Adaptive Cross-Generation Differential Evolution (ACGDE) is a recently-introduced algorithm for solving multi-objective problems with remarkable performance compared to other evolutionary algorithms (EAs). However, its convergence and diversity are not satisfactory compared with the latest algorithms. In order to adapt to the current environment, ACGDE requires improvements in many aspects, such as its initialization and mutant operator. In this paper, an enhanced version is proposed, namely SIACGDE. It incorporates a strengthened initialization strategy and optimized parameters in contrast to its predecessor. These improvements make the direction of cross-generation mutation more clearly and the ability of searching more efficiently. The experiments show that the new algorithm has better diversity and improves convergence to a certain extent. At the same time, SIACGDE outperforms other state-of-the-art algorithms on four metrics of 24 test problems.
Differential Evolution, conceptualized by Storn and Price [
DE has gained immense popularity for its simple structure and the ability of dealing with complex problems. However, the original DE is too simple to satisfy our current environment used in the actual problem [
However, few variants of the DE have considered the relationship between nondominated the sorting genetic algorithm II (NSGAII) [
On the other hand, in order to enhance the difference between populations, we need to optimize the population in the early stage. Similar to most algorithms, the effect of initialization on DE performance is ignored. Specifically, ACGDE tries to estimate the promising searching directions by analyzing the differences between consecutive generations so that the mutation strategy does not work for the first few generations. And the parameters describing the relationship between individuals are not optimal, though the parameters of the algorithm are adaptive. These problems greatly affect the correctness of the solution.
Opposition-based learning (OBL), introduced by Tizhoosh [
As an improved version, the ACGDE algorithm inherits the experiences of previous algorithms to improve DE and adds adaptive and cross-generational strategy. Two important mutation factors were considered simultaneously, such as population-based cross-generation (PCG) and neighborhood-based cross-generation (NCG), to expand the range of selected individuals while using cross-generational crossover to realize previous generation information. At the same time, the scaling factor F and crossover probability Cr of DE were adjusted adaptively to ensure that the algorithm has a suitable evolution rate in different periods. The specific steps of ACGDE will be discussed in detail in
Similar to most algorithms, the effect of initialization on DE performance is ignored. Specifically, ACGDE tries to estimate the promising searching directions by analyzing the differences between consecutive generations so that the mutation strategy does not work for the first few generations. And the parameters describing the relationship between individuals are not optimal, though the parameters of the algorithm are adaptive. These problems greatly affect the correctness of the solution.
In view of the nature of ACGDE, the solutions get increasingly better, while the number of evolving generations grows. The movement of solutions across generations may, therefore, provide the true PF for the next generation of evolution. However, this evolutionary strategy does not work for the early generations.
To address the above issues, we present Strengthened Initialization of Adaptive Cross-Generation Differential Evolution (SIACGDE) in attempt to eliminate the potential weaknesses of ACGDE, which lacks an obviously Pareto dominant relationship between early generations. In addition, the parameters are optimized in order to design a powerful and up-to-date version of DE. In SIACGDE, there are obviously individual differences between the two generations, that is, the offspring individuals better than the parent individuals. In this work, initial stage is divided into two parts. First, opposition-based computation learning is applied to the stochastic initial phase. Second, population-based cross-generation (PCG) and neighborhood-based cross-generation (NCG), proposed by ACGDE, are screened out in early generations. Then, we optimize the parameter neighbor size in the original ACGDE algorithm. Experiments show that parameter optimization is necessary in the paper of last section. And the results indicate that SIACGDE significantly outperforms its predecessor in terms of inverted generational distance (IGD) [
This paper is further organized as follows:
In order to explain SIACGDE more deeply, we introduce the three algorithms mentioned in SIACGDE.
Differential Evolution (DE) was originally an excellent algorithm for single-object numerical optimization, inspired by the simulated annealing algorithm. DE is based on the principle of natural selection to make choices, like other evolutionary algorithms, using three unique operations (mutation, crossover, and selection) to guide the individuals to find promising solutions. Research has shown that DE show dominance in multi-objective problems [
where the scaling factor
where
ACGDE has been improved in many aspects on the basis of DE, such as its mutant operators, which are subsequently discussed in detail.
