We present and discuss philosophy and methodology of chaotic evolution that is theoretically supported by chaos theory. We introduce four chaotic systems, that is, logistic map, tent map, Gaussian map, and Hénon map, in a well-designed chaotic evolution algorithm framework to implement several chaotic evolution (CE) algorithms. By comparing our previous proposed CE algorithm with logistic map and two canonical differential evolution (DE) algorithms, we analyse and discuss optimization performance of CE algorithm. An investigation on the relationship between optimization capability of CE algorithm and distribution characteristic of chaotic system is conducted and analysed. From evaluation result, we find that distribution of chaotic system is an essential factor to influence optimization performance of CE algorithm. We propose a new interactive EC (IEC) algorithm, interactive chaotic evolution (ICE) that replaces fitness function with a real human in CE algorithm framework. There is a paired comparison-based mechanism behind CE search scheme in nature. A simulation experimental evaluation is conducted with a pseudo-IEC user to evaluate our proposed ICE algorithm. The evaluation result indicates that ICE algorithm can obtain a significant better performance than or the same performance as interactive DE. Some open topics on CE, ICE, fusion of these optimization techniques, algorithmic notation, and others are presented and discussed.
Philosophy of determinism comes from the development of classic mechanic that was originally established and studied by Isaac Newton, Pierre-Simon Laplace, Gottfried Wilhelm Leibniz, and so forth. Strict determinism indicates that causality can be expressed and implemented by mathematical calculation and logical reasoning. As Pierre-Simon Laplace said, “we may regard the present state of the universe as the effect of its past and the cause of its future” [
Most of evolutionary computation (EC) algorithms are inspired from natural phenomena, such as genetic algorithm that mimics the process of natural selection and survival of the fittest. EC can be involved in continuous and combinational optimization problems, and its algorithm has a metaheuristic or stochastic characteristic in their search mechanisms [
The study on chaos theory comes from the real problems of physics, ecology, and mathematics since three-body problem was studied by Poincare et al. [
There are typically three applications in which chaos is used in optimization area, that is, a local search method, a parameter tuning technique, and the new EC algorithm inspiration resource. Some chaotic systems are introduced into conventional EC algorithm frameworks to make population diversity in local search and to tune the parameters of algorithm [
Chaotic evolution (CE) is a population-based algorithm framework that simulates chaotic motion behaviour in a search space [
This paper extends the work of [
Following this introductory section, an overview of the CE algorithm framework and chaotic systems used in this paper are reported in Section
Deterministic system and stochastic system are two views in which we understand and describe the nature in philosophy and in science. They lead to two corresponding algorithm methodologies in optimization field, that is, deterministic and stochastic optimization algorithms. Evolutionary computation can be partially categorized into the later one, that is, stochastic optimization algorithm. However, since chaotic phenomenon and mechanism were found [
The concepts of “evolution” and “chaos” have more the same characteristics in common [
Inspired from chaotic motion of nonlinear system that has an ergodicity property, a chaotic ergodicity based EC algorithm,
Suppose that an array on the left side of Figure Choose one individual as a target vector. Obtain a chaotic parameter from a chaotic system Make a mutant vector by Generate a chaotic vector by crossing the target vector and the mutant vector Compare the target vector and the chaotic vector and choose whichever a better one as offspring in the next generation. Go to (1) and generate other offspring until all individuals are replaced with offspring in the next generation.
Chaotic evolution algorithm framework; there are three special vectors in chaotic evolution algorithm, a target vector that is an individual in the population, a mutant vector that simulates the chaotic motion from a chaotic system, and a chaotic vector that is made by conducting crossover operation on a target vector and a mutation vector.
The terms of
Interactive evolutionary computation (IEC) is a niche research field in EC community [
From a framework viewpoint, there are three main parts in an IEC based optimization system. They are (1) a target system that is optimized, (2) an IEC algorithm (including an IEC interface) that conducts actual optimization function, and (3) a real human who provides his or her evaluation. The target system and the real human are fixed in an IEC optimization application, so there is a study subjective to relieve human’s fatigue in the IEC algorithm part. One is to enhance optimization performance of IEC algorithm. The other is to improve IEC interface for a user friendly interaction.
