponential function defined as:

  $$f\left(x\right)=-exp\left(-0.5\sum _{i=1}^n{x}_i^2\right),-1\le x_i\le 1$$
  In the current work, the following values were used for the conducted experiments: $n=4,8,16,32$.
• Gkls function [66]. The $f\left(x\right)=\mathrm{G}\mathrm{k}\mathrm{l}\mathrm{s}(x,n,w)$ is defined as a function with w local minima, and the dimension of the function was n. For the conducted experiments, the cases of $n=2,3$ and $w=50$ were used.
• Goldstein and Price function

  $$\begin{array}{ccc}\hfill f\left(x\right)& =& \left[1+{\left(x_1+x_2+1\right)}^2\right.\hfill \\ & & \left(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2\right)]\times \hfill \\ & & [30+{\left(2x_1-3x_2\right)}^2\hfill \\ & & \left(18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2\right)]\hfill \end{array}$$
• Griewank2 function, that has the following definition:

  $$f\left(x\right)=1+\frac{1}{200}\sum _{i=1}^2{x}_i^2-\prod _{i=1}^2\frac{cos\left(x_i\right)}{\sqrt{\left(i\right)}},x\in {[-100,100]}^2$$
• Griewank10 function defined as:

  $$f\left(x\right)=\sum _{i=1}^n\frac{x_i^2}{4000}-\prod _{i=1}^{n}cos\left(\frac{x_i}{\sqrt{i}}\right)+1$$

  with $n=10$.
• Hansen function. $f\left(x\right)={\sum }_{i=1}^{5}icos\left[(i-1)x_1+i\right]{\sum }_{j=1}^{5}jcos\left[(j+1)x_2+j\right]$, $x\in {[-10,10]}^2$.
• Hartman 3 function defined as:

  $$f\left(x\right)=-\sum _{i=1}^4{c}_{i}exp\left(-\sum _{j=1}^3{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$$

  with $x\in {[0,1]}^3$ and $a=\left(\begin{array}{ccc}3& 10& 30\\ 0.1& 10& 35\\ 3& 10& 30\\ 0.1& 10& 35\end{array}\right),c=\left(\begin{array}{c}1\\ 1.2\\ 3\\ 3.2\end{array}\right)$ and

  $$p=\left(\begin{array}{ccc}0.3689& 0.117& 0.2673\\ 0.4699& 0.4387& 0.747\\ 0.1091& 0.8732& 0.5547\\ 0.03815& 0.5743& 0.8828\end{array}\right)$$
• Hartman 6 function given by:

  $$f\left(x\right)=-\sum _{i=1}^4{c}_{i}exp\left(-\sum _{j=1}^6{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$$

  with $x\in {[0,1]}^6$ and $a=\left(\begin{array}{cccccc}10& 3& 17& 3.5& 1.7& 8\\ 0.05& 10& 17& 0.1& 8& 14\\ 3& 3.5& 1.7& 10& 17& 8\\ 17& 8& 0.05& 10& 0.1& 14\end{array}\right),c=\left(\begin{array}{c}1\\ 1.2\\ 3\\ 3.2\end{array}\right)$ and

  $$p=\left(\begin{array}{cccccc}0.1312& 0.1696& 0.5569& 0.0124& 0.8283& 0.5886\\ 0.2329& 0.4135& 0.8307& 0.3736& 0.1004& 0.9991\\ 0.2348& 0.1451& 0.3522& 0.2883& 0.3047& 0.6650\\ 0.4047& 0.8828& 0.8732& 0.5743& 0.1091& 0.0381\end{array}\right)$$
• Potential function, the well-known Lennard–Jones potential [67] defined as:

  $$V_{LJ}\left(r\right)=4ϵ\left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right]$$

  (2)

  is adopted as a test case here with $N=3,5$.
• Rastrigin function defined as:

  $$f\left(x\right)=x_1^2+x_2^2-cos\left(18x_1\right)-cos\left(18x_2\right),x\in {[-1,1]}^2$$
• Rosenbrock function.

  $$f\left(x\right)=\sum _{i=1}^{n-1}\left(100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right),-30\le x_i\le 30.$$
  For the conducted experiments, the values $n=4,8,16$ were utilized.
• Shekel 7 function.

$$f\left(x\right)=-\sum _{i=1}^7\frac{1}{(x-a_i){(x-a_i)}^T+c_i}$$

with $x\in {[0,10]}^4$ and $a=\left(\begin{array}{cccc}4& 4& 4& 4\\ 1& 1& 1& 1\\ 8& 8& 8& 8\\ 6& 6& 6& 6\\ 3& 7& 3& 7\\ 2& 9& 2& 9\\ 5& 3& 5& 3\end{array}\right),c=\left(\begin{array}{c}0.1\\ 0.2\\ 0.2\\ 0.4\\ 0.4\\ 0.6\\ 0.3\end{array}\right)$.