During initialization,
where
According to the ranking of individual distances around individuals,
where
The definition of PCG operation, which utilizes information from two successive generations but from a wider range than NCG uses in the same generation, is as follows:
where
In light of the concept of Yin and Yang originating from ancient Chinese philosophy [
In this section, an enhanced version of the ACGDE algorithm with strengthened initialization is proposed. The detailed initialization and the whole process of SIACGDE are introduced in the following subsections.
As mentioned above, OBL can accelerate the evolutionary process without any prior knowledge. This advantage makes up for the slow progress of the ACGDE algorithm in the early generations. Therefore, two main steps are distinguishable for the original ACGDE, including initialization and the different mutant operator in the first five percent generations. Like most algorithms utilizing OBL for optimization, initialization in this algorithm is divided into four steps:
1) A population
2) According to
3) The candidate’s fitness is measured, which is performed for all
4) The first
The corresponding pseudocode is given in
When the initialization provides better individuals that are closer to the Pareto frontier, we can iterate over them. Since ACGDE also implements nondominated the sorting genetic algorithm II (NSGAII) [
Although we have optimized the parameters of the algorithm at the same time, the experiments in the later section show that the strengthened initialization is the part with the biggest improvement. Therefore, the optimization of initialization is the most important step for SIACGDE.
Here, the paper will focus on how the strengthened initialization and ACGDE mutant operator interact with each other. With better initial conditions, the cross-generation mutant operators, NCG and PCG, begin to play a stronger role than its predecessor in SIACGDE algorithm. The flow of SIACGDE is presented in
At the beginning, the individuals will be iterated rapidly in previous generations by NSGAII, and the obtained offspring can be used in SIACGDE. According to our experiment, it can be concluded that the iteration speed of NSGAII is obviously faster than the original algorithm in the initial stage. Furthermore, the addition of OBL makes the first few generations of individuals significantly better than the random initial individuals. In the process of evolution, GA mutates through two random parents, which does not depend on the individual differences between the two parents. This is actually an advantage in the early stage of evolution, compared with mutation NCG and PCG. If the previous operators are still used at early stage of evolution, the characteristics of these two operators will not play out. After several generations, the connection between populations is slowly established. At this time, this connection can not be used effectively by GA operator. Therefore, each generation is better than the previous generation, which can exactly match the cross-generation mechanism. This step is mainly for the two mutation NCG and PCG to obtain more obvious difference between generations.
NCG is a mutation operator in which all mutated individuals are selected from the range of the
Then, the process is repeated until the termination condition is reached.
In this section, we discuss the performance of SIACGDE in the experiments. To fairly show performance comparisons, the same parameters are used in both algorithms, except for the change in the size of the
To date, there are a number of metrics to quantify the optimality of a solution set and the performance of multi-objective evolutionary algorithms (MOEAs) in multi-objective optimization (MOO). Each metric has its own set of performance criteria, which are divided into four categories: Capacity, Convergence, Diversity, and Convergence-Diversity. In this study, three metrics, including inverted generational distance (IGD) [
To evaluate the competitiveness of the algorithm, we selected six state-of-the-art algorithms in addition to ACGDE for comparison. These include SIACGDE, ACGDE, and NSGAII, which were introduced earlier, and two common improved DE algorithms, GDE3 [
And the results were 30 independently run tests on each test problem for each MOEA algorithm. And the termination condition of eight algorithms is reaching maximum evolution of 10,000.