If we implement a CE algorithm in an IEC application and replace the fitness function with a real human’s evaluation, the IEC framework is an implementation of interactive chaotic evolution (ICE). In the nature of ICE algorithm, there is a paired comparison-based mechanism when surviving an offspring between chaotic vector and target vector. In an IEC application, these two vectors present two IEC solutions for a real human to perform evaluations. Comparing with other IEC algorithms, such as interactive genetic algorithm (IGA), this paired comparison mechanism benefits real human’s evaluation rather than evaluating all individuals together as in IGA. If the ICE algorithm can present significantly optimization performance compared to other paired comparison-based IEC algorithms, such as interactive differential evolution (IDE) [
The search function of CE is implemented by generating a chaotic vector. It is decided by chaotic parameter from a chaotic system, which has the ergodicity property. By applying a variety of chaotic systems in the CE algorithm framework, we can implement its variations in different way. In this study, we introduce four chaotic systems, that is, logistic map, tent map, Gaussian map, and Hénon map to investigate the optimization performances of CE with these four chaotic maps. Here, we make a brief review on these chaotic maps.
The logistic map is a polynomial mapping with two degrees that can arise from a chaotic phenomenon by a simple nonlinear system easily [
Bifurcation digrams of (a) logistic map, (b) tent map, (c) Gaussian map, and (d) Hénon map. Parameter
Logistic map
Tent map
Gaussian map
Hénon map
The tent map is defined as in (
The Gaussian map is a nonlinear map that is given by a Gaussian function (equation (
The Hénon map has two points (
We use 25 benchmark functions from [
Test functions (Uni = unimodal, Multi = multimodal, Sh = shifted, Rt = rotated, GB = global on Bounds, HC = hybrid composition, and NM = number matrix).
Number | Type | Characteristic | Bounds | Optimum fitness |
---|---|---|---|---|
|
Uni | Sh sphere |
|
−450 |
|
Sh Schwefel 1.2 | −450 | ||
|
Sh Rt Elliptic | −450 | ||
|
|
−450 | ||
|
Schwefel 2.6 GB | −310 | ||
|
||||
|
Multi | Sh Rosenbrock |
|
390 |
|
Sh Rt Griewank |
|
−180 | |
|
Sh Rt Ackley GB |
|
−140 | |
|
Sh Rastrigin |
|
−330 | |
|
Sh Rt Rastrigin |
|
−330 | |
|
Sh Rt Weierstrass |
|
90 | |
|
Schwefel 2.13 |
|
−460 | |
|
Sh expanded F8F2 |
|
−130 | |
|
Sh Rt Scaffer F6 |
|
−300 | |
|
||||
|
Hybrid | HC function |
|
120 |
|
Rt HC function 1 | 120 | ||
|
|
120 | ||
|
Rt HC function 2 | 10 | ||
|
|
10 | ||
|
|
10 | ||
|
Rt HC function 3 | 360 | ||
|
|
360 | ||
|
NC Rt |
360 | ||
|
Rt HC function 4 | 260 | ||
|
|
260 |
There are five CE algorithms in our evaluation, that is, CE-logistic, CE-tent, CE-Gauss, CE-Hénon-rand, and CE-Hénon-attrac. Two DE algorithms (DE/best/1/bin and DE/rand/1/bin) are used to be compared with CE algorithms [
Algorithm parameter setting and abbreviations of the algorithms used in evaluation.
