One of the challenging problems with evolutionary computing algorithms is to maintain the balance between exploration and exploitation capability in order to search global optima. A novel convergence track based adaptive differential evolution (CTbADE) algorithm is presented in this research paper. The crossover rate and mutation probability parameters in a differential evolution algorithm have a significant role in searching global optima. A more diverse population improves the global searching capability and helps to escape from the local optima problem. Tracking the convergence path over time helps enhance the searching speed of a differential evolution algorithm for varying problems. An adaptive powerful parameter-controlled sequences utilized learning period-based memory and following convergence track over time are introduced in this paper. The proposed algorithm will be helpful in maintaining the equilibrium between an algorithm's exploration and exploitation capability. A comprehensive test suite of standard benchmark problems with different natures, i.e., unimodal/multimodal and separable/non-separable, was used to test the convergence power of the proposed CTbADE algorithm. Experimental results show the significant performance of the CTbADE algorithm in terms of average fitness, solution quality, and convergence speed when compared with standard differential evolution algorithms and a few other commonly used state-of-the-art algorithms, such as jDE, CoDE, and EPSDE algorithms. This algorithm will prove to be a significant addition to the literature in order to solve real time problems and to optimize computational models with a high number of parameters to adjust during the problem-solving process.

Differential evolution (DE) is a stochastic algorithm introduced by Storn and Price [

In a DE algorithm, all population members have the same chance of selection to be a parent but one or more amplified weighted different vector is added in the third vector to generate a new vector [

The deadly Covid-19 pandemic has badly affected the whole world in a very short span of time. One of the major issues with Covid-19 is the time taken for the diagnostic process [

This sections contains details of the original DE algorithm and a literature review.

A DE algorithm is a population based stochastic algorithm used to evolve a population of individuals over time by applying various operators, such as the selection operator, crossover operator and mutation operator. The members of a population are initialized by means of small random values from a search space of n-dimensions. This paper uses D-Dimensional minimizations optimization problems without loss of generality:
_{DIM}) represents DIM dimension members of populations and

The details of DE algorithm operators are as follows

This operator is used to generate a donor vector, sometimes called a mutant vector, which is generated by scaling the difference vector of different members of population. A number of mutation strategies are available in the literature, the most commonly used mutation strategy has the following equation:
^{th} iteration,

In this operation, a trial vector is generated by utilizing the donor vector _{k}

The term rand [0, 1] is used to generate random numbers of a uniform nature from the range [0–1], rnbr will generate the index of a random vector in the range ^{th} target vector for Gn^{th} generation, ^{th} donor vector for G_{n} _{+} _{1}^{th} generation

After mutation and crossover operations, a trial vector is evaluated by using the fitness function according to the optimization problem. It is basically a greedy method based on the concept of survival of the fittest that will select the best vector of the target vector and trial vector, based on their fitness value. The equation of the selection operator is
^{th} target vector,

The performance of DE algorithms is affected by the selection of mutation variant, crossover variant and the control parameters of the CR, mutation probability, and population size. A large number of studies have been carried out by researchers on various aspects of DE algorithms, such as improvements in control parameter (such as deterministic, adaptive, self-adaptive, fuzzy-based, etc.), enhancement of mutation variants, crossover strategies, population-based variations, and hybridization with other algorithms. Reference [

A memory-based self-adaptive control parameter with a strategy adaption concept is introduced in [

Reference [

A mechanism for adapting strategy parameters and populations in DE based on multivariate probabilistic self-adaptive control parameters was presented in [

Mutation strategies and control parameters have a major impact on DE algorithms’ performance, which can be enhanced by following a convergence track during the evolutionary process. A smaller value of mutation probability F focuses on exploitation, and a bigger value of F focuses on the exploration capability of a DE algorithm [

The value of F is increased so that, from a very diverse population, we should start by focusing on the exploration so that we should move in the direction of convergence and its values are used in the increasing sequence. This algorithm will be able to escape from local optima in a controlled manner and in a controlled range which is described in the following range. By default, the sequence of control parameter F is increased using

The mutation strategy

Initialize the control parameter for the range of time/iteration based controlled sequence:

[F__{min}= MIN

_{1 }=

_{ }0.5, F_

_{max}= MAX

_{1 }=

_{ }0.7], [CR_

_{min}= MIN

_{2 }=

_{ }0.7, CR_

_{max}= MAX

_{2 }=

_{ }0.9]

In the first step, generate

Initialize mutation probability F and CR for all individuals of population

FOR

Calculate fitness

END FOR

Initialize the starting value of mutation probability F_cur with an initial value of the mutation probability sequence and the current value of crossover probability CR_cur with an initial value of CR rate using following

WHILE

Select three individual parent individuals that will be used in the following mutation strategy from the current population

FOR

To get a donor vector ^{th} given vector ^{th} Mutation Probability F__{cur}

END FOR

FOR

To get trial vector ^{th} given vector _{_cur} and

END FOR

FOR

Use fitness function

IF

IF

END IF

ELSE

_{k,Gn+1=X,Gn}

END IF

END FOR

Update the LP based tracking for adaption of control parameters.

Continue with the same sequence increasing/decreasing control parameters F and CR using

ELSE

Reverse the sequence of CR and F control parameters (increasing to decreasing and decreasing to increasing) using

END IF

Increase the generation number Gn = Gn + 1

END WHILE

Comprehensive standard problems to test the convergence power of the CTbADE algorithm are taken from [

This section contains the average fitness performance of the original DE and a few more commonly used state-of-the-art algorithms, such as CoDE, EPSDE, jDE, and the proposed CTbADE. These experimental results are generated by implementing the algorithm in C++ using Core-i3 machine 4 GB RAM, 4 GHZ CPU, Windows 10 and by using the performance parameters given in Section 4 of this paper. The benchmark functions of varying nature are used to test the convergence power of the proposed CTbADE algorithm. The experimental results of average fitness values are reported in

Control parameter | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Factor | ADE |
FADE |
jDE |
JADE |
SaDE |
NSDE |
EPSDE |
CoDE |
MDE- |
GOjDE |
iJADE |
ZEPDE |
SLADE |
SADEHH |
SAKPDE |
SaDSDE |
MSaDE |
SLCIDE |
Proposed CTbADE |

Diversity | Lo | Md | Lo | Hg | Hg | Hg | Hg | Hg | Hg | Hg | Hg | Hg | Hg | Hg | Hg | Hg | Hg | Hg | Hg |

Local optima handling ability | Lo | Md | Md | Md | Md | Md | Md | Md | Md | Md | Md | Md | Md | Md | Hg | Hg | Md | Md | Hg |

Convergence speed focus | Lo | Md | Md | Md | Md | Md | Hg | Hg | Md | Md | Md | Md | Md | Hg | Hg | Hg | Hg | Hg | Hg |

Impact of dimensionality | Hg | Md | Md | Md | Md | Md | Md | Md | Md | Md | Md | Md | Md | Md | Lo | Lo | Md | Lo | Lo |

Deterministic | Ys | Ys | Ys | No | No | Ys | Ys | Ys | Ys | No | No | No | No | No | No | No | No | No | No |

Self-adaptive | No | No | No | No | Md | No | No | No | No | No | Md | No | Md | Md | Hg | Hg | No | No | Hg |

Parameter memorization | No | No | No | No | Ys | No | No | No | No | No | No | No | No | No | Ys | Ys | No | No | No |

Convergence tracking over time | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | Ys |

Time tracking | |||||||||||||||||||

controlled sequencing | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | Ys |

Time tracking adaptive | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | No | Ys |