The values IGD of these seven algorithms for test functions can be shown in
Problem | GDE3 | MOEADDE | NSGAII | LCSA | LMEA | SPEA2SDE | ACGDENSGAII | SIACGDE | ||
---|---|---|---|---|---|---|---|---|---|---|
VNT1 | 3 | 1.4863e −1 (5.82e − 3) − | 1.8309e −1 (3.72e − 3) − | 1.5733e −1 (8.92e − 3) − | 1.6604e −1 (8.76e − 3) − | 1.6719e −1 (1.04e − 2) − | 1.4709e −1 (9.11e − 3) − | 1.6256e −1 (9.78e − 3) − | ||
VNT2 | 3 | 1.9500e − 2 (1.33e − 3) − | 4.5856e − 2 (2.02e − 3) − | 2.1276e − 2 (1.35e − 3) − | 4.6614e − 2 (1.73e − 2) − | 3.1701e − 2 (3.18e − 3) − | 3.3974e − 2 (4.82e − 3) − | 2.2588e − 2 (1.53e − 3) − | ||
VNT3 | 3 | 3.5789e − 2 (1.62e − 3) − | 7.4637e − 1 (2.06e − 1) − | 4.1020e − 2 (3.03e − 3) − | 4.0975e − 1 (2.29e − 1) − | 4.7143e − 2 (2.86e − 3) − | 3.1026e − 1 (3.91e − 1) − | 4.7454e − 2 (2.75e − 3) − | ||
VNT4 | 3 | 1.6878e − 1 (1.17e − 2) + | 1.7716e + 0 (1.83e − 2) − | 2.4646e − 1 (6.33e − 2) + | 1.1262e + 0 (5.87e − 2) − | 1.2892e + 0 (9.71e − 2) − | 1.6922e − 1 (1.36e − 2) + | 1.0586e + 0 (5.48e − 2) | ||
WFG1 | 3 | 1.7454e + 0 (3.37e − 2) − | 1.5538e + 0 (2.89e − 2) = | 9.7203e − 1 (1.01e − 1) + | 1.4164e + 0 (6.24e − 2) + | 2.1652e + 0 (8.51e − 2) − | 1.5774e + 0 (7.65e − 2) = | 1.5366e + 0 (8.60e − 2) | ||
WFG2 | 3 | 2.8458e − 1 (1.34e − 2) − | 3.5741e − 1 (2.17e − 2) − | 2.2539e − 1 (1.08e − 2) − | 1.9848e − 1 (1.17e − 2) = | 3.3906e − 1 (6.72e − 2) − | 2.0770e − 1 (1.12e − 2) − | 2.4585e − 1 (1.47e − 2) − | ||
WFG3 | 3 | 3.6291e − 1 (3.04e − 2) − | 2.8588e − 1 (4.29e − 2) − | 1.2993e − 1 (1.48e − 2) + | 2.2058e − 1 (2.62e − 2) = | 7.4598e − 2 (7.14e − 3) + | 2.2440e − 1 (3.38e − 2) = | 2.2401e − 1 (2.00e − 2) | ||
WFG4 | 3 | 3.5795e − 1 (1.09e − 2) − | 3.9845e − 1 (1.39e − 2) − | 2.8143e − 1 (9.94e − 3) − | 2.4739e − 1 (3.71e − 3) = | 2.8696e − 1 (3.73e − 2) − | 3.2620e − 1 (1.52e − 2) − | 3.1193e − 1 (1.03e − 2) − | ||
WFG5 | 3 | 2.7651e − 1 (1.28e − 2) − | 3.3628e − 1 (3.59e − 3) − | 2.8818e − 1 (1.07e − 2) − | 2.4460e − 1 (2.26e − 3) − | 2.4425e − 1 (1.29e − 2) − | 3.2974e − 1 (1.76e − 2) − | 2.7462e − 1 (9.73e − 3) − | ||
WFG6 | 3 | 3.8259e − 1 (1.95e − 2) − | 4.3594e − 1 (2.22e − 2) − | 3.3000e − 1 (1.78e − 2) = | 3.3775e − 1 (2.70e − 2) = | 3.5611e − 1 (1.91e − 2) − | 3.9231e − 1 (2.16e − 2) − | 3.3005e − 1 (1.69e − 2) | ||
WFG7 | 3 | 3.7149e − 1 (1.19e − 2) − | 3.8492e − 1 (1.49e − 2) − | 2.8177e − 1 (1.14e − 2) − | 2.5822e − 1 (8.61e − 3) − | 3.2507e − 1 (1.58e − 2) − | 3.3227e − 1 (1.89e − 2) − | 2.5074e − 1 (7.07e − 3) | ||