Abbreviation | Meaning |
---|---|
CE-logistic | Chaotic evolution with logistic map (random initialization within (0, 1 |
CE-tent | Chaotic evolution with tent map (random initialization within (0, 1 |
CE-Gauss | Chaotic evolution with Gaussian map (random initialization within (0, 1 |
CE-Hénon-rand | Chaotic evolution with Hénon map (random initialization within (0, 1 |
CE-Hénon-attrac. | Chaotic evolution with Hénon map (initialization with attractors explained in Figure |
CE crossover rate | 1 |
|
|
DE/best/1/bin | DE with the best individual as base vector. |
DE/rand/1/bin | DE with random individual as base vector. |
DE scale factor |
1 |
DE crossover rate | 1 |
We apply Wilcoxon sign-ranked test and Friedman test on the fitness value at 1000th generation and make two groups with and without DE algorithms to rank the algorithms. We use Bonferroni-Dunn’s test to check the significance of algorithm rank in both groups. Tables
Mean fitness values of F1–F25 with 10D. The fitness values in bold and italic font are the best and worst optimization results among the five chaotic evolution algorithms, respectively. The (†) and (
Function | DE-best | DE-rand | CE-logistic | CE-tent | CE-Gauss | CE-Hénon-rand | CE-Hénon-attrac. |
---|---|---|---|---|---|---|---|
F1 | −449.998 | −414.47 | 2998.806 |
|
1925.366 | 1768.834 |
|
F2 | −440.516 | −421.736 | 4219.061 |
|
|
3855.883 | 3568.253 |
F3 | 1583.076 | 215058.7 | 3624018 | 26036921 |
|
4846787 |
|
F4 | −283.931 | −402.19 | 6487.652 |
|
|
4723.953 | 5358.734 |
F5 | −310 | −309.972 | 3775.085 |
|
3986.86 | 4148.611 |
|
F6 | 446.6181 | 35496.74 | 287017050.30 | 628335744.87 | 272134693.96 |
|
|
F7 | −177.747 | −177.865 | −97.4097 | − |
|
−105.609 | −121.71 |
F8 | −119.992 | −119.893 | −119.635 | − |
−119.62 | −119.604 |
|
F9 | −308.575 | −306.239 |
|
− |
−282.345 | −284.484 | −282.21 |
F10 | −291.876 | −282.316 | −254.408 | − |
−261.758 | −258.089 |
|
F11 | 98.34748 | 94.5809 |
|
|
|
|
97.52258 |
F12 | 321.4523 | 4968.164 | 20706.57 | 53091.15 | 33802.12 | 28609.12 |
|
F13 | −126.975 | −126.213 | −125.596 | − |
|
−124.246 |
|
F14 | −296.159 | −296.567 |
|
− |
|
|
|
F15 | 476.4005 | 557.9363 |
|
|
|
|
|
F16 | 302.9829 | 337.3688 | 378.9592 |
|
377.3656 | 391.311 |
|
F17 | 323.7027 | 326.4775 | 409.4423 |
|
401.6362 | 426.4379 |
|
F18 | 869.8405 | 578.9428 |
|
|
|
|
|
F19 | 869.4931 | 591.2802 | 910 | 910 |
|
|
910 |
F20 | 866.7625 | 610.5591 | 910 | 910 |
|
|
910 |
F21 | 1079.758 | 929.7804 | 1623.005 |
|
|
1657.526 | 1574.536 |
F22 | 1222.383 | 1164.546 | 1320.278 |
|
1307.327 | 1309.518 |
|
F23 | 1066.17 | 970.7627 | 1647.835 |
|
|
1649.8 | 1563.626 |
F24 | 824.7887 | 494.8996 | 1450.171 |
