1 Introduction
2 Background: recent criticisms of optimization metaheuristics
Abbreviation | Full name | References | Comments |
---|---|---|---|
PSO-basic | Particle swarm optimization (first version) |
Eberhart and Kennedy (1995) | The initial particle velocities are randomly generated within 20% of the bounds span. During run velocities are also restricted to be not higher than 20% of the bounds span. The global version of the model is used, as it was proposed first in Eberhart and Kennedy (1995). \(c1=c2 = 2\), as suggested in Kennedy and Eberhart (1995), even though this choice was later dismissed by Clerc and Kennedy (2002) |
ALC-PSO | PSO with an ageing leader and challengers |
Chen et al. (2013) | The initial particle velocities are set to 0. Velocities are restricted to be not larger than 50% of the bounds span. Control parameters are set according to Chen et al. (2013) as follows: \(c1=c2 = 2\), lifespan \(\varTheta _{0} = 60\), static inertia weight \({\omega } = 0.4\), \(T = 2\), \(\textit{pro} = 1/D\) (D is the problem dimensionality) |
CLPSO | Comprehensive learning PSO |
Liang et al. (2006) | The source code has been obtained from co-author of Liang et al. (2006). Velocities are restricted to be not larger than 20% of the bounds span. The initial PSO velocities are generated randomly from uniform distribution within 20% of the bounds span. Control parameters are as follows: \(c = 1.49445\), inertia weight decreases linearly during run from 0.9 to 0.2 |
DNS-PSO | Diversity-enhanced PSO with neighbourhood search |
Wang et al. (2013) | Velocities are restricted to be not larger than 50% of the bounds span. The initial velocities are generated randomly from uniform distribution within 50% of the bounds span. Control parameters are as follows (Wang et al. (2013)): \(c1=c2 = 1.49618\), \(w = 0.7298\), \(\textit{pr} = 0.9\), \(\textit{pns} = 0.6\), \(k = 2\)
|
PSO-init-weight | PSO with inertia weight |
Shi and Eberhart (1998) | The initial particle velocities are randomly generated within 20% of the bounds span. During run velocities are also restricted to be not higher than 20% of the bounds span. \(c1=c2 = 1.49\), w decreases linearly from 0.9 to 0.4 during run |
DE-basic | Differential evolution (first version) |
Storn and Price (1995) | The simplest version from the first report is used here (with scheme called there DE1 that uses the mutation strategy known in later papers by DE/rand/1), with \(F = 0.8\), \(\textit{Cr} = 0.5\) (also considered in Storn and Price 1995) |
CDE | Clustering DE |
Cai et al. (2011) | The first DE algorithm based on clustering that uses DE/rand/1/exp variant of mutation and crossover. Control parameters are set as follows (Cai et al. 2011): \(F = 0.5\), \(\textit{Cr} = 0.9\), clustering period \(m = 10\)
|
DEGL | DE with global and local neighbourhood mutation operators |
Das et al. (2009) | DEGL variant with self-adaptive weight factor is used (DEGL/SAW), as suggested in Das et al. (2009). Control parameters are as follows (Das et al. 2009): \(F = 0.8\), \(\textit{Cr} = 0.9\), neighbourhood size \(k = 5\%\) (such that total number of individuals in the neighbourhood defined on the ring topology equals \(2k = 10\%\) of PopSize; note, however, that in our application k cannot be lower than 2) |
MDE-pBX | Modified DE with p-best crossover |
Islam et al. (2012) | |
SADE | Self-adaptive DE |
Qin et al. (2009) | One of the most popular self-adaptive evolutionary algorithm. Variant with four strategies proposed in Qin et al. (2009) is used. Control parameters are set as follows: learning period \(\textit{LP} = 40\), F is generated from normal distribution N(0.5, 0.3), Cr is generated from normal distribution N(Crm, 0.1), when Crm is initialized to 0.5 and varies during search (see Qin et al. 2009), \(\varepsilon = 0.01\)
|
CLPSO-DEGL | CLPSO-DEGL hybrid algorithm |
Epitropakis et al. (2012) | |
GA-MPC | Genetic algorithm with multi-parent crossover | ||
NMA | Nelder–Mead simplex |
Nelder and Mead (1965) | The variant based on Lagarias et al. (1998) is used. When solving benchmark problems, a re-initialization approach defined as NMA1 in Piotrowski (2015) has been used. All solutions are re-initialized randomly (without keeping the best one) when two criteria are met: 1. the maximum difference between coordinates of the best and the worst point is lower than \(10^{-4}\); 2. difference in fitness between the best and the worst point is lower than \(10^{-4}\). Following control parameters are used: \(\alpha = 1\), \(\gamma = 2\), \({\beta } = \sigma = 0.5\)
|
RA | Rosenbrock’s algorithm |
Rosenbrock (1960) | The following parameter values are used: \(\alpha = 3\), \(\beta = -0.5\), initial step \(=\) 0.1 (variable during search). When solving benchmark problems, a re-initialization approach defined as RA2 in Piotrowski (2015) has been used. It works as follows: after every 100D function calls, it is verified whether during last 100D function calls the solution was improved by more than \(10^{-4}\). If it does not happen, the location of RA point is randomly re-initialized, the value of step is reset to 0.1 and the coordinates are reset to initial system |
3 Methods for structural bias detection
-
the maximum number of allowed function calls (MNFC) is set to 10,000 and 300,000;
-
the population size (PopSize) of each algorithm, except for RA and NMA, is set to 20, 50, 100 and 500 (by definition, RA, which is the only non-population-based heuristic chosen to be used in this study, may only be tested with \(\textit{PopSize} = 1\), and for NMA the PopSize is always related to D by \(\textit{PopSize} = D+ 1\)).
4 Experiments
D
| PopSize | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10,000 function calls
| |||||||||||||||
2 | 20 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 50.0 | 50.0 | 0.0 | 100.0 | 100.0 | 0.0 | 100.0 |