WFG8 | 3 | 4.8305e − 1 (1.67e − 2) − | 4.8505e − 1 (2.60e − 2) − | 3.7766e − 1 (1.11e − 2) − | 3.5792e − 1 (1.07e − 2) − | 3.5559e − 1 (1.94e − 2) − | 3.6489e − 1 (1.13e − 2) − | 4.1494e − 1 (1.28e − 2) − | ||
WFG9 | 3 | 3.9682e − 1 (2.39e − 2) − | 3.4876e − 1 (1.05e − 2) − | 2.8313e − 1 (1.36e − 2) − | 2.7494e − 1 (2.94e − 2) = | 3.0684e − 1 (1.85e − 2) − | 3.8478e − 1 (2.16e − 2) − | 2.7643e − 1 (4.90e − 2) | ||
DTLZ1 | 3 | 4.7230e + 0 (2.37e + 0) − | 8.0096e − 1 (1.00e + 0) + | 2.8234e − 1 (2.80e − 1) + | 1.1343e + 0 (5.72e − 1) + | 1.7428e − 1 (1.54e − 1) + | 1.4780e + 0 (1.08e + 0) = | 1.6547e + 0 (8.79e − 1) | ||
DTLZ2 | 3 | 8.2221e − 2 (2.87e − 3) − | 7.7747e − 2 (1.08e − 3) − | 6.9602e − 2 (2.05e − 3) − | 5.8403e − 2 (1.05e − 3) − | 5.7224e − 2 (1.89e − 3) − | 7.6377e − 2 (2.11e − 3) − | 7.0173e − 2 (2.59e − 3) − | ||
DTLZ3 | 3 | 2.8245e + 1 (3.01e + 1) = | 2.4663e + 1 (2.68e + 1) = | 8.0783e + 0 (3.58e + 0) + | 3.5322e + 1 (1.04e + 1) − | 8.3273e + 0 (4.94e + 0) + | 3.9442e + 1 (1.45e + 1) − | 2.8848e + 1 (1.12e + 1) | ||
DTLZ4 | 3 | 1.0927e − 1 (1.96e − 2) − | 1.7462e − 1 (7.13e − 2) − | 8.4867e − 2 (8.63e − 2) − | 5.9820e − 2 (1.39e − 3) − | 3.5164e − 1 (1.07e − 1) − | 1.3918e − 1 (1.61e − 1) − | 7.0611e − 2 (2.34e − 3) − | ||
DTLZ5 | 3 | 9.0909e − 3 (8.54e − 4) − | 1.4360e − 2 (2.83e − 4) − | 6.0132e − 3 (2.35e − 4) + | 1.3822e − 2 (1.57e − 3) − | 1.0016e − 2 (1.27e − 3) − | 6.8304e − 3 (4.47e − 4) = | 7.0664e − 3 (6.95e − 4) | ||
DTLZ6 | 3 | 1.0593e + 0 (6.49e − 1) − | 1.4073e − 2 (1.95e − 4) − | 6.0103e − 3 (3.89e − 4) − | 1.9972e − 2 (1.84e − 3) − | 7.9189e − 1 (3.67e − 1) − | 1.0060e − 2 (2.18e − 3) − | 6.7055e − 3 (4.11e − 4) − | ||
ZDT1 | 2 | 8.5193e − 1 (1.12e − 1) − | 4.3540e − 1 (1.17e − 1) − | 1.2231e − 2 (1.98e − 3) − | 4.1716e − 3 (3.88e − 4) − | 2.2423e + 0 (2.06e − 1) − | 1.0466e − 2 (1.87e − 3) − | 6.2812e − 3 (3.47e − 4) − | ||
ZDT2 | 2 | 1.8384e + 0 (2.12e − 1) − | 7.1837e − 1 (1.39e − 1) − | 2.8127e − 2 (5.00e − 2) − | 1.1065e − 2 (1.22e − 2) − | 3.5480e + 0 (2.61e − 1) − | 1.6330e − 2 (4.44e − 3) − | 6.3051e − 3 (3.80e − 4) − | ||
ZDT3 | 2 | 9.1497e − 1 (1.28e − 1) − | 5.0854e − 1 (9.10e − 2) − | 1.3545e − 2 (9.63e − 3) − | 2.1206e − 2 (1.49e − 2) − | 1.9628e + 0 (1.96e − 1) − | 9.9197e − 3 (4.11e − 3) − | 7.1517e − 3 (2.81e − 4) − | ||