Function | DE | CoDE | EPSDE | jDE | CTbADE |
---|---|---|---|---|---|

_{1} |
6.02E − 217 (0) | 7.63E − 230 (0) | 1.64E − 227 (0) | 2.17E − 223 (0) | |

_{2} |
5.18E − 239 (0) | 5.49E − 257 (0) | 1.37E − 245 (0) | 9.12E − 241 (0) | |

_{3} |
2.82E − 159 (2.30E − 316) | 1.00E − 46 (2.56E − 46) | 3.58E − 80 (6.14E − 80) | 1.22E − 81 (6.14E − 81) | |

_{4} |
2.85E + 00 (1.82E + 00) | 1.15E − 01 (8.70E − 02) | 3.99E − 01 (1.20E + 00) | ||

_{5} |
9.15E − 01 (8.70E − 01) | 0 (0) | 1.85E + 01 (8.78E + 00) | ||

_{6} |
2.18E − 02 (1.85E − 02) | 9.04E − 04 (2.81E − 03) | 2.91E − 01 (3.12E − 01) | ||

_{7} |
1.45E − 51 (7.46E − 51) | 0 (0) | 5.41E − 291 (0) | ||

_{8} |
8.85E − 02 (8.32E − 02) | 1.38E − 12 (1.94E − 12) | 4.18E − 03 (2.25E − 02) | 9.09E − 01 (3.26E − 01) | |

_{9} |
5.25E − 07 (6.70E − 07) | 1.06E − 07 (1.15E − 07) | 2.37E − 08 (3.83E − 08) | 2.91E − 02 (9.58E − 02) | |

_{10} |
9.87E − 160 (5.31E − 318) | 2.92E − 64 (4.44E − 64) | 3.95E − 96 (7.57E − 96) | 1.73E − 98 (5.00E − 98) | |

_{11} |
1.63E − 107 (3.40E − 107) | 6.81E − 129 (5.85E − 129) | 2.74E − 115 (3.57E − 115) | 1.26E − 124 (4.19E − 124) | |