2 | 50 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 0.0 | 50.0 | 0.0 | 100.0 | 100.0 | ||
2 | 100 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 0.0 | 50.0 | 0.0 | 100.0 | 100.0 | ||
2 | 500 | 100.0 | 50.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 0.0 | 100.0 | 0.0 | 100.0 | 100.0 | ||
10 | 20 | 70.0 | 60.0 | 90.0 | 100.0 | 100.0 | 20.0 | 10.0 | 80.0 | 90.0 | 40.0 | 100.0 | 100.0 | 10.0 | 30.0 |
10 | 50 | 80.0 | 20.0 | 60.0 | 100.0 | 100.0 | 0.0 | 0.0 | 30.0 | 50.0 | 40.0 | 100.0 | 80.0 | ||
10 | 100 | 90.0 | 40.0 | 50.0 | 100.0 | 100.0 | 0.0 | 0.0 | 80.0 | 80.0 | 20.0 | 90.0 | 90.0 | ||
10 | 500 | 100.0 | 80.0 | 70.0 | 100.0 | 100.0 | 0.0 | 10.0 | 30.0 | 90.0 | 20.0 | 80.0 | 60.0 | ||
30 | 20 | 53.3 | 13.3 | 30.0 | 100.0 | 100.0 | 10.0 | 10.0 | 46.7 | 66.7 | 73.3 | 100.0 | 100.0 | 0.0 | 20.0 |
30 | 50 | 96.7 | 40.0 | 33.3 | 100.0 | 100.0 | 6.7 | 3.3 | 46.7 | 40.0 | 53.3 | 100.0 | 86.7 | ||
30 | 100 | 100.0 | 80.0 | 16.7 | 100.0 | 100.0 | 3.3 | 3.3 | 50.0 | 70.0 | 43.3 | 96.7 | 83.3 | ||
30 | 500 | 100.0 | 70.0 | 10.0 | 100.0 | 100.0 | 0.0 | 6.7 | 46.7 | 93.3 | 36.7 | 93.3 | 76.7 | ||
100 | 20 | 77.0 | 20.0 | 10.0 | 100.0 | 100.0 | 9.0 | 13.0 | 42.0 | 81.0 | 56.0 | 100.0 | 98.0 | 4.0 | 6.0 |
100 | 50 | 95.0 | 30.0 | 21.0 | 100.0 | 100.0 | 5.0 | 8.0 | 55.0 | 57.0 | 47.0 | 98.0 | 95.0 | ||
100 | 100 | 96.0 | 72.0 | 17.0 | 100.0 | 100.0 | 9.0 | 4.0 | 41.0 | 56.0 | 39.0 | 97.0 | 83.0 | ||
100 | 500 | 100.0 | 57.0 | 9.0 | 100.0 | 100.0 | 5.0 | 7.0 | 47.0 | 86.0 | 24.0 | 85.0 | 82.0 | ||
1000 | 20 | 72.0 | 19.1 | 9.5 | 100.0 | 100.0 | 5.1 | 11.6 | 44.8 | 77.7 | 56.5 | 97.0 | 98.9 | 4.9 | 4.8 |
1000 | 50 | 94.9 | 32.7 | 8.8 | 100.0 | 100.0 | 5.2 | 7.2 | 56.2 | 56.2 | 52.0 | 98.3 | 91.5 | ||
1000 | 100 | 99.2 | 64.3 | 9.3 | 100.0 | 100.0 | 4.6 | 4.5 | 44.8 | 54.4 | 40.1 | 93.6 | 80.4 | ||
1000 | 500 | 99.8 | 64.4 | 8.7 | 100.0 | 100.0 | 4.7 | 5.0 | 43.9 | 86.1 | 28.9 | 84.4 | 69.3 | ||
300,000 function calls
| |||||||||||||||
2 | 20 | 50.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 0.0 | 50.0 | 50.0 | 100.0 | 100.0 | 0.0 | 100.0 |
2 | 50 | 50.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 50.0 | 0.0 | 0.0 | 50.0 | 100.0 | 100.0 | ||
2 | 100 | 0.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 50.0 | 0.0 | 50.0 | 100.0 | 50.0 | ||
2 | 500 | 50.0 | 50.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 50.0 | 0.0 | 50.0 | 100.0 | 100.0 | ||
10 | 20 | 20.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 30.0 | 30.0 | 50.0 | 100.0 | 100.0 | 10.0 | 30.0 |
10 | 50 | 50.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 10.0 | 40.0 | 0.0 | 70.0 | 100.0 | 90.0 | ||
10 | 100 | 80.0 | 100.0 | 100.0 | 100.0 | 100.0 | 10.0 | 0.0 | 10.0 | 30.0 | 20.0 | 100.0 | 80.0 | ||
10 | 500 | 100.0 | 30.0 | 100.0 | 100.0 | 100.0 | 0.0 | 10.0 | 10.0 | 30.0 | 30.0 | 100.0 | 80.0 | ||
30 | 20 | 33.3 | 60.0 | 60.0 | 100.0 | 100.0 | 6.7 | 20.0 | 36.7 | 26.7 | 73.3 | 100.0 | 100.0 | 6.7 | 13.3 |
30 | 50 | 56.7 | 80.0 | 60.0 | 100.0 | 100.0 | 6.7 | 6.7 | 16.7 | 20.0 | 73.3 | 100.0 | 93.3 | ||
30 | 100 | 46.7 | 56.7 | 43.3 | 100.0 | 100.0 | 6.7 | 3.3 | 16.7 | 13.3 | 43.3 | 96.7 | 76.7 | ||
30 | 500 | 83.3 | 46.7 | 26.7 | 100.0 | 100.0 | 6.7 | 10.0 | 20.0 | 20.0 | 60.0 | 96.7 | 83.3 | ||
100 | 20 | 23.0 | 52.0 | 19.0 | 100.0 | 100.0 | 5.0 | 29.0 | 14.0 | 28.0 | 61.0 | 99.0 | 100.0 | 6.0 | 12.0 |
100 | 50 | 53.0 | 70.0 | 21.0 | 100.0 | 100.0 | 3.0 | 9.0 | 23.0 | 23.0 | 66.0 | 99.0 | 95.0 | ||
100 | 100 | 63.0 | 66.0 | 25.0 | 100.0 | 100.0 | 6.0 | 6.0 | 13.0 | 16.0 | 57.0 | 92.0 | 93.0 | ||
100 | 500 | 91.0 | 27.0 | 7.0 | 100.0 | 100.0 | 3.0 | 6.0 | 18.0 | 25.0 | 55.0 | 90.0 | 84.0 | ||
1000 | 20 | 27.1 | 44.4 | 9.0 | 100.0 | 100.0 | 5.4 | 29.7 | 11.7 | 33.8 | 75.1 | 95.4 | 99.6 | 5.0 | 4.9 |
1000 | 50 | 54.3 | 59.8 | 9.3 | 100.0 | 100.0 | 5.3 | 13.6 | 25.1 | 24.0 | 63.1 | 96.1 | 92.7 | ||
1000 | 100 | 66.9 | 47.4 | 8.3 | 100.0 | 100.0 | 5.4 | 12.0 | 14.6 | 19.5 | 62.7 | 89.3 | 86.9 | ||
1000 | 500 | 91.6 | 34.7 | 9.3 | 100.0 | 100.0 | 5.2 | 4.6 | 24.2 | 18.8 | 58.5 | 88.6 | 82.9 |
D
| PopSize | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE | DEGL | MDE- pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Initialization for tests with 10,000 function calls
| |||||||||||||||
2 | 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50.0 |
2 | 50 | 0.0 | 0.0 | 100.0 | 0.0 | 50.0 | 0.0 | 50.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||
2 | 100 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||
2 | 500 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||
10 | 20 | 0.0 | 0.0 | 0.0 | 10.0 | 10.0 | 20.0 | 0.0 | 10.0 | 0.0 | 0.0 | 10.0 | 10.0 | 0.0 | 0.0 |
10 | 50 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 20.0 | 0.0 | 10.0 | 10.0 | 10.0 | 0.0 | 10.0 | ||
10 | 100 | 10.0 | 0.0 | 10.0 | 20.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 10.0 | 0.0 | ||
10 | 500 | 0.0 | 0.0 | 40.0 | 0.0 | 0.0 | 30.0 | 0.0 | 0.0 | 0.0 | 0.0 | 30.0 | 30.0 | ||
30 | 20 | 3.3 | 3.3 | 6.7 | 6.7 | 0.0 | 0.0 | 10.0 | 3.3 | 6.7 | 0.0 | 0.0 | 16.7 | 3.3 | 13.3 |
30 | 50 | 0.0 | 3.3 | 0.0 | 0.0 | 3.3 | 3.3 | 0.0 | 3.3 | 0.0 | 0.0 | 6.7 | 10.0 | ||
30 | 100 | 6.7 | 0.0 | 6.7 | 6.7 | 6.7 | 3.3 | 0.0 | 13.3 | 0.0 | 3.3 | 3.3 | 20.0 | ||
30 | 500 | 6.7 | 3.3 | 10.0 | 0.0 | 6.7 | 6.7 | 6.7 | 0.0 | 10.0 | 0.0 | 10.0 | 16.7 | ||
100 | 20 | 3.0 | 7.0 | 7.0 | 6.0 | 9.0 | 4.0 | 4.0 | 7.0 | 5.0 | 2.0 | 10.0 | 9.0 | 6.0 | 5.0 |
100 | 50 | 6.0 | 7.0 | 8.0 | 6.0 | 0.0 | 7.0 | 6.0 | 3.0 | 6.0 | 3.0 | 10.0 | 14.0 | ||
100 | 100 | 2.0 | 8.0 | 7.0 | 4.0 | 1.0 | 6.0 | 5.0 | 6.0 | 4.0 | 3.0 | 13.0 | 10.0 | ||
100 | 500 | 5.0 | 5.0 | 9.0 | 7.0 | 0.0 | 2.0 | 2.0 | 4.0 | 1.0 | 3.0 | 6.0 | 9.0 | ||
1000 | 20 | 5.0 | 5.2 | 8.7 | 3.2 | 6.3 | 5.9 | 6.7 | 4.9 | 4.4 | 5.7 | 10.2 | 7.1 | 5.7 | 4.7 |
1000 | 50 | 4.4 | 4.3 | 9.1 | 5.1 | 5.8 | 4.8 | 4.2 | 4.9 | 4.0 | 4.3 | 7.3 | 7.1 | ||