ZDT4 | 2 | 3.2734e + 1 (5.52e + 0) − | 3.2828e + 0 (1.42e + 0) − | 2.7115e − 1 (1.22e − 1) + | 2.7563e + 0 (1.27e + 0) − | 5.4520e − 1 (1.82e − 1) + | 2.1177e + 0 (1.08e + 0) = | 1.7511e + 0 (7.61e − 1) | ||
ZDT6 | 2 | 1.4138e − 1 (2.13e − 1) − | 7.8136e − 2 (1.52e − 1) − | 5.8193e − 2 (3.01e − 2) − | 3.5168e − 3 (4.41e − 4) − | 4.8868e − 1 (1.99e − 1) − | 6.2477e − 2 (4.17e − 2) − | 5.2366e − 3 (2.19e − 4) − | ||
+/-/= | 0/23/1 | 1/21/2 | 7/16/1 | 3/16/5 | 7/16/1 | 5/19/0 | 1/18/5 |
In order to show the performance of the algorithms in more angles, two metrics that are rarely considered in most experiments are also added in this experiment. As shown in
The bold font represents the best performance on this problem. Where the symbols ‘+’, ‘ −’ and ‘=’ indicate the results of compared algorithm is significantly better, significantly worse, and statistically similar to that obtained by SIACGDED at a 0.05 level by the Wilcoxon’s rank sum test, respectively.
Among the metrics that show convergence-diversity performance, the most effective and popular is HV. And we showed the performance comparison of eight algorithms in
In addition, we compared the runtime of eight algorithms on VNT1
Problem | M | GDE3 | MOEADDE | NSGAII | LCSA | LMEA | SPEA2SDE | ACGDENSGAII | SIACGDE | |
---|---|---|---|---|---|---|---|---|---|---|
VNT1 | 3 | 3.3556e − 1 (4.11e − 4) − | 3.3358e − 1 (2.34e − 4) − | 3.3430e − 1 (5.93e − 4) − | 3.3308e − 1 (1.19e − 3) − | 3.3291e − 1 (1.20e − 3) − | 3.3297e − 1 (1.44e − 3) − | 3.3373e − 1 (7.81e − 4) − | ||
VNT2 | 3 | 3.3216e − 1 (2.60e − 4) − | 3.3090e − 1 (9.65e − 5) − | 3.3173e − 1 (2.84e − 4) − | 3.3336e − 1 (2.58e − 3) = | 3.3071e − 1 (4.75e − 4) − | 3.3136e − 1 (3.12e − 4) − | 3.3302e − 1 (9.14e − 5) | ||
VNT4 | 3 | 2.0967e − 1 (1.84e − 3) − | 2.5748e − 1 (4.57e − 4) + | 2.5650e − 1 (5.98e − 4) + | 2.4082e − 1 (3.61e − 3) = | 2.3497e − 1 (4.64e − 3) − | 2.5726e − 1 (4.34e − 4) + | 2.4129e − 1 (3.50e − 3) | ||
DTLZ1 | 3 | 0.0000e + 0 (0.00e + 0) − | 2.5207e − 1 (3.42e − 1) + | 3.7401e − 1 (3.35e − 1) + | 2.5425e − 3 (8.73e − 3) = | 4.8542e − 1 (3.24e − 1) + | 5.2865e − 2 (1.86e − 1) = | 1.9110e − 2 (6.68e − 2) | ||
DTLZ2 | 3 | 4.8579e − 1 (5.74e − 3) − | 5.1497e − 1 (4.79e − 3) − | 5.2977e − 1 (4.26e − 3) − | 5.4412e − 1 (2.69e − 3) − | 5.4192e − 1 (6.12e − 3) − | 5.2004e − 1 (5.25e − 3) − | 5.4712e − 1 (2.44e − 3) | ||
DTLZ3 | 3 | 2.2170e − 2 (8.54e − 2) = | 0.0000e + 0 (0.00e + 0) = | 0.0000e + 0 (0.00e + 0) = | 0.0000e + 0 (0.00e + 0) = | 0.0000e + 0 (0.00e + 0) = | 0.0000e + 0 (0.00e + 0) = | 0.0000e + 0 (0.00e + 0) | ||