|
1376.52 | 1398.889 |
|
F25 | 1989.595 | 2001.341 | 2010.443 |
|
2018.292 |
|
2014.102 |
Mean fitness values of F1–F25 with 30D. The explanations of special marks are as the same as in Table
Function | DE-best | DE-rand | CE-logistic | CE-tent | CE-Gauss | CE-Hénon-rand | CE-Hénon-attrac. |
---|---|---|---|---|---|---|---|
F1 | 97.24339 | 10626.17 | 33437.54 |
|
|
34997.74 | 27669.15 |
F2 | 1513.336 | 22808.96 | 28931.33 |
|
|
31553.89 | 26444.3 |
F3 | 3203063 | 91581615 | 155259859.97 |
|
|
212799570.04 | 139163745.08 |
F4 | 10475.36 | 31198.64 |
|
|
|
37707.88 | 33388.45 |
F5 | 3753.477 | 12014.48 | 18617.58 |
|
19258.04 | 21922.04 |
|
F6 | 10041703 | 917911267.58 | 11016735211.66 |
|
6390791853.98 | 11349392992.19 |
|
F7 | −165.651 | 98.12331 | 1042.799 |
|
|
1145.698 | 996.1877 |
F8 | −119.769 | −119.03 |
|
− |
|
|
|
F9 | −224.472 | −80.3587 | −4.98262 |
|
|
−15.6137 | −20.0503 |
F10 | −157.64 | −16.854 | 170.6209 |
|
156.3653 | 133.2562 |
|
F11 | 124.2699 | 129.963 |
|
|
|
|
|
F12 | 39573.98 | 945070.8 |
|
|
1097621 |
|
|
F13 | −94.4806 | −72.5022 |
|
−64.6016 |
|
− |
|
F14 | −286.7 | −286.454 |
|
− |
|
|
|
F15 | 712.6463 | 679.5186 | 907.8008 |
|
|
943.7358 | 899.282 |
F16 | 394.5135 | 445.3347 | 645.2657 |
|
633.273 | 684.447 |
|
F17 | 456.3692 | 489.36 | 698.1601 |
|
725.3444 | 715.6857 |
|
F18 | 924.3655 | 976.0293 |
|
|
|
|
|
F19 | 927.3029 | 973.5324 |
|
|
|
|
|
F20 | 927.4663 | 971.5176 |
|
|
|
|
|
F21 | 1335.783 | 1319.538 | 1665.937 |
|
|
1657.629 | 1657.872 |
F22 | 1260.922 | 1312.237 | 1552.663 |
|
|
1569.841 | 1559.818 |
F23 | 1418.319 | 1374.563 | 1667.109 |
|
|
1659.605 | 1662.561 |
F24 | 1218.33 | 1210.63 | 1618.306 |
|
1614.015 |
|
1621.32 |
F25 | 1890.716 | 1944.179 |
|
|
|
|
|
One of the objectives in this paper is to investigate the optimization performance by using different chaotic systems in CE algorithm framework. Tables
We apply Wilcoxon sign-ranked test (
Figure
System output total number and its percentage in each interval of logistic map, tent map, Gaussian map, and Hénon map with attractor initiation after
Interval | Logistic | Tent | Gaussian | Hénon-attrac. | ||||
---|---|---|---|---|---|---|---|---|
|
20552 | 20.55% | 99952 | 99.95% | 7319 | 10.77% | 2193 | 4.52% |
|
9051 | 9.05% | 6 | 0.01% | 13830 | 20.36% | 3272 | 6.74% |
|
7396 | 7.40% | 9 | 0.01% | 10366 | 15.26% | 3263 | 6.72% |
|
6674 | 6.67% | 3 | 0.00% | 13236 | 19.48% | 7039 | 14.49% |
|
6353 | 6.35% | 7 | 0.01% | 23191 | 34.13% | 5410 | 11.14% |
|
6473 | 6.47% | 6 | 0.01% | 0 | 0.00% | 5505 | 11.34% |
|
6691 | 6.69% | 1 | 0.00% | 0 | 0.00% | 7601 | 15.65% |
|
7296 | 7.30% | 3 | 0.00% | 0 | 0.00% | 5606 | 11.54% |
|