• Shekel 5 function.

$$f\left(x\right)=-\sum _{i=1}^5\frac{1}{(x-a_i){(x-a_i)}^T+c_i}$$

with $x\in {[0,10]}^4$ and $a=\left(\begin{array}{cccc}4& 4& 4& 4\\ 1& 1& 1& 1\\ 8& 8& 8& 8\\ 6& 6& 6& 6\\ 3& 7& 3& 7\end{array}\right),c=\left(\begin{array}{c}0.1\\ 0.2\\ 0.2\\ 0.4\\ 0.4\end{array}\right)$.

• Shekel 10 function.

$$f\left(x\right)=-\sum _{i=1}^{10}\frac{1}{(x-a_i){(x-a_i)}^T+c_i}$$

with $x\in {[0,10]}^4$ and $a=\left(\begin{array}{cccc}4& 4& 4& 4\\ 1& 1& 1& 1\\ 8& 8& 8& 8\\ 6& 6& 6& 6\\ 3& 7& 3& 7\\ 2& 9& 2& 9\\ 5& 5& 3& 3\\ 8& 1& 8& 1\\ 6& 2& 6& 2\\ 7& 3.6& 7& 3.6\end{array}\right),c=\left(\begin{array}{c}0.1\\ 0.2\\ 0.2\\ 0.4\\ 0.4\\ 0.6\\ 0.3\\ 0.7\\ 0.5\\ 0.6\end{array}\right)$.

• Sinusoidal function defined as:

  $$f\left(x\right)=-\left(2.5\prod _{i=1}^{n}sin\left(x_i-z\right)+\prod _{i=1}^{n}sin\left(5\left(x_i-z\right)\right)\right),0\le x_i\le \pi .$$
  For the current series of experiments the values $n=4,8,16$ and $z=\frac{\pi }{6}$ were used.
• Test2N function defined as:

  $$f\left(x\right)=\frac{1}{2}\sum _{i=1}^n{x}_i^4-16x_i^2+5x_i,x_i\in [-5,5].$$
  The function has $2^n$ local minima, and for the conducted experiments, the cases of $n=4,5,6,7$ were used.
• Test30N function defined as:

  $$f\left(x\right)=\frac{1}{10}{sin}^2\left(3\pi x_1\right)\sum _{i=2}^{n-1}\left({\left(x_i-1\right)}^2\left(1+{sin}^2\left(3\pi x_{i+1}\right)\right)\right)+{\left(x_n-1\right)}^2\left(1+{sin}^2\left(2\pi x_n\right)\right)$$

  where $x\in [-10,10]$. This function has ${30}^n$ local minima, and for the conducted experiments, the cases $n=3,4$ were used.

3.2. Experimental Results

The software used in the experiments was coded in ANSI-C++. Also, the freely available Optimus optimization environment was incorporated. The software can be downloaded from https://github.com/itsoulos/GlobalOptimus/ (accessed on 14 April 2024). Optimus is entirely written in ANSI-C++ and was prepared using the freely available QT library. All the experiments were executed on an AMD Ryzen 5950X with 128 GB of RAM. The operating system used was Debian Linux. In all experimental tables, the numbers in cells denote average function calls for 30 runs. In each run, a different seed for the random number generator was used. The decimal numbers enclosed in parentheses represent the success rate of the method in finding the global minimum of the corresponding function. If this number does not appear, then the method managed to discover the global minimum in each run. The simulation parameters for the used optimization techniques are listed in

Table 1. The experimental values for each parameter used in the conducted experiments.

PARAMETER	MEANING	VALUE
$N_c$	Number of armadillos or chromosomes	100
$N_g$	Maximum number of performed iterations	200
$p_l$	Local Search rate	0.05
$p_s$	Selection rate in genetic algorithm	0.10
$p_m$	Mutation rate in genetic algorithm	0.05

. The values for these parameters were chosen to strike a balance between the expected efficiency of the optimization methods and their speed. All techniques used uniform distribution to initialize the corresponding population.

The results from the conducted experiments are outlined in Table 2. The following applies to this table:

1.

The column PROBLEM denotes the objective problem.

2.

The column GENETIC stands for the average function calls for the Genetic algorithm. The same number of armadillos and chromosomes and particles was used in the experiments conducted to make a fair comparison between the algorithms. Also, the same number of maximum generations and the same stopping criteria were utilized among the different optimization methods.