DTLZ4 | 3 | 4.9464e − 1 (1.06e − 2) − | 5.0047e − 1 (2.48e − 2) − | 5.2509e − 1 (3.46e − 2) − | 5.4268e − 1 (3.60e − 3) − | 3.7115e − 1 (6.59e − 2) − | 5.3191e − 1 (7.38e − 2) − | 5.1937e − 1 (5.39e − 3) − | ||
DTLZ6 | 3 | 3.4264e − 2 (7.16e − 2) − | 1.9504e − 1 (1.05e − 4) − | 1.9938e − 1 (1.56e − 4) − | 1.9226e − 1 (1.26e − 3) − | 6.5324e − 3 (3.49e − 2) − | 1.9915e − 1 (2.17e − 3) − | 1.9901e − 1 (2.26e − 4) − | ||
ZDT1 | 2 | 2.3445e − 2 (2.02e − 2) − | 2.5080e − 1 (9.70e − 2) − | 7.0661e − 1 (2.73e − 3) − | 7.1991e − 1 (4.07e − 4) − | 0.0000e + 0 (0.00e + 0) − | 7.0881e − 1 (2.63e − 3) − | 7.1799e − 1 (3.67e − 4) − | ||
ZDT2 | 2 | 0.0000e + 0 (0.00e + 0) − | 4.7134e − 3 (1.19e − 2) − | 4.1355e − 1 (4.54e − 2) − | 4.3691e − 1 (1.34e − 2) − | 0.0000e + 0 (0.00e + 0) − | 4.2470e − 1 (7.43e − 3) − | 4.4272e − 1 (3.37e − 4) − | ||
ZDT3 | 2 | 3.5922e − 2 (3.78e − 2) − | 2.4763e − 1 (8.18e − 2) − | 6.0197e − 1 (2.73e − 2) + | 0.0000e + 0 (0.00e + 0) − | 5.9310e − 1 (6.41e − 3) − | 5.9907e − 1 (1.44e − 4) − | 5.9979e − 1 (2.38e − 4) | ||
ZDT4 | 2 | 0.0000e + 0 (0.00e + 0) − | 0.0000e + 0 (0.00e + 0) − | 4.7546e − 1 (1.12e − 1) + | 0.0000e + 0 (0.00e + 0) − | 1.6205e − 1 (1.03e − 1) + | 1.5596e − 2 (5.06e − 2) = | 1.1261e − 2 (4.03e − 2) | ||
ZDT6 | 2 | 3.2134e − 1 (1.01e − 1) − | 3.3490e − 1 (9.45e − 2) − | 3.1506e − 1 (3.70e − 2) − | 3.8824e − 1 (5.93e − 4) − | 5.6432e − 2 (8.44e − 2) − | 3.1161e − 1 (4.73e − 2) − | 3.8678e − 1 (2.26e − 4) − | ||
+/-/= | 1/11/1 | 2/11/0 | 4/8/1 | 2/8/3 | 2/9/2 | 4/8/1 | 1/9/3 |
In this experiment, the original algorithm ACGDE is based on NSGAII framework to select offspring. By comparing ACGDE with SIACGDE, we can see that the new algorithm has been greatly improved in terms of both diversity and finding solutions due to the difference between successive generations generated in the initial stage. In the first few generations of the original algorithm, the use of cross-generation information to make mutations could not find an effective way forward. At this point, the original mutation operator can be replaced by genetic operator to make better use of excellent individuals for searching. At the same time, opposition-based learning is applied to provide more excellent individuals for genetic operator and, thus, differences between individuals when the algorithm produces cross-generational replacement.