8996 | 9.00% | 6 | 0.01% | 0 | 0.00% | 3402 | 7.01% |
|
20518 | 20.52% | 7 | 0.01% | 0 | 0.00% | 5272 | 10.86% |
There are some relationships between output distribution characteristic of chaotic system and optimization performance of CE. In logistic map, most of the system outputs cover the intervals of
We select two DE algorithms (DE/best/1/bin and DE/rand/1/bin) as two competitors to make a comparative evaluation with our proposed five chaotic evolution algorithms. The
We apply Friedman test on our proposed five CE algorithms and two DE algorithms to make an algorithm rank. Table
Algorithms’ rank with DE by Friedman test for F1–F25 of 10D and 30D, respectively. The abbreviation meanings are in Table
Function | DE-best | DE-rand | Logistic | Tent | Gauss | Hénon-rand | Hénon-attrac. |
---|---|---|---|---|---|---|---|
10-dimensional function | |||||||
F1 | 1.00 | 2.00 | 5.07 | 6.93 | 4.33 | 4.47 | 4.20 |
F2 | 1.03 | 1.97 | 5.17 | 6.87 | 3.73 | 4.73 | 4.50 |
F3 | 1.03 | 1.97 | 4.40 | 6.90 | 3.77 | 5.33 | 4.60 |
F4 | 1.70 | 1.30 | 5.30 | 6.60 | 4.03 | 4.50 | 4.57 |
F5 | 1.00 | 2.00 | 4.63 | 6.83 | 4.83 | 5.00 | 3.70 |
F6 | 1.00 | 2.00 | 5.20 | 6.77 | 4.50 | 5.03 | 3.50 |
F7 | 1.33 | 1.67 | 5.17 | 6.77 | 3.83 | 5.03 | 4.20 |
F8 | 1.07 | 2.23 | 4.30 | 6.70 | 4.43 | 4.93 | 4.33 |
F9 | 1.53 | 1.67 | 4.37 | 6.70 | 4.70 | 4.27 | 4.77 |
F10 | 1.33 | 2.17 | 5.07 | 6.83 | 4.13 | 4.50 | 3.97 |
F11 | 4.37 | 1.23 | 4.37 | 6.47 | 2.80 | 5.63 | 3.13 |
F12 | 1.13 | 2.00 | 3.87 | 6.83 | 5.63 | 5.10 | 3.43 |
F13 | 2.43 | 3.53 | 4.43 | 6.67 | 2.50 | 5.83 | 2.60 |
F14 | 5.17 | 2.80 | 3.50 | 6.70 | 2.47 | 4.30 | 3.07 |
F15 | 2.43 | 3.83 | 3.37 | 6.77 | 4.03 | 4.50 | 3.07 |
F16 | 1.43 | 2.60 | 4.53 | 6.90 | 4.13 | 5.10 | 3.30 |
F17 | 1.67 | 1.60 | 4.70 | 6.87 | 4.27 | 5.30 | 3.60 |
F18 | 4.90 | 1.27 | 3.83 | 5.38 | 3.83 | 4.95 | 3.83 |
F19 | 4.67 | 1.23 | 3.92 | 5.53 | 3.78 | 4.95 | 3.92 |
F20 | 4.70 | 1.50 | 3.87 | 5.40 | 3.77 | 4.90 | 3.87 |
F21 | 1.67 | 1.63 | 5.03 | 6.57 | 3.43 | 5.47 | 4.20 |
F22 | 2.17 | 1.10 | 4.97 | 6.97 | 4.37 | 4.37 | 4.07 |
F23 | 1.37 | 1.77 | 5.17 | 6.67 | 3.90 | 5.07 | 4.07 |
F24 | 2.07 | 1.20 | 5.43 | 6.17 | 4.60 | 4.73 | 3.80 |
F25 | 1.17 | 2.70 | 4.07 | 3.80 | 5.70 | 5.93 | 4.63 |
Ave. |
|
|
|
|
|
|
|
|
|||||||
30-dimensional function | |||||||
F1 | 1.00 | 2.00 | 5.20 | 6.93 | 3.10 | 5.77 | 4.00 |
F2 | 1.00 | 2.87 | 4.77 | 6.97 | 2.70 | 5.77 | 3.93 |
F3 | 1.00 | 2.50 | 4.63 | 6.93 | 2.97 | 5.80 | 4.17 |
F4 | 1.03 | 3.33 | 3.97 | 6.87 | 2.57 | 5.80 | 4.43 |
F5 | 1.00 | 2.03 | 4.03 | 6.90 | 4.33 | 5.67 | 4.03 |
F6 | 1.00 | 2.00 | 5.47 | 6.90 | 3.50 | 5.43 | 3.70 |
F7 | 1.00 | 2.00 | 4.63 | 6.93 | 3.60 | 5.33 | 4.50 |