_{12} |
1.12E + 01 (1.35E + 01) | ||||

_{13} |
6.73E − 88 (3.62E − 87) | ||||

_{14} |
1.05E − 04 (1.25E − 04) | 8.98E − 06 (1.04E − 05) | 2.00E − 05 (1.99E − 05) | 1.19E − 05 (1.38E − 05) | |

_{15} |
1.55E − 01 (3.38E − 01) | ||||

_{16} |
|||||

_{17} |
2.41E − 01 (1.50E − 01) | ||||

_{18} |
1.08E − 210 (0) | 1.41E − 224 (0) | 7.01E − 222 (0) | 5.92E − 215 (0) | |

_{19} |
5.99E − 217 (0) | 3.55E − 230 (0) | 7.14E − 228 (0) | 2.75E − 223 (0) | |

_{20} |
5.02E − 214 (0) | 1.95E − 227 (0) | 2.56E − 225 (0) | 7.64E − 222 (0) | |

_{21} |
6.04E − 213 (0) | 2.75E − 227 (0) | 5.39E − 225 (0) | 2.83E − 219 (0) | |

_{22} |
|||||

_{23} |
3.50E − 06 (3.65E − 06) | 1.89E − 07 (1.66E − 07) | 7.74E − 07 (6.72E − 07) | 4.38E − 08 (1.50E − 07) | |

_{24} |
1.63E − 15 (2.14E − 15) | ||||

_{25} |
2.21E − 15 (2.46E − 15) | ||||

_{26} |
|||||

_{27} |
1.94E − 15 (1.47E − 16) | 1.34E − 15 (3.89E − 16) | 1.99E − 15 (1.16E − 16) | 1.90E − 15 (2.13E − 16) | |

_{28} |
5.47E − 01 (8.65E − 02) | 2.79E − 41 (5.19E − 41) | 1.35E − 63 (6.98E − 63) | 5.78E − 01 (4.05E − 01) | |

_{29} |
2.42E − 08 (6.82E − 08) | 5.96E − 09 (1.10E − 08) | 2.32E − 10 (4.09E − 10) | 4.67E − 29 (4.90E − 29) | |

_{30} |
2.13E − 04 (1.80E − 05) | 3.49E − 04 (6.03E − 05) |

Function | DE | CoDE | EPSDE | jDE | CTbADE |
---|---|---|---|---|---|

_{1} |
1.72E − 226 (0) | 1.99E − 191 (0) | 5.06E − 255 (0) | 1.70E − 241 (0) | |

_{2} |
3.35E − 234 (0) | 8.43E − 204 (0) | 5.29E − 263 (0) | 1.07E − 252 (0) | |

_{3} |
9.33E − 66 (4.99E − 65) | 1.24E − 05 (6.91E − 06) | 1.72E − 37 (2.73E − 37) | 8.01E − 39 (3.05E − 38) | |

_{4} |
6.21E + 00 (1.90E + 00) | 4.34E − 30 (1.46E − 29) | 1.93E + 00 (1.29E + 00) | 1.33E + 00 (1.88E + 00) | |

_{5} |
7.06E + 00 (2.48E + 00) | 3.32E − 01 (5.93E − 01) | 5.39E + 01 (1.41E + 01) | ||

_{6} |
1.31E − 03 (2.97E − 03) | 8.22E − 04 (2.49E − 03) | 3.39E − 01 (6.38E − 01) | ||

_{7} |
3.40E − 17 (1.83E − 16) | 1.49E − 162 (6.42e − 323) | |||

_{8} |
2.15E − 01 (5.32E − 02) | 1.22E − 12 (1.09E − 12) | 3.96E − 02 (4.57E − 02) | 1.99E + 00 (3.33E − 01) | |

_{9} |
2.11E − 07 (2.19E − 07) | 2.64E − 08 (2.02E − 08) | 8.12E − 08 (8.05E − 08) | 2.61E + 00 (1.33E + 00) | |

_{10} |
3.40E − 83 (1.63E − 82) | 4.80E − 18 (3.96E − 18) | 1.61E − 47 (2.37E − 47) | 3.44E − 57 (8.83E − 57) | |

_{11} |
1.40E − 112 (3.86E − 112) | 1.89E − 110 (1.22E − 110) | 7.92E − 126 (1.02E − 125) | 1.10E − 138 (2.13E − 138) | |