1000 | 100 | 4.9 | 5.2 | 10.9 | 4.9 | 5.3 | 5.9 | 4.9 | 4.4 | 3.6 | 5.1 | 9.0 | 7.4 | ||
1000 | 500 | 5.7 | 4.9 | 7.2 | 4.8 | 4.6 | 4.1 | 3.7 | 6.0 | 4.3 | 5.4 | 8.5 | 8.0 | ||
Initialization for tests with 300,000 function calls
| |||||||||||||||
2 | 20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50.0 | 0.0 | 0.0 | 0.0 |
2 | 50 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||
2 | 100 | 0.0 | 0.0 | 0.0 | 50.0 | 0.0 | 0.0 | 0.0 | 50.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||
2 | 500 | 0.0 | 50.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50.0 | 0.0 | 0.0 | ||
10 | 20 | 10.0 | 0.0 | 30.0 | 0.0 | 10.0 | 10.0 | 10.0 | 10.0 | 20.0 | 0.0 | 30.0 | 30.0 | 10.0 | 20.0 |
10 | 50 | 0.0 | 10.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 10.0 | 0.0 | 0.0 | ||
10 | 100 | 0.0 | 10.0 | 0.0 | 0.0 | 0.0 | 20.0 | 10.0 | 0.0 | 0.0 | 0.0 | 0.0 | 10.0 | ||
10 | 500 | 0.0 | 20.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 20.0 | 0.0 | 20.0 | 10.0 | ||
30 | 20 | 6.7 | 3.3 | 6.7 | 3.3 | 6.7 | 3.3 | 3.3 | 3.3 | 6.7 | 6.7 | 10.0 | 10.0 | 6.7 | 6.7 |
30 | 50 | 0.0 | 0.0 | 3.3 | 10.0 | 0.0 | 6.7 | 0.0 | 0.0 | 0.0 | 10.0 | 6.7 | 0.0 | ||
30 | 100 | 3.3 | 0.0 | 3.3 | 6.7 | 3.3 | 0.0 | 0.0 | 3.3 | 10.0 | 6.7 | 6.7 | 0.0 | ||
30 | 500 | 3.3 | 3.3 | 20.0 | 3.3 | 3.3 | 3.3 | 10.0 | 3.3 | 3.3 | 3.3 | 6.7 | 0.0 | ||
100 | 20 | 7.0 | 2.0 | 3.0 | 3.0 | 7.0 | 4.0 | 1.0 | 6.0 | 5.0 | 3.0 | 7.0 | 12.0 | 3.0 | 9.0 |
100 | 50 | 5.0 | 4.0 | 8.0 | 3.0 | 5.0 | 9.0 | 6.0 | 2.0 | 4.0 | 8.0 | 4.0 | 11.0 | ||
100 | 100 | 5.0 | 6.0 | 4.0 | 3.0 | 5.0 | 4.0 | 0.0 | 6.0 | 8.0 | 5.0 | 6.0 | 13.0 | ||
100 | 500 | 6.0 | 3.0 | 3.0 | 6.0 | 6.0 | 6.0 | 5.0 | 6.0 | 4.0 | 2.0 | 6.0 | 9.0 | ||
1000 | 20 | 6.1 | 5.6 | 7.2 | 4.8 | 6.1 | 4.9 | 5.4 | 4.6 | 4.6 | 6.0 | 6.1 | 6.9 | 5.0 | 4.3 |
1000 | 50 | 4.3 | 5.5 | 7.6 | 6.0 | 4.3 | 5.3 | 5.3 | 5.6 | 5.5 | 5.3 | 7.5 | 7.2 | ||
1000 | 100 | 5.0 | 4.5 | 6.5 | 5.4 | 5.0 | 5.3 | 4.1 | 4.8 | 4.3 | 5.9 | 6.9 | 7.4 | ||
1000 | 500 | 3.7 | 4.6 | 7.7 | 4.4 | 3.7 | 5.6 | 4.0 | 4.3 | 4.9 | 5.5 | 7.3 | 7.8 |
D
| PopSize | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE |
---|---|---|---|---|---|---|---|---|
10,000 function calls
| ||||||||
2 | 20 | 5.40E−03 | 1.30E−03 | 3.95E−04 | 9.87E−144 | 2.51E−10 | 3.19E−01 | 1.27E−01 |
2 | 50 | 1.36E−03 | 8.74E−03 | 2.43E−06 | 2.62E−93 | 2.08E−09 | 1.38E−01 | 2.10E−01 |
2 | 100 | 5.97E−04 | 5.88E−03 | 1.64E−04 | 1.57E−78 | 1.35E−09 | 3.31E−01 | 2.21E−01 |
2 | 500 | 7.04E−06 | 7.29E−03 | 1.10E−02 | 4.24E−45 | 2.30E−09 | 3.69E−01 | 3.41E−01 |
10 | 20 | 4.63E−03 | 4.03E−04 | 5.57E−05 | 4.12E−103 | 2.24E−09 | 8.41E−03 | 3.11E−02 |
10 | 50 | 7.21E−05 | 1.14E−02 | 2.34E−03 | 3.80E−83 | 3.26E−10 | 8.36E−02 | 2.31E−01 |
10 | 100 | 1.14E−05 | 4.52E−04 | 8.31E−04 | 7.63E−71 | 6.70E−14 | 1.57E−01 | 6.29E−02 |
10 | 500 | 8.21E−07 | 1.14E−03 | 1.56E−06 | 2.39E−51 | 1.04E−11 | 1.00E−01 | 3.61E−03 |
30 | 20 | 4.18E−05 | 1.20E−03 | 2.52E−04 | 6.15E−96 | 4.34E−13 | 1.61E−02 | 1.82E−02 |
30 | 50 | 4.31E−05 | 9.11E−04 | 6.03E−04 | 1.77E−79 | 4.24E−12 | 7.97E−03 | 2.24E−02 |
30 | 100 | 6.89E−07 | 4.44E−04 | 9.76E−03 | 3.98E−67 | 2.04E−13 | 2.66E−02 | 1.96E−03 |
30 | 500 | 8.62E−08 | 3.82E−04 | 5.83E−03 | 1.64E−38 | 1.63E−13 | 6.97E−02 | 1.95E−02 |
100 | 20 | 3.67E−05 | 1.05E−03 | 5.30E−04 | 7.45E−96 | 2.77E−12 | 6.24E−03 | 8.25E−04 |
100 | 50 | 3.66E−07 | 1.89E−05 | 3.53E−04 | 3.15E−75 | 9.89E−15 | 8.94E−03 | 1.31E−03 |
100 | 100 | 6.97E−08 | 1.04E−06 | 4.36E−03 | 6.12E−77 | 1.14E−13 | 1.04E−02 | 9.64E−03 |
100 | 500 | 1.18E−08 | 1.13E−05 | 2.65E−04 | 7.65E−42 | 6.32E−15 | 1.96E−02 | 1.71E−02 |
1000 | 20 | 5.93E−06 | 2.64E−05 | 6.32E−06 | 1.58E−98 | 3.66E−14 | 2.71E−04 | 4.27E−04 |
1000 | 50 | 4.71E−08 | 1.07E−04 | 1.84E−04 | 2.46E−92 | 1.59E−15 | 2.62E−04 | 2.28E−04 |
1000 | 100 | 3.27E−09 | 2.53E−06 | 9.63E−05 | 2.30E−73 | 4.22E−17 | 1.54E−04 | 8.53E−04 |
1000 | 500 | 5.91E−10 | 2.16E−06 | 5.39E−05 | 3.96E−46 | 1.27E−16 | 2.94E−03 | 4.88E−04 |
D
| PopSize | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|---|
10,000 function calls
| ||||||||
2 | 20 | 4.45E−02 | 3.40E−02 | 5.92E−02 | 4.11E−05 | 3.31E−06 | 3.02E−01 | 4.64E−07 |
2 | 50 | 2.46E−01 | 7.73E−03 | 8.73E−02 | 1.00E−04 | 7.82E−03 | ||
2 | 100 | 5.45E−02 | 8.22E−03 | 2.10E−01 | 1.25E−06 | 4.65E−04 | ||
2 | 500 | 1.45E−01 | 1.17E−03 | 2.14E−01 | 8.34E−05 | 3.83E−04 | ||
10 | 20 | 5.58E−03 | 7.65E−04 | 5.82E−03 | 1.06E−06 | 3.06E−06 | 1.67E−02 | 1.07E−05 |
10 | 50 | 1.62E−02 | 7.83E−03 | 3.43E−03 | 4.72E−08 | 7.54E−05 | ||
10 | 100 | 1.20E−03 | 2.20E−03 | 4.33E−03 | 3.17E−05 | 2.94E−04 | ||
10 | 500 | 5.48E−05 | 4.18E−04 | 3.93E−02 | 1.21E−04 | 2.24E−03 | ||
30 | 20 | 1.88E−03 | 3.13E−04 | 6.11E−04 | 1.26E−06 | 5.14E−06 | 6.18E−02 | 3.79E−03 |
30 | 50 | 6.66E−04 | 7.42E−05 | 1.94E−04 | 9.68E−08 | 1.33E−05 | ||
30 | 100 | 1.77E−04 | 2.22E−04 | 3.44E−04 | 3.31E−06 | 6.30E−07 | ||
30 | 500 | 1.49E−03 | 8.77E−05 | 2.84E−03 | 3.03E−06 | 1.87E−05 | ||
100 | 20 | 7.39E−05 | 7.29E−06 | 2.54E−04 | 1.43E−07 | 1.21E−06 | 7.45E−04 | 7.56E−04 |
100 | 50 | 3.73E−04 | 2.09E−06 | 1.32E−05 | 2.45E−06 | 2.54E−08 | ||
100 | 100 | 1.13E−04 | 3.57E−05 | 6.68E−05 | 4.65E−08 | 5.61E−08 | ||
100 | 500 | 3.76E−05 | 1.61E−05 | 2.15E−03 | 1.44E−06 | 2.35E−05 | ||
1000 | 20 | 5.04E−05 | 1.13E−06 | 7.47E−05 | 2.46E−07 | 6.85E−08 | 5.50E−04 | 6.82E−04 |
1000 | 50 | 3.94E−07 | 1.69E−06 | 1.15E−05 | 1.49E−09 | 1.07E−06 | ||
1000 | 100 | 3.58E−06 | 1.71E−05 | 3.08E−06 | 2.55E−07 | 2.92E−07 | ||
1000 | 500 | 1.56E−05 | 7.08E−07 | 2.48E−06 | 6.28E−07 | 1.44E−05 |
D
| PopSize | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE |
---|---|---|---|---|---|---|---|---|
300,000 function calls
| ||||||||
2 | 20 | 2.71E−02 | 2.02E−07 | 4.51E−06 | 7.17E−199 | 5.54E−06 | 1.53E−01 | 6.29E−01 |
2 | 50 | 4.07E−02 | 3.10E−07 | 1.75E−06 | 4.35E−157 | 2.79E−06 | 1.15E−01 | 7.00E−03 |