In this subsection, we describe the experiments that were performed to verify the effectiveness of the cross-generation mutation proposed in SIACGDE. That is to say, we used the adjacent individuals of the parent and to select offspring. The validity of this strategy must be carried out under the premise that there is an obvious evolution in the two generations, so we suspended the experiment in the middle of evolution to observe. When SIACGDE ran the process of ZDT1 solution, we randomly selected two generations of experimental data. According to the results shown in
From the above experiments, we can see that the NCG mutation factor plays a key role in finding the optimal solution, and the size of the
In the last subsection of the experiment, we tested the GA factor that has the greatest impact on the original ACGDE algorithm, the GA replaces the initial stage of mutation. The purpose of this experiment is to find how many generations in initial stage of GA will maximize the algorithm convergence of performance. Before the experiment, for the convenience of comparison, we named different GA proportions of SIACGDE: When the SIACGDE just used OBL in initialization, this model is SIACGDEI. When the proportion of GA in whole algorithm is 10%, this model is SIACGDEII. When the proportion of GA in whole algorithm is 15%, this model is SIACGDEIII. The SIACGDE algorithm, used in this paper, the proportion of GA in whole algorithm is 5%.
This paper describes the improvement of the ACGDE algorithm by introducing the GA mutant operator in the initial stage, which corrects for the lack of search ability. SIACGDE specifically developed a new initialization method for two cross-generation mutations, NCG and PCG, so that the new alternative individuals are more suitable for the two mutations. Furthermore, the OBL mechanism is added to improve the efficiency of evolution. Experimental results demonstrate the significant improvements of two adjusted versions of ACGDE and that SIACGDE is obviously superior to the other six state-of-the-art algorithms. However, our proposed algorithm has its limitations. Firstly, when the real Pareto front is not continuous, the solving ability is reduced, which is also caused by the characteristics of cross-generation mutation. Secondly, the algorithm has no ability to deal with the problem of constraints.
In the future, the first plan is to find a method to deal with constraints matching the algorithm, so as to solve the real-world problems [
The authors thank the Ocean University of China for their support in this work.
Problem | GDE3 | MOEADDE | NSGAII | LCSA | LMEA | SPEA2SDE | ACGDENSGAII | SIACGDE | ||
---|---|---|---|---|---|---|---|---|---|---|
VNT1 | 3 | 3.8583e − 1 (3.05e − 2) − | 6.9251e − 1 (2.73e − 2) − | 4.8022e − 1 (4.93e − 2) − | 6.4162e − 1 (6.97e − 2) − | 4.5108e − 1 (4.25e − 2) − | 4.5857e − 1 (3.65e − 2) − | 5.6532e − 1 (5.97e − 2) − | ||