3.

The column PSO stands for the application of a Particle Swarm Optimization method in the objective problem. The number of particles and the stopping rule in the PSO method are the same as in the proposed method.

4.

The column GWO stands for the application of Gray Wolf Optimizer [68] on the benchmark functions.

5.

The column PROPOSED represents the experimental results for the Gao method with the suggested modifications.

6.

The final row, denoted as SUM, stands for the sum of the function calls and the average success rate for all the used objective functions.

The statistical comparison for the previous experimental results is depicted in Figure 2. The previous experiments and their subsequent statistical processing demonstrate that the proposed method significantly outperforms Particle Swarm Optimization when a comparison is conducted for the average number of function calls since it requires 20% fewer function calls on average to efficiently find the global minimum. In addition, the proposed method appears to have similar efficiency in terms of required function calls to that of the Genetic Algorithm.

Table 2. Experimental results and comparison against other methods. The stopping rule used is the Similarity stopping rule.

PROBLEM	Genetic	PSO	GWO	PROPOSED
BF1	2179	2364 (0.97)	2927	2239
BF2	1944	2269 (0.90)	2893	1864
BRANIN	1177	2088	2430	1179
CAMEL	1401	2278	1533	1450
EASOM	979	2172	2074 (0.93)	886
EXP4	1474	2231	1640	1499
EXP8	1551	2256	2392	1539
EXP16	1638	2165	3564	1581
EXP32	1704	2106	5631	1567
GKLS250	1195	2113	1407	1292
GKLS350	1396 (0.87)	1968	1889	1510
GOLDSTEIN	1878	2497	1820	1953
GRIEWANK2	2360 (0.87)	3027 (0.97)	1914	2657
GRIEWANK10	3474 (0.87)	3117 (0.87)	3427 (0.13)	4064 (0.97)
HANSEN	1761 (0.97)	2780	2290 (0.60)	1885
HARTMAN3	1404	2086	1497	1448
HARTMAN6	1632	2213 (0.87)	1616 (0.53)	1815
POTENTIAL3	2127	3557	1520	1942
POTENTIAL5	3919	7132	2544	3722
RASTRIGIN	2438 (0.97)	2754	1858	2411
ROSENBROCK4	1841	2909	4925	2690
ROSENBROCK8	2570	3382	5662	3573
ROSENBROCK16	4331	3780	6752	5085
SHEKEL5	1669 (0.97)	2700	1297 (0.53)	1911
SHEKEL7	1696	2612	1472 (0.60)	1930
SHEKEL10	1758	2594	1224 (0.57)	1952
TEST2N4	1787 (0.97)	2285	1680 (0.80)	1840 (0.83)
TEST2N5	2052 (0.93)	2368 (0.97)	1791 (0.73)	2029 (0.63)
TEST2N6	2216 (0.73)	2330 (0.73)	1710 (0.53)	2438 (0.80)
TEST2N7	2520 (0.73)	2378 (0.63)	1631 (0.47)	2567 (0.60)
SINU4	1514	2577	1241 (0.70)	1712
SINU8	1697	2527	1184 (0.83)	1992
SINU16	2279 (0.97)	2657	1296 (0.67)	2557
TEST30N3	1495	3302	1527	1749
TEST30N4	1897	3817	2520	2344
SUM	68,953 (0.97)	95,391 (0.97)	82,778 (0.85)	74,982 (0.97)

Table 2. Experimental results and comparison against other methods. The stopping rule used is the Similarity stopping rule.