_{12} |
1.31E + 03 (8.83E + 02) | ||||

_{13} |
5.81E − 14 (3.02E − 13) | ||||

_{14} |
4.36E − 05 (4.61E − 05) | 1.76E − 05 (1.71E − 05) | 1.23E − 05 (1.56E − 05) | 4.18E − 06 (6.47E − 06) | |

_{15} |
1.65E − 31 (4.23E − 33) | 1.19E − 01 (1.59E − 01) | |||

_{16} |
4.58E − 05 (5.02E − 05) | 7.37E − 06 (5.73E − 06) | 3.76E − 05 (3.55E − 05) | 5.28E − 06 (1.34E − 05) | |

_{17} |
1.02E + 00 (2.92E − 01) | ||||

_{18} |
5.07E − 221 (0) | 2.07E − 185 (0) | 3.14E − 249 (0) | 6.65E − 235 (0) | |

_{19} |
8.78E − 226 (0) | 2.17E − 191 (0) | 2.88E − 255 (0) | 2.21E − 242 (0) | |

_{20} |
1.74E − 223 (0) | 1.02E − 188 (0) | 6.25E − 253 (0) | 7.45E − 241 (0) | |

_{21} |
8.78E − 224 (0) | 1.69E − 188 (0) | 6.99E − 252 (0) | 3.54E − 238 (0) | |

_{22} |
3.07E − 31 (4.72E − 31) | 0 (0) | 0 (0) | 6.82E − 31 (7.82E − 31) | |

_{23} |
1.63E − 06 (2.18E − 06) | 2.51E − 07 (2.73E − 07) | 8.82E − 07 (7.42E − 07) | 2.03E − 07 (2.69E − 07) | |

_{24} |
1.01E − 14 (7.36E − 15) | ||||

_{25} |
9.28E − 15 (3.34E − 15) | ||||

_{26} |
|||||

_{27} |
3.78E − 15 (2.92E − 16) | 2.35E − 15 (5.75E − 16) | 4.12E − 15 (9.41E − 17) | 3.74E − 15 (2.68E − 16) | |

_{28} |
4.01E + 00 (3.84E − 01) | 1.93E − 30 (3.75E − 30) | 8.10E − 57 (2.08E − 56) | 2.97E + 00 (9.45E − 01) | |

_{29} |
1.31E − 08 (2.85E − 08) | 3.84E − 09 (6.54E − 09) | 7.36E − 11 (1.21E − 10) | 4.97E − 29 (4.89E − 29) | |

_{30} |
2.52E − 08 (4.26E − 09) | 3.93E − 08 (6.55E − 09) |

Overall, from the experimental results it is clear that the CTbADE algorithm has a dominating performance over that of the standard DE, CoDE, EPSDE, and jDE algorithms. The CTbADE algorithm outperforms other well-known and state-of-the-art algorithms in most cases. The CTbADE algorithm's convergence is better than the other algorithms for unimodal, multimodal, separable and non-separable functions. It shows that we can apply the CTbADE algorithm to solve problems of varying natures. In cases of 10D, 20D, and 30D, CTbADE outperforms other algorithms for unimodal problems, such as the axis parallel hyperellipsoid, Schwefel's problem, Schwefel's problem 2.22, ellipse function and tablet function, as well as the Neumaier-2 problem in the cases of 20D and 30D. The performance of CTbADE is better in multimodal problems, such as the sphere model in cases of 10D, 20D, and 30D; sum of difference power in the case of 10D; the Zakharov function in cases of 10D, 20D and 30D; cigar function in cases of 10D, 20D, and 30D; function-15 in cases of 10D, 20D, and 30D; the problem of the deflected corrugated spring in cases of 10D, 20D, and 30D; the problem of multi-model global optimization in the case of 20D; and the Quintic global optimization problem also in the case of 20D. The performance of CTbADE is better in separable problems, such as the sphere model in cases of 10D, 20D, and 30D; the axis parallel hyperellipsoid in cases of 10D, 20D, and 30D; the Neumaier-2 problem in 20D and 30D; cigar in cases of 10D, 20D, and 30D; function-15 in cases of 10D, 20D, and 30D; the ellipse function in case of 10D, 20D, and 30D; the tablet function in case of 10D, 20D, and 30D; the deflected corrugated spring in cases of 10D, 20D, and 30D; the multimodal global optimization problem and the Quintic global optimization problem, both in the case of 20D. The performance of CTbADE is better for non-separable problems, such as Schwefel's problem 1.2 in cases of 10D, 20D, and 30D; sum of different power in the case of 10D; Zakharov function in 10D, 20D, and 30D; Schwefel's problem 2.22 in cases of 10D, 20D, and 30D. Although the fitness value of the given average value fitness tables are same, the convergence speed of CTbADE is better for the unimodal problem De Jong's no-noise function-4 in cases of 10D and 20D. The performance of CTbADE is better for unimodal problems in 10D and 20D. The convergence speed of CTbADE is better for multimodal problems, such as Alpine function in the case of 10D; Schewel function in case of 10D; the multimodal global optimization problem in the case of 30D; the Quintic global optimization problem in case of 10D and 30D; the stretched_V_global_optimization in cases of 20D and 30D. The performance of CTbADE is better for separable problems, such as the sum of different power in the case of 20D; De Jong's no-noise function-4 in cases of 10D and 20D; Alpine function in the case of 10D; Schewel function in the case of 10D; the multimodal global optimization problem in the case of 30D and the Quintic global optimization problem in cases of 10D and 30D. The performance of CTbADE is better for non-separable problems, such as the sum of different power problem in the case of 20D and the stretched_V_global_optimization problem in the case of 20D and 30D.

The performance of the CTbADE algorithm is competitive compared to other algorithms for unimodal functions: De Jong's no-noise function-4 in the case of 30D and the Neumaier-2 problem in the case of 10D; for multimodal functions sum of different problem in case of 30D; Alpine function in case of 20D and 30D; multimodal global optimization problem in case of 10D and 20D; Quintic global optimization problem in the case of 30D; stretched V global optimization problem in the case of 20D; Xin-SheYang in the case of 30D; separable functions for De Jong's no noise function-4 in the case of 30D; Neumaier-2 Problem in the case of 10D; Alpine function in cases of 20D and 30D; multimodal global optimization problem in cases of 10D and 20D; Quintic global optimization problem in the case of 30D; stretched_V_global_optimization problem in the case of 20D; for the non-separable problem sum of different power in case of 30D; Xin-SheYang in the case of 30D.