F8 | 1.00 | 3.93 | 4.37 | 6.77 | 4.27 | 3.60 | 4.07 |
F9 | 1.00 | 2.10 | 5.37 | 6.60 | 3.73 | 4.73 | 4.47 |
F10 | 1.03 | 1.97 | 5.23 | 6.87 | 4.90 | 3.90 | 4.10 |
F11 | 2.77 | 5.73 | 3.30 | 6.20 | 1.90 | 5.77 | 2.33 |
F12 | 1.00 | 4.80 | 2.80 | 6.33 | 6.20 | 4.37 | 2.50 |
F13 | 2.60 | 4.60 | 4.13 | 5.83 | 1.60 | 6.57 | 2.67 |
F14 | 3.93 | 5.67 | 2.90 | 6.20 | 2.67 | 4.80 | 1.83 |
F15 | 2.03 | 1.47 | 4.67 | 6.70 | 3.20 | 5.40 | 4.53 |
F16 | 1.53 | 1.70 | 4.37 | 6.80 | 4.40 | 5.33 | 3.87 |
F17 | 1.43 | 1.70 | 4.40 | 6.73 | 5.03 | 4.80 | 3.90 |
F18 | 5.00 | 6.30 | 2.02 | 3.95 | 2.02 | 6.70 | 2.02 |
F19 | 5.20 | 6.23 | 2.02 | 3.95 | 2.02 | 6.57 | 2.02 |
F20 | 5.17 | 6.33 | 2.02 | 3.95 | 2.02 | 6.50 | 2.02 |
F21 | 1.70 | 1.30 | 5.23 | 6.87 | 3.47 | 4.70 | 4.73 |
F22 | 1.07 | 1.93 | 4.30 | 6.97 | 4.27 | 4.87 | 4.60 |
F23 | 1.63 | 1.37 | 5.27 | 6.63 | 3.40 | 4.83 | 4.87 |
F24 | 1.37 | 1.63 | 5.10 | 6.30 | 4.67 | 3.77 | 5.17 |
F25 | 2.53 | 6.37 | 2.80 | 2.07 | 4.97 | 6.63 | 2.63 |
Ave. |
|
|
|
|
|
|
|
Algorithms’ rank without DE by Friedman test for F1–F25 of 10D and 30D, respectively. The abbreviation meanings are in Table
Function | Logistic | Tent | Gauss | Hénon-rand | Hénon-attrac. |
---|---|---|---|---|---|
10-dimensional function | |||||
F1 | 3.07 | 4.93 | 2.33 | 2.47 | 2.20 |
F2 | 3.17 | 4.87 | 1.73 | 2.73 | 2.50 |
F3 | 2.40 | 4.90 | 1.77 | 3.33 | 2.60 |
F4 | 3.30 | 4.60 | 2.03 | 2.50 | 2.57 |
F5 | 2.63 | 4.83 | 2.83 | 3.00 | 1.70 |
F6 | 3.20 | 4.77 | 2.50 | 3.03 | 1.50 |
F7 | 3.17 | 4.77 | 1.83 | 3.03 | 2.20 |
F8 | 2.33 | 4.73 | 2.57 | 2.97 | 2.40 |
F9 | 2.40 | 4.70 | 2.77 | 2.37 | 2.77 |
F10 | 3.13 | 4.83 | 2.20 | 2.57 | 2.27 |
F11 | 2.90 | 4.73 | 1.53 | 3.97 | 1.87 |
F12 | 1.93 | 4.83 | 3.63 | 3.10 | 1.50 |
F13 | 3.00 | 4.70 | 1.50 | 4.03 | 1.77 |
F14 | 2.67 | 5.00 | 1.73 | 3.27 | 2.33 |
F15 | 2.27 | 4.90 | 2.77 | 3.13 | 1.93 |
F16 | 2.67 | 4.90 | 2.47 | 3.23 | 1.73 |
F17 | 2.73 | 4.87 | 2.33 | 3.30 | 1.77 |
F18 | 2.47 | 4.02 | 2.47 | 3.58 | 2.47 |
F19 | 2.48 | 4.10 | 2.42 | 3.52 | 2.48 |
F20 | 2.50 | 4.03 | 2.43 | 3.53 | 2.50 |
F21 | 3.07 | 4.57 | 1.53 | 3.57 | 2.27 |
F22 | 3.00 | 4.97 | 2.43 | 2.43 | 2.17 |
F23 | 3.17 | 4.67 | 1.97 | 3.07 | 2.13 |
F24 | 3.43 | 4.17 | 2.70 | 2.77 | 1.93 |
F25 | 2.27 | 2.10 | 3.73 | 4.03 | 2.87 |
Ave. |
|
|
|
|
|
|
|||||
30-dimensional function | |||||
F1 | 3.20 | 4.93 | 1.10 | 3.77 | 2.00 |
F2 | 2.87 | 4.97 | 1.20 | 3.83 | 2.13 |
F3 | 2.67 | 4.93 | 1.33 | 3.80 | 2.27 |
F4 | 2.37 | 4.87 | 1.17 | 3.87 | 2.73 |
F5 | 2.07 | 4.90 | 2.33 | 3.67 | 2.03 |
F6 | 3.47 | 4.90 | 1.50 | 3.43 | 1.70 |
F7 | 2.63 | 4.93 | 1.60 | 3.33 | 2.50 |
F8 | 2.73 | 4.80 | 2.70 | 2.17 | 2.60 |
F9 | 3.37 | 4.60 | 1.73 | 2.77 | 2.53 |