PROBLEM	Genetic	PSO	GWO	PROPOSED
BF1	2179	2364 (0.97)	2927	2239
BF2	1944	2269 (0.90)	2893	1864
BRANIN	1177	2088	2430	1179
CAMEL	1401	2278	1533	1450
EASOM	979	2172	2074 (0.93)	886
EXP4	1474	2231	1640	1499
EXP8	1551	2256	2392	1539
EXP16	1638	2165	3564	1581
EXP32	1704	2106	5631	1567
GKLS250	1195	2113	1407	1292
GKLS350	1396 (0.87)	1968	1889	1510
GOLDSTEIN	1878	2497	1820	1953
GRIEWANK2	2360 (0.87)	3027 (0.97)	1914	2657
GRIEWANK10	3474 (0.87)	3117 (0.87)	3427 (0.13)	4064 (0.97)
HANSEN	1761 (0.97)	2780	2290 (0.60)	1885
HARTMAN3	1404	2086	1497	1448
HARTMAN6	1632	2213 (0.87)	1616 (0.53)	1815
POTENTIAL3	2127	3557	1520	1942
POTENTIAL5	3919	7132	2544	3722
RASTRIGIN	2438 (0.97)	2754	1858	2411
ROSENBROCK4	1841	2909	4925	2690
ROSENBROCK8	2570	3382	5662	3573
ROSENBROCK16	4331	3780	6752	5085
SHEKEL5	1669 (0.97)	2700	1297 (0.53)	1911
SHEKEL7	1696	2612	1472 (0.60)	1930
SHEKEL10	1758	2594	1224 (0.57)	1952
TEST2N4	1787 (0.97)	2285	1680 (0.80)	1840 (0.83)
TEST2N5	2052 (0.93)	2368 (0.97)	1791 (0.73)	2029 (0.63)
TEST2N6	2216 (0.73)	2330 (0.73)	1710 (0.53)	2438 (0.80)
TEST2N7	2520 (0.73)	2378 (0.63)	1631 (0.47)	2567 (0.60)
SINU4	1514	2577	1241 (0.70)	1712
SINU8	1697	2527	1184 (0.83)	1992
SINU16	2279 (0.97)	2657	1296 (0.67)	2557
TEST30N3	1495	3302	1527	1749
TEST30N4	1897	3817	2520	2344
SUM	68,953 (0.97)	95,391 (0.97)	82,778 (0.85)	74,982 (0.97)

The reliability of the termination techniques was tested with one more experiment, in which both proposed termination rules were used, and the produced results for the benchmark functions are presented in

Table 3. Average number of function calls for the proposed method using the two suggested termination rules.

PROBLEM	Similarity	Doublebox
BF1	2239	2604
BF2	1974	1864
BRANIN	1179	1179
CAMEL	1450	1245
EASOM	886	775
EXP4	1499	1332
EXP8	1539	1371
EXP16	1581	1388
EXP32	1567	1384
GKLS250	1292	1483
GKLS350	1510	2429
GOLDSTEIN	1953	2019
GRIEWANK2	2657	5426
GRIEWANK10	4064 (0.97)	4940 (0.97)
HANSEN	1885	4482
HARTMAN3	1448	1458
HARTMAN6	1815	1625
POTENTIAL3	1942	1700
POTENTIAL5	3722	3395
RASTRIGIN	2411	4591
ROSENBROCK4	2690	2371
ROSENBROCK8	3573	3166
ROSENBROCK16	5085	4386
SHEKEL5	1911	1712
SHEKEL7	1930	1722
SHEKEL10	1952	1956
TEST2N4	1840 (0.83)	3103 (0.83)
TEST2N5	2029 (0.63)	3375 (0.67)
TEST2N6	2438 (0.80)	4458 (0.83)
TEST2N7	2567 (0.60)	4425 (0.63)
SINU4	1712	1657
SINU8	1992	1874
SINU16	2557	2612
TEST30N3	1749	1483
TEST30N4	2344	2737
SUM	74,982 (0.97)	87,727 (0.97)

. Also, the statistical comparison for the experiment is shown graphically in Figure 3.

From the statistical processing of the experimental results, one can find that the termination method using the Similarity criterion demands a lower number of function calls than the DoubleBox stopping rule to achieve the goal, which is to effectively find the global minimum. Furthermore, there is no significant difference in the success rate of the two termination techniques, as reflected in the success rate in finding the global minimum, which remains high for both techniques (around 97%).

Moreover, the effect of the application of the local search technique is explored in the experiments shown in

Table 4. Experimental results using different values for the local search rate and the proposed method.