2 | 100 | 9.79E−02 | 1.36E−05 | 1.25E−05 | 5.17E−146 | 2.97E−09 | 1.17E−01 | 5.44E−01 |
2 | 500 | 1.89E−05 | 2.54E−03 | 2.87E−07 | 1.44E−126 | 1.59E−10 | 2.21E−01 | 5.44E−01 |
10 | 20 | 1.95E−02 | 6.77E−05 | 8.58E−08 | 5.98E−159 | 1.33E−06 | 6.67E−02 | 2.34E−01 |
10 | 50 | 1.77E−05 | 2.51E−04 | 4.20E−05 | 3.29E−139 | 1.40E−08 | 7.64E−02 | 2.54E−02 |
10 | 100 | 3.96E−04 | 1.66E−05 | 1.42E−07 | 2.56E−127 | 3.14E−12 | 2.31E−02 | 6.19E−02 |
10 | 500 | 5.34E−05 | 1.96E−02 | 4.04E−04 | 7.02E−100 | 3.49E−13 | 2.82E−01 | 2.89E−02 |
30 | 20 | 4.84E−03 | 9.32E−04 | 5.99E−04 | 1.51E−146 | 1.30E−08 | 1.16E−02 | 1.34E−03 |
30 | 50 | 8.15E−05 | 1.46E−05 | 1.28E−04 | 5.42E−119 | 1.17E−09 | 1.54E−03 | 3.09E−02 |
30 | 100 | 2.14E−03 | 9.44E−05 | 4.65E−04 | 3.68E−120 | 2.21E−10 | 2.05E−02 | 2.83E−02 |
30 | 500 | 5.28E−05 | 9.18E−06 | 9.16E−03 | 2.22E−90 | 9.87E−14 | 8.25E−03 | 1.61E−02 |
100 | 20 | 9.70E−04 | 3.26E−05 | 1.98E−04 | 3.33E−150 | 5.07E−10 | 2.42E−02 | 7.98E−04 |
100 | 50 | 1.41E−05 | 2.64E−04 | 1.91E−06 | 6.52E−136 | 4.96E−10 | 2.20E−02 | 3.12E−03 |
100 | 100 | 2.05E−04 | 1.53E−04 | 5.88E−05 | 2.21E−123 | 8.46E−13 | 1.85E−02 | 4.92E−05 |
100 | 500 | 2.59E−06 | 2.20E−04 | 1.18E−02 | 1.91E−93 | 4.74E−14 | 3.78E−03 | 1.73E−02 |
1000 | 20 | 6.54E−05 | 2.01E−05 | 1.43E−05 | 2.31E−157 | 1.22E−09 | 2.33E−03 | 5.00E−05 |
1000 | 50 | 3.70E−07 | 2.47E−06 | 6.59E−04 | 5.45E−144 | 1.10E−11 | 1.34E−03 | 3.15E−04 |
1000 | 100 | 7.96E−06 | 3.15E−05 | 5.51E−04 | 2.60E−126 | 3.43E−13 | 1.18E−03 | 3.41E−06 |
1000 | 500 | 5.14E−08 | 3.83E−05 | 1.76E−05 | 5.35E−102 | 1.24E−16 | 7.29E−04 | 1.71E−04 |
D
| PopSize | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|---|
300,000 function calls
| ||||||||
2 | 20 | 3.28E−01 | 3.74E−02 | 9.39E−03 | 3.24E−06 | 1.33E−04 | 2.28E−01 | 7.37E−07 |
2 | 50 | 5.86E−02 | 1.29E−01 | 1.86E−02 | 4.41E−05 | 4.04E−06 | ||
2 | 100 | 6.68E−03 | 4.61E−01 | 2.75E−02 | 4.51E−05 | 4.32E−04 | ||
2 | 500 | 1.17E−02 | 1.86E−01 | 4.36E−02 | 2.92E−06 | 2.37E−03 | ||
10 | 20 | 2.15E−02 | 1.52E−03 | 5.63E−04 | 9.88E−06 | 1.15E−05 | 1.81E−02 | 2.78E−07 |
10 | 50 | 6.56E−03 | 1.26E−01 | 3.74E−04 | 9.50E−07 | 8.84E−04 | ||
10 | 100 | 1.82E−02 | 4.50E−03 | 1.02E−03 | 1.76E−07 | 5.72E−04 | ||
10 | 500 | 3.42E−02 | 1.48E−02 | 2.63E−03 | 1.11E−05 | 1.69E−05 | ||
30 | 20 | 3.75E−03 | 1.58E−03 | 4.01E−04 | 7.06E−07 | 1.17E−06 | 2.76E−02 | 9.90E−05 |
30 | 50 | 1.68E−03 | 8.24E−05 | 7.79E−05 | 2.25E−08 | 6.22E−05 | ||
30 | 100 | 1.02E−02 | 2.18E−03 | 3.62E−04 | 2.02E−05 | 1.15E−04 | ||
30 | 500 | 7.08E−03 | 7.05E−03 | 1.58E−03 | 1.09E−07 | 1.77E−04 | ||
100 | 20 | 1.42E−04 | 1.07E−03 | 7.14E−05 | 3.61E−07 | 2.56E−07 | 4.73E−03 | 1.72E−04 |
100 | 50 | 3.49E−04 | 4.49E−03 | 1.72E−05 | 1.86E−06 | 3.97E−07 | ||
100 | 100 | 2.94E−03 | 6.00E−03 | 5.39E−06 | 6.43E−06 | 1.33E−05 | ||
100 | 500 | 1.50E−03 | 3.04E−04 | 7.12E−05 | 4.68E−05 | 2.54E−06 | ||
1000 | 20 | 2.59E−04 | 4.10E−05 | 1.06E−06 | 1.05E−07 | 1.13E−08 | 8.07E−06 | 4.98E−10 |
1000 | 50 | 4.92E−05 | 2.89E−04 | 8.48E−06 | 1.20E−08 | 2.50E−07 | ||
1000 | 100 | 1.08E−03 | 1.06E−05 | 4.98E−06 | 6.33E−08 | 1.14E−06 | ||
1000 | 500 | 2.04E−04 | 7.49E−05 | 2.11E−06 | 3.59E−07 | 1.48E−06 |
D
| PopSize | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE |
---|---|---|---|---|---|---|---|---|
Initialization for tests with 10,000 function calls
| ||||||||
2 | 20 | 2.25E−01 | 6.23E−01 | 5.62E−02 | 7.81E−01 | 5.34E−01 | 7.26E−01 | 3.08E−01 |
2 | 50 | 4.47E−01 | 8.54E−01 | 1.16E−03 | 1.56E−01 | 2.49E−03 | 3.75E−01 | 2.93E−02 |
2 | 100 | 1.43E−01 | 4.08E−01 | 1.00E−01 | 2.15E−01 | 1.38E−01 | 6.32E−01 | 5.57E−01 |
2 | 500 | 5.38E−01 | 4.75E−01 | 2.72E−01 | 1.47E−01 | 3.78E−01 | 4.39E−01 | 1.50E−02 |
10 | 20 | 1.43E−01 | 1.66E−01 | 6.42E−02 | 3.96E−02 | 6.82E−03 | 1.20E−02 | 1.78E−01 |
10 | 50 | 6.65E−02 | 1.25E−01 | 5.08E−02 | 3.07E−01 | 1.49E−01 | 1.34E−02 | 1.11E−01 |
10 | 100 | 4.60E−02 | 1.13E−01 | 1.29E−02 | 3.26E−02 | 6.66E−02 | 9.11E−02 | 1.02E−01 |
10 | 500 | 5.61E−02 | 6.55E−02 | 6.92E−04 | 1.30E−01 | 1.03E−01 | 1.88E−02 | 3.28E−01 |
30 | 20 | 2.58E−02 | 2.50E−02 | 1.61E−02 | 2.49E−02 | 6.00E−02 | 1.00E−01 | 2.23E−02 |
30 | 50 | 7.71E−02 | 2.39E−02 | 7.87E−02 | 5.65E−02 | 3.60E−02 | 2.14E−03 | 7.90E−02 |
30 | 100 | 1.15E−02 | 7.21E−02 | 1.06E−02 | 1.65E−02 | 3.26E−02 | 3.78E−02 | 1.03E−01 |
30 | 500 | 4.43E−03 | 3.72E−02 | 2.19E−03 | 6.32E−02 | 1.60E−02 | 1.75E−03 | 4.60E−02 |
100 | 20 | 3.67E−03 | 1.29E−02 | 3.19E−03 | 1.15E−02 | 2.88E−04 | 1.90E−03 | 1.27E−02 |
100 | 50 | 2.08E−03 | 4.91E−03 | 4.80E−04 | 4.11E−03 | 6.30E−02 | 7.56E−04 | 2.84E−03 |
100 | 100 | 2.85E−02 | 2.01E−02 | 4.18E−03 | 2.51E−02 | 1.74E−02 | 9.31E−03 | 1.41E−02 |
100 | 500 | 6.52E−03 | 3.06E−03 | 1.48E−03 | 9.62E−03 | 6.41E−02 | 4.37E−03 | 2.51E−02 |
1000 | 20 | 3.39E−04 | 2.11E−04 | 4.51E−04 | 3.36E−03 | 4.60E−04 | 1.75E−03 | 4.31E−05 |
1000 | 50 | 3.13E−04 | 1.21E−03 | 8.58E−04 | 1.44E−03 | 6.91E−04 | 2.01E−04 | 2.24E−03 |
1000 | 100 | 1.80E−03 | 6.75E−04 | 2.61E−05 | 1.53E−03 | 4.76E−05 | 2.30E−04 | 1.93E−03 |
1000 | 500 | 1.58E−04 | 6.50E−04 | 9.75E−04 | 3.28E−04 | 4.15E−04 | 7.17E−04 | 1.15E−04 |
D
| PopSize | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|---|
Initialization for tests with 10,000 function calls
| ||||||||
2 | 20 | 3.23E−01 | 1.37E−01 | 3.19E−01 | 4.04E−01 | 6.18E−01 | 8.03E−01 | 3.59E−02 |
2 | 50 | 4.68E−01 | 4.57E−01 | 5.11E−02 | 9.44E−02 | 3.16E−01 | ||
2 | 100 | 3.47E−01 | 1.80E−01 | 5.02E−01 | 3.50E−01 | 2.17E−01 | ||
2 | 500 | 7.86E−02 | 4.77E−01 | 7.52E−01 | 9.38E−02 | 8.37E−02 | ||
10 | 20 | 4.81E−02 | 9.57E−02 | 2.47E−01 | 3.54E−03 | 3.31E−02 | 7.13E−02 | 5.10E−02 |
10 | 50 | 4.51E−02 | 2.92E−02 | 8.77E−03 | 1.20E−01 | 3.29E−02 | ||
10 | 100 | 1.24E−01 | 1.18E−01 | 5.99E−02 | 3.20E−02 | 8.25E−02 | ||
10 | 500 | 1.00E−01 | 6.32E−02 | 2.03E−01 | 3.27E−04 | 2.23E−03 | ||
30 | 20 | 2.31E−02 | 9.12E−03 | 1.24E−01 | 5.74E−02 | 5.31E−04 | 4.80E−02 | 1.38E−02 |