F10 | 3.23 | 4.87 | 2.90 | 1.90 | 2.10 |
F11 | 2.70 | 4.67 | 1.50 | 4.33 | 1.80 |
F12 | 1.73 | 4.47 | 4.33 | 3.03 | 1.43 |
F13 | 3.03 | 4.23 | 1.07 | 4.73 | 1.93 |
F14 | 2.50 | 4.77 | 2.23 | 3.87 | 1.63 |
F15 | 2.73 | 4.70 | 1.50 | 3.43 | 2.63 |
F16 | 2.43 | 4.80 | 2.43 | 3.33 | 2.00 |
F17 | 2.43 | 4.77 | 3.03 | 2.80 | 1.97 |
F18 | 2.02 | 3.95 | 2.02 | 5.00 | 2.02 |
F19 | 2.02 | 3.95 | 2.02 | 5.00 | 2.02 |
F20 | 2.02 | 3.95 | 2.02 | 5.00 | 2.02 |
F21 | 3.23 | 4.87 | 1.47 | 2.70 | 2.73 |
F22 | 2.30 | 4.97 | 2.27 | 2.87 | 2.60 |
F23 | 3.27 | 4.63 | 1.40 | 2.83 | 2.87 |
F24 | 3.10 | 4.30 | 2.67 | 1.77 | 3.17 |
F25 | 2.27 | 1.60 | 4.00 | 5.00 | 2.13 |
Ave. |
|
|
|
|
|
We apply Bonferroni-Dunn’s tests on the results of Friedman test to evaluate our hypothesis whether these algorithms have a significant difference. The tests are applied on a group with DE to compare CE and DE and on a group without DE to compare the CE algorithms with different chaotic systems. Both tests are for the 10D and 30D benchmark tasks. The evaluation metrics of critical difference (
In our evaluation,
We propose using CE algorithm framework to implement a new IEC algorithm, interactive chaotic evolution (ICE). It is one of originalities and contributions in this paper. We use a Gaussian mixture model (equation (
Means of Gaussian mixture models with 3D, 5D, 7D, 10D, 13D, 15D, 17D, and 20 D. The abbreviation meanings are in Table
DE-best | DE-rand | Logistic | Tent | Gauss | Hénon-rand | Hénon-attrac. | |
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3D | −5.32 | −5.05 |
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5D | −2.47 | −2.23 |
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7D | −1.50 | −1.00 |
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10D | −0.34 | −0.17 |
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13D | −5.20 | −5.17 |
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15D | −2.59 | −2.13 |
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17D | −1.32 | −0.87 |
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20D | −0.35 | −0.22 |
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Interactive chaotic evolution convergent curves of 5D, 10D, 15D, and 20D Gaussian mixture model. All the ICE algorithms present fast convergent speed, especially in 10D and 20D tasks. IDE/best can obtain better final results than ICE, but it needs more fitness evaluation (i.e., more human subjective evaluation in IEC application) to find the best base vector. From the metric of fitness evaluation time, ICE is significantly better than IDE/best when these two algorithms call the same fitness function time.