PROBLEM	${\mathit{p}}_{\mathit{l}}=0.005$	${\mathit{p}}_{\mathit{l}}=0.01$	${\mathit{p}}_{\mathit{l}}=0.05$
BF1	1531 (0.97)	1559	2239
BF2	1457 (0.97)	1319	1864
BRANIN	921	913	1179
CAMEL	1037	1022	1450
EASOM	871	850	886
EXP4	942	926	1499
EXP8	930	936	1539
EXP16	1020	961	1581
EXP32	1005	982	1567
GKLS250	1197	1106	1292
GKLS350	1256	1221	1510
GOLDSTEIN	1124	1146	1953
GRIEWANK2	1900 (0.93)	1976 (0.97)	2657
GRIEWANK10	1444 (0.40)	1963 (0.70)	4064 (0.97)
HANSEN	1872	1726 (0.93)	1885
HARTMAN3	1005	967	1448
HARTMAN6	976 (0.87)	1052 (0.97)	1815
POTENTIAL3	1018	1081	1942
POTENTIAL5	1313	1439	3722
RASTRIGIN	1614 (0.97)	1687 (0.97)	2411
ROSENBROCK4	1097	1203	2690
ROSENBROCK8	1179	1403	3573
ROSENBROCK16	1437	1801	5085
SHEKEL5	1070 (0.97)	1073	1911
SHEKEL7	1076 (0.93)	1124	1930
SHEKEL10	1152 (0.97)	1170 (0.97)	1952
TEST2N4	1409 (0.80)	1285 (0.87)	1840 (0.83)
TEST2N5	1451 (0.53)	1350 (0.63)	2029 (0.63)
TEST2N6	1417 (0.60)	1529 (0.67)	2438 (0.80)
TEST2N7	1500 (0.47)	1451 (0.33)	2567 (0.60)
SINU4	1210	1199	1712
SINU8	1163	1145	1992
SINU16	1377	1296	2557
TEST30N3	1057	1189	1749
TEST30N4	1897	3817	2344
SUM	43,213 (0.92)	44,331 (0.94)	74,982 (0.97)

, where the local search rate increases from 0.5% to 5%.

As expected, the success rate in discovering the global minimum increases as the rate of application of the local minimization technique increases. For the case of the current method, this rate increases from 92% to 97% in the experimental results. This finding demonstrates that if this method is combined with effective local minimization techniques, it can lead to a more efficient finding of the global minimum for the objective function.

Also, to measure the time complexity of the proposed work, the ELP (High Elliptic Function) function was employed with arbitrary dimensions. The function is defined as:

$$f\left(x\right)=\sum _{i=1}^n{\left({10}^6\right)}^{\frac{i-1}{n-1}}{x}_i^2$$

In this test, the dimension of the function (n) increased from 1 to 15, and the average execution time was measured. The results obtained for the similarity termination rule are outlined in Figure 4, and the results for the DoubleBox termination rule are graphically shown in Figure 5.

As expected, the execution time increases as the dimensions of the function increase, but there are no significant differences between the execution times of the three optimization methods.

Furthermore, as a practical application, consider the training of an artificial neural network for classification or data fitting problems [69,70]. Neural networks are non-linear parametric tools with many applications in real-world problems [71,72,73]. Neural networks can be defined as functions $N(\overrightarrow{x},\overrightarrow{w})$. The vector $\overrightarrow{x}$ represents the input pattern, while the vector $\overrightarrow{w}$ represents the weight vector of the neural network that should be estimated. Optimization methods can be used to estimate the set of weights by minimizing the following equation:

$$E\left(N\left(\overrightarrow{x},\overrightarrow{w}\right)\right)=\sum _{i=1}^M{\left(N\left({\overrightarrow{x}}_i,\overrightarrow{w}\right)-y_i\right)}^2$$

(3)

The quantity of Equation (3) was minimized using the mentioned algorithm of this work for the BK dataset [74], which is used to estimate the points in a basketball game. The average test error using the four methods presented in this article is shown graphically in Figure 6. To validate the results, the well-known ten-fold cross method was applied. The current work has the same performance as the PSO algorithm and significantly outperforms the Genetic algorithm.

4. Conclusions

Two modifications for the Giant-Armadillo Optimization method were suggested in this article. These modifications aimed to improve the efficiency and the speed of the underlying global optimization algorithm. The first modification suggested the periodic application of a local optimization procedure to randomly selected armadillos from the current population. The second modification utilized some stopping rules from the recent bibliography to stop the more efficient optimization method and to avoid unnecessary iterations when the global minimum was already discovered. The modified global optimization method was tested against two other global optimization methods from the relevant literature and, more specifically, an implementation of the Genetic Algorithm and a Particle Swarm Optimization variant on a series of well-known test functions. To make a fair comparison between these methods, the same number of test solutions (armadillo or chromosomes) and the same termination rule were used. The present technique, after comparing the experimental results, clearly outperforms particle optimization and has a similar behavior to that of the genetic algorithm. Also, after a series of experiments, it was shown that the Similarity termination rule outperforms the DoubleBox termination rule in terms of function calls without reducing the effectiveness of the proposed method in the task of locating the global minimum.

Since the experimental results have been shown to be extremely promising, further efforts can be made to develop the technique in various fields. For example, an extension could be to develop a termination rule that exploits the particularities of the particular global optimization technique. Among the future extensions of the application may be the use of parallel computing techniques to speed up the optimization process, such as the incorporation of the MPI [75] or the OpenMP library [76]. For example, in this direction, it could be investigated to parallelize the technique in a similar way as genetic algorithms using islands [77,78].