30 | 50 | 1.03E−02 | 6.14E−02 | 6.93E−02 | 2.71E−02 | 1.51E−02 | ||
30 | 100 | 1.75E−03 | 5.59E−02 | 4.14E−02 | 3.03E−02 | 2.51E−03 | ||
30 | 500 | 7.76E−02 | 2.19E−02 | 6.12E−02 | 2.77E−03 | 1.61E−03 | ||
100 | 20 | 4.21E−03 | 1.63E−03 | 9.95E−03 | 1.51E−03 | 6.33E−04 | 5.72E−03 | 2.26E−02 |
100 | 50 | 6.80E−03 | 8.77E−03 | 1.25E−02 | 1.84E−03 | 4.04E−03 | ||
100 | 100 | 1.79E−03 | 6.59E−03 | 1.14E−02 | 1.25E−04 | 3.82E−04 | ||
100 | 500 | 3.63E−03 | 1.40E−02 | 3.06E−03 | 5.58E−04 | 7.33E−04 | ||
1000 | 20 | 5.99E−05 | 2.91E−03 | 1.13E−04 | 4.83E−05 | 3.79E−04 | 2.77E−04 | 1.02E−03 |
1000 | 50 | 2.71E−04 | 3.34E−03 | 9.80E−05 | 5.42E−04 | 5.70E−04 | ||
1000 | 100 | 2.26E−04 | 1.14E−03 | 5.30E−04 | 7.28E−05 | 1.16E−03 | ||
1000 | 500 | 2.40E−05 | 6.55E−04 | 6.51E−04 | 1.18E−04 | 3.27E−05 |
D
| PopSize | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE |
---|---|---|---|---|---|---|---|---|
Initialization for tests with 300,000 function calls
| ||||||||
2 | 20 | 8.18E−01 | 3.26E−01 | 8.35E−01 | 2.81E−01 | 8.18E−01 | 5.85E−01 | 5.70E−01 |
2 | 50 | 2.03E−01 | 8.17E−01 | 3.94E−01 | 4.25E−01 | 2.03E−01 | 7.58E−01 | 3.65E−02 |
2 | 100 | 5.71E−01 | 2.66E−01 | 6.46E−01 | 2.15E−03 | 5.71E−01 | 1.53E−01 | 4.61E−01 |
2 | 500 | 2.14E−01 | 3.32E−02 | 2.38E−01 | 2.29E−01 | 2.14E−01 | 4.76E−01 | 4.61E−01 |
10 | 20 | 4.41E−02 | 6.47E−02 | 6.91E−03 | 8.34E−02 | 4.41E−02 | 3.33E−02 | 4.03E−02 |
10 | 50 | 1.75E−01 | 4.32E−02 | 1.94E−01 | 2.45E−01 | 1.75E−01 | 6.97E−02 | 1.22E−01 |
10 | 100 | 1.36E−01 | 3.94E−02 | 1.20E−01 | 9.33E−02 | 1.36E−01 | 1.15E−02 | 3.20E−03 |
10 | 500 | 1.78E−01 | 1.83E−02 | 1.58E−01 | 5.52E−02 | 1.78E−01 | 6.99E−02 | 6.58E−02 |
30 | 20 | 1.84E−02 | 1.70E−02 | 3.07E−02 | 3.81E−02 | 1.84E−02 | 3.53E−02 | 3.78E−02 |
30 | 50 | 7.98E−02 | 1.98E−01 | 4.80E−02 | 5.13E−03 | 7.98E−02 | 1.53E−02 | 7.39E−02 |
30 | 100 | 2.92E−02 | 8.03E−02 | 3.36E−02 | 1.23E−02 | 2.92E−02 | 5.02E−02 | 5.08E−02 |
30 | 500 | 4.91E−02 | 3.42E−02 | 9.29E−03 | 1.75E−04 | 4.91E−02 | 1.57E−02 | 1.71E−02 |
100 | 20 | 1.10E−02 | 2.41E−03 | 1.57E−02 | 2.84E−03 | 1.10E−02 | 5.27E−03 | 3.97E−03 |
100 | 50 | 2.98E−03 | 2.04E−03 | 8.32E−03 | 2.21E−03 | 2.98E−03 | 3.45E−03 | 2.20E−03 |
100 | 100 | 1.32E−02 | 6.55E−03 | 5.17E−03 | 8.92E−03 | 1.32E−02 | 1.53E−02 | 6.93E−02 |
100 | 500 | 2.02E−02 | 6.28E−03 | 4.12E−03 | 1.39E−03 | 2.02E−02 | 3.37E−03 | 3.44E−03 |
1000 | 20 | 8.27E−04 | 1.17E−03 | 1.77E−04 | 1.23E−04 | 8.27E−04 | 1.55E−04 | 2.96E−06 |
1000 | 50 | 4.45E−03 | 4.65E−04 | 1.74E−03 | 1.63E−03 | 4.45E−03 | 1.56E−03 | 1.54E−04 |
1000 | 100 | 1.56E−04 | 2.63E−04 | 9.25E−04 | 2.55E−03 | 1.56E−04 | 8.96E−04 | 1.45E−03 |
1000 | 500 | 5.00E−04 | 3.71E−04 | 9.56E−04 | 2.46E−03 | 5.00E−04 | 8.47E−04 | 7.42E−04 |
D
| PopSize | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|---|
Initialization for tests with 300,000 function calls
| ||||||||
2 | 20 | 2.49E−01 | 1.68E−01 | 4.12E−01 | 3.00E−02 | 5.79E−01 | 1.15E−01 | 1.18E−01 |
2 | 50 | 1.20E−01 | 4.71E−01 | 3.44E−01 | 4.21E−01 | 3.55E−01 | ||
2 | 100 | 1.51E−02 | 4.59E−01 | 8.41E−01 | 1.14E−01 | 5.87E−01 | ||
2 | 500 | 6.36E−01 | 8.55E−01 | 2.60E−02 | 4.03E−01 | 2.69E−01 | ||
10 | 20 | 4.28E−02 | 1.89E−02 | 5.33E−02 | 2.72E−02 | 1.78E−03 | 3.05E−02 | 2.06E−02 |
10 | 50 | 1.61E−01 | 1.52E−01 | 1.13E−02 | 6.92E−02 | 1.26E−01 | ||
10 | 100 | 1.34E−01 | 7.21E−02 | 1.16E−01 | 9.07E−02 | 4.70E−02 | ||
10 | 500 | 2.66E−01 | 8.54E−03 | 9.86E−02 | 3.00E−03 | 4.61E−02 | ||
30 | 20 | 3.00E−02 | 4.61E−02 | 2.11E−02 | 4.39E−03 | 2.42E−02 | 2.44E−02 | 6.10E−03 |
30 | 50 | 6.83E−02 | 1.38E−01 | 2.22E−02 | 1.38E−02 | 5.39E−02 | ||
30 | 100 | 3.02E−02 | 2.49E−02 | 2.88E−02 | 1.27E−02 | 6.77E−02 | ||
30 | 500 | 2.95E−02 | 1.52E−02 | 4.97E−02 | 1.97E−02 | 1.16E−01 | ||
100 | 20 | 9.50E−03 | 2.20E−02 | 5.32E−03 | 2.49E−03 | 6.04E−04 | 1.64E−02 | 6.14E−03 |
100 | 50 | 3.79E−03 | 2.45E−02 | 1.18E−03 | 6.75E−03 | 3.30E−03 | ||
100 | 100 | 5.76E−03 | 1.43E−02 | 2.96E−02 | 1.81E−03 | 2.61E−03 | ||
100 | 500 | 6.91E−03 | 1.24E−02 | 2.65E−02 | 1.17E−02 | 1.21E−04 | ||
1000 | 20 | 1.79E−04 | 3.40E−03 | 1.25E−03 | 1.47E−03 | 4.29E−04 | 1.83E−03 | 1.87E−03 |
1000 | 50 | 4.34E−04 | 2.82E−04 | 1.85E−04 | 3.99E−05 | 5.49E−05 | ||
1000 | 100 | 5.83E−04 | 5.56E−04 | 1.15E−04 | 1.25E−03 | 1.07E−04 | ||
1000 | 500 | 1.09E−03 | 3.54E−04 | 2.64E−03 | 2.02E−04 | 1.36E−03 |
4.1 The presence of structural bias
-
immediately after initialization (the presence of such bias cannot be attributed to the algorithm);
-
after the algorithm was run for a relatively small number of function calls set to 10,000;
-
after the algorithm was run for a number of function calls set to a more frequently used value of 300,000;
-
or such bias is not observed at all.
D
| PopSize | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 20 | 100.0 | 100.0 | 50.0 | 100.0 | 100.0 | 0.0 | 0.0 | 50.0 | 50.0 | 0.0 | 100.0 | 50.0 | 0.0 | 50.0 |
2 | 50 | 100.0 | 0.0 | 100.0 | 100.0 | 100.0 | 0.0 | 50.0 | 0.0 | 50.0 | 50.0 | 0.0 | 50.0 | ||
2 | 100 | 100.0 | 50.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 0.0 | 100.0 | 0.0 | 100.0 | 100.0 | ||
2 | 500 | 100.0 | 50.0 | 0.0 | 100.0 | 100.0 | 0.0 | 0.0 | 50.0 | 50.0 | 0.0 | 100.0 | 0.0 | ||
10 | 20 | 30.0 | 20.0 | 30.0 | 100.0 | 100.0 | 20.0 | 10.0 | 40.0 | 50.0 | 0.0 | 80.0 | 60.0 | 10.0 | 10.0 |
10 | 50 | 40.0 | 20.0 | 30.0 | 100.0 | 100.0 | 0.0 | 0.0 | 20.0 | 30.0 | 40.0 | 100.0 | 50.0 | ||
10 | 100 | 100.0 | 30.0 | 50.0 | 100.0 | 100.0 | 0.0 | 0.0 | 30.0 | 40.0 | 20.0 | 80.0 | 70.0 | ||
10 | 500 | 70.0 | 30.0 | 70.0 | 100.0 | 100.0 | 10.0 | 10.0 | 30.0 | 60.0 | 0.0 | 50.0 | 50.0 | ||
30 | 20 | 33.3 | 13.3 | 13.3 | 100.0 | 100.0 | 0.0 | 13.3 | 6.7 | 23.3 | 26.7 | 93.3 | 86.7 | 3.3 | 3.3 |
30 | 50 | 60.0 | 20.0 | 16.7 | 100.0 | 100.0 | 6.7 | 6.7 | 23.3 | 16.7 | 23.3 | 83.3 | 73.3 | ||
30 | 100 | 76.7 | 26.7 | 6.7 | 100.0 | 100.0 | 0.0 | 6.7 | 36.7 | 36.7 | 13.3 | 80.0 | 60.0 | ||