Logarithmic convergence graphs of some sampled problems have been shown in

Function | DE | CoDE | EPSDE | jDE | CTbADE |
---|---|---|---|---|---|

_{1} |
9.98E − 237 20) | 7.84E − 164 (0) | 1.48E − 282 (0) | 7.08E − 259 (0) | |

_{2} |
2.12E − 242 (0) | 3.30E − 171 (0) | 1.20E − 288 (0) | 3.42E − 265 (0) | |

_{3} |
4.52E − 39 (2.17E − 38) | 3.52E + 01 (2.38E + 01) | 7.37E − 22 (2.41E − 21) | 3.37E − 22 (8.11E − 22) | |

_{4} |
1.28E + 01 (6.32E + 00) | 2.28E − 19 (2.45E − 19) | 1.01E + 01 (3.91E + 00) | 6.64E − 01 (1.49E + 00) | |

_{5} |
1.35E + 01 (5.01E + 00) | 1.29E + 00 (1.12E + 00) | 7.66E + 01 (1.62E + 01) | ||

_{6} |
2.05E − 03 (4.76E − 03) | 1.81E − 21 (9.73E − 21) | 2.47E − 04 (1.33E − 03) | 2.43E − 01 (8.99E − 01) | |

_{7} |
1.01E − 10 (5.26E − 10) | 0 (0) | 5.75E − 12 (3.10E − 11) | 0 (0) | |

_{8} |
2.53E − 01 (4.49E − 02) | 2.25E − 13 (2.07E − 13) | 1.68E − 12 (1.58E − 12) | 8.75E − 02 (4.78E − 02) | 2.36E + 00 (3.47E − 01) |

_{9} |
7.68E − 08 (5.84E − 08) | 1.67E − 08 (1.27E − 08) | 7.43E − 08 (6.60E − 08) | 9.01E + 00 (3.92E + 00) | |

_{10} |
3.37E − 52 (1.35E − 51) | 1.01E − 07 (5.38E − 08) | 1.79E − 29 (2.01E − 29) | 7.57E − 40 (1.75E − 39) | |

_{11} |
6.86E − 123 (8.30E − 123) | 1.20E − 96 (7.18E − 97) | 2.84E − 137 (3.49E − 137) | 1.64E − 148 (5.54E − 148) | |

_{12} |
5.50E + 03 (2.09E + 03) | ||||

_{13} |
1.62E − 10 (8.74E − 10) | 1.06E − 271 (0) | 0 (0) | ||

_{14} |
3.43E − 05 (3.19E − 05) | 2.00E − 06 (1.84E − 06) | 8.72E − 06 (6.83E − 06) | 4.25E − 06 (3.62E − 06) | 3.37E − 06 (3.28E − 06) |

_{15} |
1.66E − 01 (2.22E − 01) | ||||

_{16} |
7.63E − 05 (5.44E − 05) | 2.75E − 05 (3.79E − 05) | 6.55E − 05 (9.10E − 05) | 7.78E − 06 (1.38E − 05) | |

_{17} |
9.85E − 03 (3.69E − 02) | 1.73E + 00 (4.77E − 01) | |||

_{18} |
4.16E − 231 (0) | 3.63E − 158 (2.57e − 315) | 2.80E − 277 (0) | 9.08E − 255 (0) | |

_{19} |
5.44E − 236 (0) | 2.19E − 163 (0) | 9.77E − 283 (0) | 1.28E − 259 (0) | |

_{20} |
2.60E − 233 (0) | 4.45E − 161 (2.33e − 321) | 2.08E − 280 (0) | 1.68E − 257 (0) | |

_{21} |
8.53E − 235 (0) | 5.16E − 161 (3.85e − 321) | 3.50E − 280 (0) | 8.76E − 255 (0) | |