We analyse the optimization results of CE algorithms at the 1000th generation in Section
One of the characteristics in IEC is that it is required to run with less generation and less population size to solve the user fatigue problem. IEC user compares each individual with others kept in their memory and the mental load and fatigue increase. Reference [
In the running process of CE algorithm framework (Section
Evaluation results indicate that the convergent speeds of CE algorithms become down in some benchmark functions, especially in 15D tasks (Figure
There are multiple implementations of the CE algorithm framework. We initially have investigated their optimization performance and output distribution of chaotic systems. Another opportunity for enhancing optimization performance of CE is to fuse multiple chaotic systems in one CE algorithm. It is as well a promising research topic for further study. There are three methods to implement this algorithm framework. First is to apply one chaotic system for a certain generation and change another for the following generations by observing the algorithm convergent speed or other evaluation metrics. Second is to fuse CE with multiple chaotic systems in individual level; that is, some of individuals are searching by the law of one chaotic system and some of the others by that of other chaotic systems. Third is to apply different chaotic systems on different dimensions of one individual. The landscape or search situation is different from all dimensions, and this must be matched by different dimensional searches with different chaotic systems. We can design a CE algorithm with multiple chaotic systems by obtaining the fitness landscape information in the whole or the particle dimensions. Some methods of analysing and approximating the fitness landscape information can be found in [
We develop an algorithmic notation system for a better explanation and further development of CE algorithms. The actual search function of chaotic algorithms is implemented by a chaotic system, in which there are some parameters. This is one element in chaotic evolution. Another parameter of chaotic evolution is the crossover rate.
We abbreviate chaotic evolution as CE and use the notation format
In this paper, we develop a chaotic optimization algorithm that is theoretically supported by the fundamental of chaos theory. We introduce four chaotic systems in a well-designed CE algorithm framework to implement several CE algorithms, that is, CE-logistic, CE-tent, CE-Gauss, CE-Hénon-rand, and CE-Hénon-attrac. We analyse the optimization performances of these developed algorithms. A comparative evaluation is conducted by applying the Wilcoxon sign-ranked test and the Friedman test with two DE algorithms. We propose a new IEC algorithm, that is, ICE that has a paired comparison mechanism in its search scheme. A series of topics on optimization performances of chaotic evolution algorithms, comparison with DE algorithms, algorithms rank, interactive chaotic evolution, and fusion of these algorithms for enhancing performance are analysed and discussed.
In this paper, we do not pursue obtaining the results of the best winner algorithm but analyse and discuss the algorithm optimization mechanism of CE and the philosophy behind it. In the modern scientific world, there are two primary philosophies and methodologies to describe and study nature world from the determinism and probability viewpoint. In the optimization field, there are corresponding deterministic and stochastic optimization algorithms that are supported by these two methodologies. EC is one of stochastic optimization algorithms. However, after the discoveries of chaotic phenomena and systems, chaos becomes the third methodology to study the nature world. Chaotic optimization algorithm should therefore be researched and developed in optimization field. Chaotic evolution is one of the implementations, although it has many disadvantages in its search scheme, such as searching without considering fitness landscape, exploration, and exploitation depending on chaotic system rather than adaptive mechanism. In the theoretical analysis of CE, we will analyse concrete CE algorithm with Markov chain [
The author declares that there is no conflict of interests regarding the publication of this paper.