30 | 500 | 100.0 | 16.7 | 10.0 | 100.0 | 100.0 | 6.7 | 0.0 | 10.0 | 53.3 | 13.3 | 60.0 | 36.7 | ||
100 | 20 | 43.0 | 15.0 | 10.0 | 100.0 | 100.0 | 8.0 | 6.0 | 21.0 | 35.0 | 29.0 | 82.0 | 66.0 | 7.0 | 1.0 |
100 | 50 | 65.0 | 18.0 | 14.0 | 100.0 | 100.0 | 5.0 | 9.0 | 26.0 | 29.0 | 29.0 | 68.0 | 60.0 | ||
100 | 100 | 66.0 | 28.0 | 10.0 | 100.0 | 100.0 | 9.0 | 3.0 | 29.0 | 33.0 | 18.0 | 74.0 | 38.0 | ||
100 | 500 | 89.0 | 34.0 | 8.0 | 100.0 | 100.0 | 5.0 | 5.0 | 20.0 | 47.0 | 12.0 | 47.0 | 46.0 | ||
1000 | 20 | 33.8 | 11.4 | 7.6 | 100.0 | 100.0 | 5.7 | 8.9 | 20.0 | 40.5 | 27.4 | 66.0 | 76.4 | 5.9 | 0.0 |
1000 | 50 | 57.0 | 14.8 | 8.0 | 100.0 | 100.0 | 4.4 | 5.5 | 24.1 | 26.8 | 26.1 | 73.9 | 56.1 | ||
1000 | 100 | 77.3 | 32.5 | 9.9 | 100.0 | 100.0 | 5.2 | 5.1 | 20.6 | 26.3 | 18.1 | 66.5 | 44.8 | ||
1000 | 500 | 88.1 | 33.1 | 8.5 | 100.0 | 100.0 | 4.1 | 5.3 | 21.5 | 49.1 | 14.5 | 49.0 | 34.8 |
D
| PopSize | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 20 | 0.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 0.0 | 50.0 |
2 | 50 | 0.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 50.0 | 0.0 | 50.0 | 100.0 | 100.0 | ||
2 | 100 | 0.0 | 100.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50.0 | 100.0 | ||
2 | 500 | 50.0 | 0.0 | 100.0 | 100.0 | 100.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50.0 | 100.0 | 100.0 | ||
10 | 20 | 20.0 | 60.0 | 70.0 | 100.0 | 100.0 | 0.0 | 10.0 | 0.0 | 20.0 | 40.0 | 100.0 | 80.0 | 0.0 | 10.0 |
10 | 50 | 20.0 | 60.0 | 30.0 | 100.0 | 100.0 | 0.0 | 10.0 | 30.0 | 0.0 | 40.0 | 100.0 | 60.0 | ||
10 | 100 | 20.0 | 70.0 | 50.0 | 100.0 | 100.0 | 20.0 | 0.0 | 0.0 | 0.0 | 10.0 | 80.0 | 60.0 | ||
10 | 500 | 40.0 | 30.0 | 30.0 | 100.0 | 100.0 | 0.0 | 0.0 | 0.0 | 30.0 | 30.0 | 80.0 | 50.0 | ||
30 | 20 | 6.7 | 20.0 | 20.0 | 100.0 | 96.7 | 3.3 | 10.0 | 16.7 | 13.3 | 43.3 | 93.3 | 90.0 | 3.3 | 0.0 |
30 | 50 | 10.0 | 40.0 | 33.3 | 100.0 | 100.0 | 6.7 | 3.3 | 13.3 | 10.0 | 36.7 | 96.7 | 53.3 | ||
30 | 100 | 30.0 | 26.7 | 26.7 | 100.0 | 100.0 | 6.7 | 6.7 | 3.3 | 10.0 | 20.0 | 83.3 | 50.0 | ||
30 | 500 | 46.7 | 30.0 | 23.3 | 100.0 | 100.0 | 6.7 | 13.3 | 3.3 | 13.3 | 20.0 | 66.7 | 50.0 | ||
100 | 20 | 8.0 | 27.0 | 18.0 | 100.0 | 94.0 | 2.0 | 15.0 | 5.0 | 11.0 | 30.0 | 76.0 | 78.0 | 5.0 | 1.0 |
100 | 50 | 29.0 | 32.0 | 12.0 | 100.0 | 99.0 | 2.0 | 5.0 | 18.0 | 14.0 | 37.0 | 66.0 | 63.0 | ||
100 | 100 | 22.0 | 29.0 | 16.0 | 100.0 | 100.0 | 4.0 | 1.0 | 12.0 | 10.0 | 31.0 | 56.0 | 59.0 | ||
100 | 500 | 50.0 | 11.0 | 7.0 | 100.0 | 100.0 | 5.0 | 5.0 | 9.0 | 12.0 | 26.0 | 45.0 | 50.0 | ||
1000 | 20 | 10.2 | 21.3 | 6.4 | 100.0 | 93.1 | 4.9 | 13.7 | 7.6 | 16.8 | 37.2 | 63.1 | 85.1 | 3.7 | 0.1 |
1000 | 50 | 16.4 | 26.9 | 7.3 | 100.0 | 99.7 | 4.6 | 7.7 | 12.9 | 13.6 | 31.4 | 67.6 | 58.0 | ||
1000 | 100 | 29.0 | 23.9 | 5.9 | 100.0 | 100.0 | 4.6 | 7.1 | 9.5 | 11.6 | 31.7 | 55.2 | 53.2 | ||
1000 | 500 | 54.5 | 18.9 | 7.2 | 100.0 | 100.0 | 5.1 | 4.3 | 12.2 | 11.1 | 28.6 | 56.4 | 47.5 |
4.2 How quickly structural bias affects the search
Number of function calls | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Initialization | 4.9 | 5.2 | 10.9 | 4.9 | 5.3 | 5.9 | 4.9 | 4.4 | 3.6 | 5.1 | 9.0 | 7.4 | 5.7 | 4.7 |
200 | 99.1 | 68.1 | 7.2 | 72.2 | 99.8 | 6.0 | 4.5 | 15.8 | 14.3 | 6.2 | 27.4 | 28.8 | ||
300 | 100.0 | 7.9 | 8.2 | 100.0 | 100.0 | 4.8 | 5.4 | 26.3 | 26.6 | 8.6 | 35.1 | 40.9 | ||
400 | 91.2 | 19.5 | 9.0 | 96.3 | 100.0 | 3.6 | 6.6 | 31.6 | 43.0 | 11.4 | 49.2 | 39.5 | ||
500 | 52.4 | 32.8 | 10.2 | 100.0 | 100.0 | 3.9 | 4.5 | 42.4 | 55.5 | 11.4 | 51.3 | 44.5 | ||
1000 | 97.7 | 43.2 | 7.6 | 100.0 | 100.0 | 5.8 | 4.8 | 57.6 | 85.2 | 21.2 | 76.2 | 65.2 | 5.2 | |
2000 | 99.8 | 59.6 | 7.3 | 100.0 | 100.0 | 4.2 | 6.3 | 56.7 | 90.4 | 26.7 | 88.6 | 78.2 | 5.9 | 5.3 |
5000 | 99.9 | 60.5 | 7.3 | 100.0 | 100.0 | 5.4 | 5.3 | 53.5 | 79.2 | 37.3 | 90.2 | 82.4 | 5.2 | 5.5 |
10,000 | 99.2 | 64.3 | 9.3 | 100.0 | 100.0 | 4.6 | 4.5 | 44.8 | 54.4 | 40.1 | 93.6 | 80.4 | 4.9 | 4.8 |
4.3 Can we identify reasons for the structural bias?
4.4 Strength of structural bias and algorithms’ performance in the case of various benchmark problems
Algorithm | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE-basic | CDE | |
---|---|---|---|---|---|---|---|---|
Structural-bias group | G2 | G3 | G4 | G1 | G1 | G5 | G4 | |
PopSize | 20 | 20 | 1D, within [10,50] | 40 | 40 | 5D | 100 | |
Problem |
D
| |||||||
1 | 6 | 1.671E\(+\)01 | 1.066E\(+\)01 | 1.547E\(+\)00 | 1.139E\(+\)01 | 1.133E\(+\)01 | 0.000E\(+\)00 | 0.000E\(+\)00 |
2 | 30 | −7.758E\(+\)00 | −1.817E\(+\)01 | −2.097E\(+\)01 | −2.717E\(+\)01 | −2.268E\(+\)01 | −5.884E\(+\)00 | −1.873E\(+\)0 |
3 | 1 | 1.151E−05 | 1.151E−05 | 1.151E−05 | 1.151E−05 | 1.151E−05 | 1.151E−05 | 1.151E−05 |
4 | 1 | 1.393E\(+\)01 | 1.394E\(+\)01 | 1.479E\(+\)01 |
1.378E
\(+\)
01
| 1.419E\(+\)01 | 2.023E\(+\)01 | 1.765E\(+\)01 |
5 | 30 | −2.170E\(+\)01 | −3.267E\(+\)01 | −3.516E
\(+\)
01
| −3.359E\(+\)01 | −3.274E\(+\)01 | −2.053E\(+\)01 | −3.277E\(+\)0 |
6 | 30 | −1.666E\(+\)01 | −2.421E\(+\)01 | −2.854E
\(+\)
01
| −2.193E\(+\)01 | −2.267E\(+\)01 | −1.503E\(+\)01 | −2.726E\(+\)0 |
7 | 20 | 1.514E\(+\)00 | 1.131E\(+\)00 | 1.079E\(+\)00 | 1.017E\(+\)00 | 9.787E−01 | 1.706E\(+\)00 | 1.360E\(+\)00 |
8 | 7 | 2.200E\(+\)02 | 2.200E\(+\)02 | 2.200E\(+\)02 | 2.220E\(+\)02 | 2.243E\(+\)02 | 2.200E\(+\)02 | 2.200E\(+\)02 |
9 | 126 | 3.366E\(+\)05 | 6.297E\(+\)05 | 2.099E\(+\)04 | 3.205E\(+\)03 | 2.827E\(+\)05 | 8.546E\(+\)05 | 2.407E\(+\)03 |
10 | 12 | −1.358E\(+\)01 | −1.581E\(+\)01 | −2.042E\(+\)01 | −1.858E\(+\)01 | −1.784E\(+\)01 | −2.043E\(+\)01 | −2.169E
\(+\)
01
|
11.1 | 120 | 1.990E\(+\)07 | 8.421E\(+\)05 | 1.277E\(+\)05 | 9.895E\(+\)05 | 2.741E\(+\)05 | 1.359E\(+\)08 |
5.189E
\(+\)
04
|
11.2 | 216 | 2.630E\(+\)07 | 1.908E\(+\)07 | 1.871E\(+\)07 | 1.794E\(+\)07 | 2.413E\(+\)07 | 3.638E\(+\)07 | 1.746E\(+\)07 |