_{22} |
5.88E − 30 (7.86E − 30) | 9.04E − 32 (1.71E − 31) | 4.48E − 30 (5.10E − 30) | 1.40E − 31 (2.65E − 31) | |

_{23} |
1.64E − 06 (1.61E − 06) | 2.87E − 07 (3.07E − 07) | 6.26E − 07 (5.38E − 07) | 2.23E − 07 (2.86E − 07) | |

_{24} |
1.00E + 09 (0) | ||||

_{25} |
1.00E + 09 (0) | ||||

_{26} |
|||||

_{27} |
5.20E − 15 (2.88E − 16) | 2.61E − 15 (6.13E − 16) | 6.18E − 15 (8.78E − 17) | 5.30E − 15 (3.48E − 16) | |

_{28} |
9.55E + 00 (5.88E − 01) | 9.50E − 27 (1.43E − 26) | 3.70E − 52 (1.85E − 51) | 7.01E + 00 (1.67E + 00) | |

_{29} |
6.87E − 09 (2.03E − 08) | 7.69E − 09 (2.09E − 08) | 4.01E − 11 (9.90E − 11) | 2.80E − 29 (4.48E − 29) | |

_{30} |
2.63E − 12 (9.83E − 13) | 3.82E − 12 (4.38E − 13) | 1.30E − 12 (2.90E − 14) | 2.81E − 12 (4.47E − 13) |

This paper presents a convergence tracking over time based parametric adaptive version of differential evolution algorithms. The two important operators of parameters of mutation probability that contribute are mutation operation and crossover rate, which are controlled by using a new sequence over time. The proposed sequences are helpful to track and follow the convergence path and remove the DE algorithm from the local optima problem. The concept of a small memory based on a user defined learning period is used in the sequence of control parameters of the algorithm to improve the convergence behavior of DE algorithms and escaping from the stagnation problem. A comprehensive suite of well-known and varying nature benchmark standard optimization functions was used to test the convergence power of the proposed convergence track based adaptive evolution algorithm (CTbADE). The experimental results are generated using the same set of parameters for a fair comparison. The results of the new CTbADE algorithm are compared with standard DE algorithms and some other well-known and commonly used state-of-the-art algorithms, such as CoDE, jDE, and EPSDE algorithms. The research result shows that the proposed CTbADE algorithm has more powerful convergence than the other algorithms used. The novel optimization algorithm presented in this research work will help the development of a fast automated diagnostics model to detect Covid-19 infection. The future work of this research work is to apply the CTbADE algorithm to optimize the hyper parameters of convolutional neural networks for Covid-19 CT/X-ray images feature extraction and classification to construct a fast automated diagnostics model.

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through project number 959.

Function | Name of function (type) | Equation | Search space | Optima |
---|---|---|---|---|

_{1} |
Sphere model |
−5.12 |
0 | |

_{2} |
Axis parallel hyperellipsoid |
−5.12 |
0 | |

_{3} |
Schwefel's problem 1.2 |
−65 |
0 | |

_{4} |
Rosenbrock's valley |
−30 |
0 | |

_{5} |
Rastrigin's function |
−5.12 |
0 | |

_{6} |
Griewank's function |
−600 |
0 | |

_{7} |
Sum of different power |
−1 |
0 | |

_{8} |
Ackley's path function |
−32 |
0 | |

_{9} |
Levy function |
−10 |
0 | |

_{10} |
Zakharov function |
−5 |
0 | |

_{11} |
Schwefel's problem 2.22 |
−10 |
0 | |

_{12} |
Step function |
−100 |
0 | |

_{13} |
De Jong's function 4 (no noise) |
−1.28 |
0 | |

_{14} |
Alpine function |
−10 |
0 | |

_{15} |
Levy and Montalvo Problem |
−10 |
0 |

Function | Name of function (type) | Equation | Search space | Optima |
---|---|---|---|---|

_{16} |
Neumaier 2 Problem |
0 | ||

_{17} |
Cosine Mixture |
|||

− 0.1x (n) | ||||

_{18} |
Cigar |
0 | ||

_{19} |
Function ‘15’ |
0 | ||

_{20} |
Ellipse Function |
0 | ||

_{21} |
Tablet Function |
0 | ||

_{22} |
Schewel |
0 | ||

_{23} |
Deflected Corrugated Spring |
0 | ||

_{24} |
Mishra 1 global optimization problem (Non-Separable, Multimodal) | 2 | ||

_{25} |
Mishra 2 global optimization problem |
2 | ||

_{26} |
MultiModal global optimization problem |
0 | ||

_{27} |
Quintic global optimization problem |
-1 | ||

_{28} |
Stochastic global optimization problem |
0 | ||

_{29} |
Stretched V global optimization problem |
0 | ||

_{30} |
XinSheYang |
0 |