11.3 | 6 | 1.546E\(+\)04 | 1.547E\(+\)04 | 1.545E\(+\)04 | 1.547E\(+\)04 | 1.547E\(+\)04 | 1.544E\(+\)04 | 1.545E\(+\)04 |
11.4 | 13 | 1.931E\(+\)04 | 1.923E\(+\)04 | 1.907E\(+\)04 | 1.919E\(+\)04 | 1.913E\(+\)04 | 1.940E\(+\)04 | 1.813E\(+\)04 |
11.5 | 15 | 4.864E\(+\)04 | 3.296E\(+\)04 | 3.298E\(+\)04 | 3.299E\(+\)04 | 3.310E\(+\)04 | 3.298E\(+\)04 | 3.277E\(+\)04 |
11.6 | 40 | 1.374E\(+\)05 | 1.406E\(+\)05 | 1.361E\(+\)05 | 1.391E\(+\)05 | 1.413E\(+\)05 | 1.387E\(+\)05 | 1.340E\(+\)05 |
11.7 | 140 | 3.586E\(+\)06 | 2.050E\(+\)06 | 2.088E\(+\)06 | 1.993E\(+\)06 | 1.972E\(+\)06 | 5.592E\(+\)07 | 1.941E\(+\)06 |
11.8 | 96 | 1.901E\(+\)06 | 1.127E\(+\)06 | 1.023E\(+\)06 | 9.749E\(+\)05 | 1.006E\(+\)06 | 9.814E\(+\)06 | 9.759E\(+\)05 |
11.9 | 96 | 2.574E\(+\)06 | 1.572E\(+\)06 | 1.314E\(+\)06 | 1.416E\(+\)06 | 1.238E\(+\)06 | 1.067E\(+\)07 | 1.432E\(+\)06 |
11.10 | 96 | 1.929E\(+\)06 | 1.178E\(+\)06 | 1.022E\(+\)06 | 9.690E\(+\)05 | 1.001E\(+\)06 | 9.711E\(+\)06 | 9.761E\(+\)05 |
12 | 26 | 2.741E\(+\)01 | 1.778E\(+\)01 | 1.547E\(+\)01 |
1.254E
\(+\)
01
| 1.662E\(+\)01 | 2.787E\(+\)01 | 1.971E\(+\)01 |
13 | 22 | 2.958E\(+\)01 | 2.379E\(+\)01 | 1.789E\(+\)01 | 2.208E\(+\)01 | 2.346E\(+\)01 | 2.468E\(+\)01 | 1.601E\(+\)01 |
Average ranks | 11.48 | 9.25 | 6.30 | 7.32 | 8.59 | 11.18 | 4.95 |
Algorithm | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA | |
---|---|---|---|---|---|---|---|---|
Structural-bias group | G3 | G3 | G3 | G2 | G2 | G5 | G6 | |
PopSize | 10D | 100 | 100 | 1D, within [10,50] | 90 | D+1 | 1 | |
Problem |
D
| |||||||
1 | 6 | 8.222E\(+\)00 | 3.318E\(+\)00 | 3.322E−01 | 1.078E\(+\)01 | 5.939E−01 | 2.031E\(+\)01 | 2.002E\(+\)01 |
2 | 30 | −1.187E\(+\)01 | −2.489E\(+\)01 | −2.108E\(+\)01 | −2.523E\(+\)01 | −2.723E
\(+\)
01
| −2.673E\(+\)01 | −1.523E\(+\)01 |
3 | 1 | 1.151E−05 | 1.151E−05 | 1.151E−05 | 1.151E−05 | 1.151E−05 | 1.151E−05 | 1.151E−05 |
4 | 1 | 2.047E\(+\)01 | 1.596E\(+\)01 | 1.694E\(+\)01 | 1.557E\(+\)01 | 1.412E\(+\)01 | 1.413E\(+\)01 | 1.382E\(+\)01 |
5 | 30 | −2.199E\(+\)01 | −3.311E\(+\)01 | −3.402E\(+\)01 | −3.455E\(+\)01 | −3.425E\(+\)01 | −3.449E\(+\)01 | −3.397E\(+\)01 |
6 | 30 | −1.924E\(+\)01 | −2.429E\(+\)01 | −2.845E\(+\)01 | −2.583E\(+\)01 | −2.739E\(+\)01 | −2.330E\(+\)01 | −2.433E\(+\)01 |
7 | 20 | 1.481E\(+\)00 | 1.132E\(+\)00 | 1.232E\(+\)00 | 9.680E−01 | 9.992E−01 |
7.794E−01
| 1.247E\(+\)00 |
8 | 7 | 2.200E\(+\)02 | 2.200E\(+\)02 | 2.200E\(+\)02 | 2.226E\(+\)02 | 2.200E\(+\)02 | 2.200E\(+\)02 | 2.365E\(+\)02 |
9 | 126 | 8.614E\(+\)03 | 2.088E\(+\)03 | 2.214E\(+\)03 | 4.809E\(+\)03 | 2.431E\(+\)03 | 2.253E\(+\)06 |
1.791E
\(+\)
03
|
10 | 12 | −2.008E\(+\)01 | −2.156E\(+\)01 | −2.159E\(+\)01 | −1.971E\(+\)01 | −2.166E\(+\)01 | −1.974E\(+\)01 | −1.020E\(+\)01 |
11.1 | 120 | 7.540E\(+\)04 | 5.251E\(+\)04 | 5.237E\(+\)04 | 5.709E\(+\)04 | 5.229E\(+\)04 | 2.503E\(+\)05 | 8.893E\(+\)04 |
11.2 | 216 | 1.837E\(+\)07 | 1.743E\(+\)07 | 1.747E\(+\)07 | 1.772E\(+\)07 |
1.079E
\(+\)
06
| 4.565E\(+\)07 | 1.773E\(+\)07 |
11.3 | 6 | 1.546E\(+\)04 | 1.545E\(+\)04 | 1.544E\(+\)04 | 1.548E\(+\)04 | 1.544E\(+\)04 | 1.548E\(+\)04 | 1.546E\(+\)04 |
11.4 | 13 | 1.814E\(+\)04 |
1.810E
\(+\)
04
| 1.811E\(+\)04 | 1.875E\(+\)04 | 1.832E\(+\)04 | 1.841E\(+\)04 | 1.839E\(+\)04 |
11.5 | 15 |
3.276E
\(+\)
04
| 3.278E\(+\)04 | 3.279E\(+\)04 | 3.304E\(+\)04 | 3.278E\(+\)04 | 3.300E\(+\)04 | 3.309E\(+\)04 |
11.6 | 40 | 1.333E\(+\)05 | 1.292E\(+\)05 | 1.297E\(+\)05 | 1.355E\(+\)05 | 1.371E\(+\)05 |
1.280E
\(+\)
05
| 1.428E\(+\)05 |
11.7 | 140 | 2.066E\(+\)06 | 1.916E\(+\)06 |
1.906E
\(+\)
06
| 2.000E\(+\)06 | 2.150E\(+\)06 | 1.396E\(+\)07 | 2.275E\(+\)06 |
11.8 | 96 | 1.259E\(+\)06 |
9.385E
\(+\)
05
| 9.402E\(+\)05 | 9.548E\(+\)05 | 1.048E\(+\)06 | 2.460E\(+\)06 | 1.812E\(+\)06 |
11.9 | 96 | 1.756E\(+\)06 |
9.506E
\(+\)
05
| 1.037E\(+\)06 | 1.090E\(+\)06 | 1.250E\(+\)06 | 3.090E\(+\)06 | 2.555E\(+\)06 |
11.10 | 96 | 1.252E\(+\)06 |
9.388E
\(+\)
05
| 9.404E\(+\)05 | 9.563E\(+\)05 | 1.045E\(+\)06 | 2.378E\(+\)06 | 1.625E\(+\)06 |
12 | 26 | 1.622E\(+\)01 | 1.595E\(+\)01 | 1.750E\(+\)01 | 1.486E\(+\)01 | 1.324E\(+\)01 | 2.154E\(+\)01 | 4.975E\(+\)01 |
13 | 22 | 1.742E\(+\)01 | 1.867E\(+\)01 | 1.669E\(+\)01 | 2.118E\(+\)01 |
1.070E
\(+\)
01
| 2.647E\(+\)01 | 3.613E\(+\)01 |
Average ranks | 7.98 | 4.16 | 4.11 | 6.18 | 4.52 | 9.39 | 9.59 |
Algorithm | PSO-basic | ALC-PSO | CLPSO | DNS-PSO | PSO-init-weight | DE- basic | CDE |
---|---|---|---|---|---|---|---|
Structural-bias group | G2 | G3 | G4 | G1 | G1 | G5 | G4 |
PopSize | 20 | 20 | 1D, within[10, 50] | 40 | 40 | 5D | 100 |
Problem | |||||||
1 | 2.304E\(+\)08 | 2.140E\(+\)06 | 1.526E\(+\)07 | 8.947E\(+\)05 | 6.993E\(+\)06 | 5.924E\(+\)08 | 2.647E\(+\)06 |
2 | 1.573E\(+\)10 | 1.156E\(+\)05 | 3.872E\(+\)01 | 7.042E\(+\)03 | 1.113E\(+\)06 | 5.280E\(+\)09 | 1.039E\(+\)03 |
3 | 3.687E\(+\)04 | 1.622E\(+\)04 | 2.120E\(+\)03 | 1.617E\(+\)03 | 5.897E\(+\)02 | 1.344E\(+\)05 | 1.394E\(+\)02 |
4 | 1.057E\(+\)03 | 9.872E\(+\)01 | 9.823E\(+\)01 | 8.125E\(+\)01 | 1.953E\(+\)02 | 7.136E\(+\)02 | 9.427E\(+\)01 |
5 | 2.113E\(+\)01 | 2.099E\(+\)01 | 2.046E\(+\)01 | 2.112E\(+\)01 | 2.096E\(+\)01 | 2.112E\(+\)01 | 2.060E\(+\)01 |
6 | 5.185E\(+\)01 | 3.605E\(+\)01 | 2.795E\(+\)01 | 3.329E\(+\)01 | 2.838E\(+\)01 | 6.532E\(+\)01 | 1.790E\(+\)01 |
7 | 1.479E\(+\)02 | 6.924E−02 | 3.199E−04 | 9.591E−03 | 8.007E−03 | 2.944E\(+\)01 | 1.112E−03 |
8 | 3.861E\(+\)02 | 3.902E−02 | 1.315E−13 | 1.898E\(+\)02 | 8.531E\(+\)01 | 3.289E\(+\)02 | 0.000E\(+\)00 |
9 | 4.342E\(+\)02 | 2.195E\(+\)02 | 1.187E\(+\)02 | 2.282E\(+\)02 | 1.430E\(+\)02 | 4.669E\(+\)02 | 1.319E\(+\)02 |
10 | 1.292E\(+\)04 | 3.316E\(+\)02 |
2.555E
\(+\)
00
| 4.107E\(+\)03 | 2.374E\(+\)03 | 9.905E\(+\)03 | 4.122E\(+\)01 |
11 | 1.306E\(+\)04 | 5.389E\(+\)03 | 4.582E\(+\)03 | 6.219E\(+\)03 | 5.611E\(+\)03 | 1.312E\(+\)04 | 7.677E\(+\)03 |
12 | 3.392E\(+\)00 | 9.989E−01 | 3.407E−01 | 2.502E\(+\)00 | 7.517E−01 | 3.174E\(+\)00 | 8.004E−01 |
13 | 1.220E\(+\)00 | 5.971E−01 | 3.996E−01 | 5.776E−01 | 5.491E−01 | 8.004E−01 | 3.875E−01 |
14 | 4.141E\(+\)01 | 5.765E−01 | 3.029E−01 | 3.858E−01 | 6.459E−01 | 3.993E\(+\)00 |
2.963E−01
|
15 | 3.053E\(+\)03 | 3.566E\(+\)01 | 1.799E\(+\)01 | 2.125E\(+\)01 | 1.572E\(+\)01 | 3.973E\(+\)03 | 1.416E\(+\)01 |
16 | 2.195E\(+\)01 | 2.105E\(+\)01 | 1.860E\(+\)01 | 2.117E\(+\)01 | 2.012E\(+\)01 | 2.244E\(+\)01 | 1.964E\(+\)01 |
17 | 1.141E\(+\)07 | 4.268E\(+\)05 | 2.415E\(+\)06 | 1.570E\(+\)05 | 1.409E\(+\)06 | 2.678E\(+\)07 | 3.231E\(+\)05 |
18 | 4.759E\(+\)08 | 2.460E\(+\)03 |
1.440E
\(+\)
02
| 1.001E\(+\)03 | 1.226E\(+\)04 | 1.091E\(+\)06 | 3.772E\(+\)02 |
19 | 1.392E\(+\)02 | 2.918E\(+\)01 | 1.773E\(+\)01 | 1.633E\(+\)01 | 5.750E\(+\)01 | 4.765E\(+\)01 | 3.473E\(+\)01 |
20 | 6.747E\(+\)03 | 5.192E\(+\)03 | 6.251E\(+\)03 | 4.519E\(+\)02 | 5.335E\(+\)02 | 4.868E\(+\)04 | 7.014E\(+\)02 |
21 | 4.788E\(+\)06 | 2.330E\(+\)05 | 1.506E\(+\)06 | 1.032E\(+\)05 | 3.545E\(+\)05 | 1.117E\(+\)07 | 2.958E\(+\)05 |
22 | 1.592E\(+\)03 | 1.257E\(+\)03 | 5.782E\(+\)02 | 1.108E\(+\)03 | 8.023E\(+\)02 | 1.425E\(+\)03 | 5.941E\(+\)02 |
23 | 4.889E\(+\)02 | 3.440E\(+\)02 | 3.440E\(+\)02 |
2.000E
\(+\)
02
| 3.462E\(+\)02 | 3.482E\(+\)02 | 3.440E\(+\)02 |
24 | 3.663E\(+\)02 | 2.769E\(+\)02 | 2.580E\(+\)02 |
2.000E
\(+\)
02
| 2.743E\(+\)02 | 3.324E\(+\)02 | 2.680E\(+\)02 |
25 | 2.515E\(+\)02 | 2.157E\(+\)02 | 2.153E\(+\)02 | 2.000E\(+\)02 | 2.206E\(+\)02 | 3.059E\(+\)02 | 2.007E\(+\)02 |
26 | 2.013E\(+\)02 | 1.482E\(+\)02 | 1.005E\(+\)02 | 1.961E\(+\)02 | 1.716E\(+\)02 | 1.008E\(+\)02 | 1.004E\(+\)02 |
27 | 1.646E\(+\)03 | 1.395E\(+\)03 | 7.626E\(+\)02 |
2.428E
\(+\)
02
| 1.126E\(+\)03 | 1.822E\(+\)03 | 4.856E\(+\)02 |
28 | 3.634E\(+\)03 | 3.273E\(+\)03 | 1.388E\(+\)03 |
2.126E
\(+\)
02
| 2.786E\(+\)03 | 1.748E\(+\)03 | 1.275E\(+\)03 |
29 | 4.474E\(+\)07 | 6.774E\(+\)06 | 1.396E\(+\)03 | 2.936E\(+\)06 | 5.216E\(+\)07 | 8.651E\(+\)05 | 1.825E\(+\)03 |
30 | 2.257E\(+\)05 | 2.270E\(+\)04 | 9.905E\(+\)03 | 1.241E\(+\)04 | 4.697E\(+\)04 | 1.475E\(+\)05 | 8.986E\(+\)03 |
Average ranks | 13.32 | 9.41 | 5.68 | 8.59 | 8.86 | 13.05 | 6.20 |
Algorithm | DEGL | MDE-pBX | SADE | CLPSO-DEGL | GA-MPC | NMA | RA |
---|---|---|---|---|---|---|---|
Structural-bias group | G3 | G3 | G3 | G2 | G2 | G5 | G6 |
PopSize | 10D | 100 | 100 | 1D, within [10, 50] | 90 | D+1 | 1 |
Problem | |||||||
1 | 4.056E\(+\)05 | 7.704E\(+\)04 | 1.461E\(+\)05 | 1.387E\(+\)05 | 3.963E\(+\)05 |
5.181E
\(+\)
04
| 2.836E\(+\)05 |
2 |
0.000E
\(+\)
00
| 3.734E\(+\)03 | 6.969E−10 | 3.652E−05 | 3.727E\(+\)03 | 9.540E\(+\)03 | 8.092E\(+\)03 |
3 |
0.000E
\(+\)
00
| 7.845E−03 | 3.000E\(+\)02 | 2.296E−03 | 1.404E\(+\)01 | 5.630E−07 | 8.917E−06 |
4 | 2.600E\(+\)01 | 3.453E\(+\)01 |
1.951E
\(+\)
01
| 5.308E\(+\)01 | 6.359E\(+\)01 | 3.235E\(+\)01 | 4.670E\(+\)01 |
5 | 2.113E\(+\)01 | 2.050E\(+\)01 | 2.060E\(+\)01 | 2.075E\(+\)01 | 2.024E\(+\)01 | 2.000E\(+\)01 | 2.000E\(+\)01 |
6 |
3.439E
\(+\)
00
| 5.805E\(+\)00 | 1.014E\(+\)01 | 3.181E\(+\)01 | 5.055E\(+\)00 | 2.759E\(+\)01 | 8.137E\(+\)01 |
7 | 1.208E−03 | 6.546E−03 | 4.301E−03 | 3.326E−02 | 3.427E−03 | 1.358E−02 |
7.023E−11
|
8 | 2.747E\(+\)02 | 5.557E\(+\)01 | 0.000E\(+\)00 | 8.455E\(+\)01 | 5.222E\(+\)01 | 1.629E\(+\)02 | 5.299E\(+\)02 |
9 | 3.367E\(+\)02 | 1.277E\(+\)02 | 8.703E\(+\)01 | 1.427E\(+\)02 |
5.651E
\(+\)
01
| 1.532E\(+\)02 | 7.983E\(+\)02 |
10 | 1.199E\(+\)04 | 1.533E\(+\)03 | 1.349E\(+\)01 | 1.739E\(+\)02 | 1.141E\(+\)03 | 6.737E\(+\)03 | 6.773E\(+\)03 |
11 | 1.287E\(+\)04 | 8.089E\(+\)03 | 6.474E\(+\)03 | 6.021E\(+\)03 |
3.657E
\(+\)
03
| 6.916E\(+\)03 | 7.011E\(+\)03 |
12 | 3.324E\(+\)00 | 7.290E−01 | 7.427E−01 | 7.693E−01 |
8.171E−02
| 2.174E−01 | 1.057E\(+\)00 |
13 | 3.464E−01 |
3.046E−01
| 3.774E−01 | 5.291E−01 | 4.241E−01 | 7.607E−01 | 3.283E−01 |
14 | 3.399E−01 | 3.221E−01 | 3.034E−01 | 3.487E−01 | 3.346E−01 | 3.061E−01 | 3.018E−01 |
15 | 2.849E\(+\)01 | 9.405E\(+\)00 | 1.397E\(+\)01 | 3.602E\(+\)01 |
5.621E
\(+\)
00
| 3.000E\(+\)01 | 7.344E\(+\)02 |
16 | 2.153E\(+\)01 | 1.979E\(+\)01 | 1.921E\(+\)01 | 1.908E\(+\)01 |
1.704E
\(+\)
01
| 2.195E\(+\)01 | 2.371E\(+\)01 |
17 |
1.753E
\(+\)
03
| 4.466E\(+\)03 | 1.071E\(+\)04 | 1.085E\(+\)04 | 2.973E\(+\)04 | 2.833E\(+\)03 | 1.600E\(+\)04 |
18 | 1.802E\(+\)02 | 3.245E\(+\)02 | 3.222E\(+\)02 | 1.010E\(+\)03 | 4.799E\(+\)02 | 2.343E\(+\)02 | 2.602E\(+\)03 |
19 | 1.272E\(+\)01 | 1.056E\(+\)01 | 1.669E\(+\)01 | 1.779E\(+\)01 |
5.767E
\(+\)
00
| 3.249E\(+\)01 | 2.840E\(+\)01 |
20 |
9.189E
\(+\)
01
| 1.698E\(+\)02 | 2.864E\(+\)02 | 2.058E\(+\)02 | 2.119E\(+\)02 | 5.474E\(+\)02 | 7.217E\(+\)02 |
21 |
6.417E
\(+\)
02
| 1.401E\(+\)03 | 4.551E\(+\)03 | 2.756E\(+\)03 | 1.684E\(+\)04 | 3.110E\(+\)03 | 1.707E\(+\)04 |
22 | 1.139E\(+\)03 | 7.680E\(+\)02 |
4.013E
\(+\)
02
| 4.925E\(+\)02 | 1.010E\(+\)03 | 1.145E\(+\)03 | 1.242E\(+\)03 |
23 | 3.440E\(+\)02 | 3.440E\(+\)02 | 3.440E\(+\)02 | 3.440E\(+\)02 | 3.370E\(+\)02 | 3.440E\(+\)02 | 3.444E\(+\)02 |
24 | 2.750E\(+\)02 | 2.781E\(+\)02 | 2.739E\(+\)02 | 2.821E\(+\)02 | 2.685E\(+\)02 | 2.830E\(+\)02 | 3.016E\(+\)02 |
25 | 2.060E\(+\)02 | 2.102E\(+\)02 | 2.096E\(+\)02 | 2.311E\(+\)02 | 2.005E\(+\)02 | 2.000E\(+\)02 | 2.208E\(+\)02 |
26 | 1.004E\(+\)02 | 1.043E\(+\)02 | 1.316E\(+\)02 | 1.143E\(+\)02 | 1.004E\(+\)02 | 1.415E\(+\)02 | 1.219E\(+\)02 |
27 | 4.696E\(+\)02 | 5.568E\(+\)02 | 5.838E\(+\)02 | 1.216E\(+\)03 | 1.239E\(+\)03 | 9.970E\(+\)02 | 1.565E\(+\)03 |
28 | 1.149E\(+\)03 | 1.165E\(+\)03 | 1.176E\(+\)03 | 2.416E\(+\)03 | 3.695E\(+\)02 | 6.928E\(+\)03 | 7.805E\(+\)03 |
29 | 8.312E\(+\)05 | 9.349E\(+\)02 | 1.008E\(+\)03 | 1.154E\(+\)03 |
2.158E
\(+\)
02
| 1.283E\(+\)03 | 1.205E\(+\)04 |
30 | 5.618E\(+\)03 | 9.436E\(+\)03 | 1.048E\(+\)04 | 1.338E\(+\)04 |
1.143E
\(+\)
03
| 2.155E\(+\)04 | 1.430E\(+\)04 |
Average ranks | 6.18 | 4.50 | 4.02 | 6.05 | 4.41 | 6.55 | 8.18 |