VNT2 | 3 | 3.4594e − 1 (3.44e − 2) − | 1.2971e + 0 (1.77e − 2) − | 5.4477e − 1 (5.32e − 2) − | 9.7740e − 1 (7.44e − 2) − | 9.6290e − 1 (6.95e − 2) − | 8.9665e − 1 (5.92e − 2) − | 8.2969e − 1 (8.36e − 2) − | ||
VNT3 | 3 | 6.5924e − 1 (2.59e − 2) − | 1.1593e + 0 (1.06e − 1) − | 7.6379e − 1 (3.42e − 2) − | 1.2834e + 0 (1.10e − 1) − | 8.8243e − 1 (3.08e − 2) − | 1.1198e + 0 (1.10e − 1) − | 1.0612e + 0 (8.30e − 2) − | ||
VNT4 | 3 | 4.6556e − 1 (4.45e − 2) − | 9.8453e − 1 (1.79e − 2) − | 6.0677e − 1 (4.61e − 2) − | 6.9888e − 1 (7.46e − 2) − | 7.1814e − 1 (3.19e − 2) − | 6.9797e − 1 (1.92e − 2) − | 8.5081e − 1 (6.69e − 2) − | ||
WFG1 | 3 | 9.5443e − 1 (2.12e − 2) = | 8.6690e − 1 (5.53e − 2) + | 7.0383e − 1 (4.52e − 2) + | 8.1751e − 1 (7.00e − 2) + | 1.3197e + 0 (2.08e − 1) − | 9.8055e − 1 (9.02e − 2) = | 9.7618e − 1 (1.68e − 1) | ||
WFG2 | 3 | 4.9531e − 1 (3.78e − 2) − | 9.3141e − 1 (8.56e − 2) − | 5.2113e − 1 (5.85e − 2) − | 4.5339e − 1 (5.40e − 2) − | 7.7922e − 1 (1.06e − 1) − | 6.1243e − 1 (4.65e − 2) − | 5.5433e − 1 (4.68e − 2) − | ||
WFG3 | 3 | 4.9286e − 1 (3.95e − 2) − | 6.7378e − 1 (6.09e − 2) − | 5.8718e − 1 (5.26e − 2) − | 6.9372e − 1 (5.80e − 2) − | 3.9573e − 1 (3.99e − 2) − | 2.7274e − 1 (3.17e − 2) − | 5.9517e − 1 (5.07e − 2) − | ||
WFG4 | 3 | 4.2163e − 1 (4.03e − 2) − | 8.2134e − 1 (1.10e − 1) − | 4.4712e − 1 (3.00e − 2) − | 3.3669e − 1 (2.28e − 2) − | 3.4790e − 1 (2.97e − 2) − | 3.7447e − 1 (2.52e − 2) − | 4.3578e − 1 (3.80e − 2) − | ||
WFG5 | 3 | 3.9367e − 1 (3.80e − 2) − | 7.8173e − 1 (4.60e − 2) − | 5.0027e − 1 (4.69e − 2) − | 3.8441e − 1 (2.42e − 2) − | 2.7601e − 1 (2.25e − 2) − | 3.7015e − 1 (2.91e − 2) − | 4.6857e − 1 (5.00e − 2) − | ||
WFG6 | 3 | 3.9921e − 1 (3.39e − 2) − | 8.8743e − 1 (8.22e − 2) − | 4.9709e − 1 (3.92e − 2) − | 4.2747e − 1 (6.09e − 2) − | 2.8059e − 1 (3.23e − 2) − | 3.8476e − 1 (3.64e − 2) − | 4.9730e − 1 (4.95e − 2) − | ||
WFG7 | 3 | 3.8759e − 1 (2.76e − 2) − | 6.7532e − 1 (8.11e − 2) − | 5.7243e − 1 (5.72e − 2) − | 3.8348e − 1 (4.50e − 2) − | 2.3517e − 1 (2.86e − 2) − | 3.7893e − 1 (2.97e − 2) − | 5.2843e − 1 (4.68e − 2) − | ||
WFG8 | 3 | 4.1099e − 1 (3.03e − 2) − | 7.4598e − 1 (9.00e − 2) − | 5.3368e − 1 (4.13e − 2) − | 4.6012e − 1 (6.34e − 2) − | 3.2062e − 1 (3.31e − 2) − | 3.6914e − 1 (2.74e − 2) − | 4.4774e − 1 (3.87e − 2) − | ||
WFG9 | 3 | 3.9499e − 1 (2.91e − 2) − | 7.8124e − 1 (6.41e − 2) − | 4.8120e − 1 (3.66e − 2) − | 3.7504e − 1 (3.91e − 2) − | 2.3126e − 1 (2.09e − 2) − | 3.5486e − 1 (2.55e − 2) − | 4.6679e − 1 (4.11e − 2) − | ||
DTLZ1 | 3 | 6.1843e − 1 (1.11e − 1) − | 1.1651e + 0 (4.06e − 1) − | 6.1903e − 1 (1.53e − 1) − | 8.1300e − 1 (7.77e − 2) − | 2.7682e − 1 (7.55e − 2) − | 4.4943e − 1 (3.69e − 1) − | 5.4460e − 1 (9.53e − 2) − | ||
DTLZ2 | 3 | 3.6511e − 1 (4.00e − 2) − | 7.2693e − 1 (4.66e − 2) − | 5.2172e − 1 (4.00e − 2) − | 3.2673e − 1 (4.67e − 2) − | 1.8948e − 1 (2.41e − 2) − | 3.2863e − 1 (2.53e − 2) − | 4.1585e − 1 (5.72e − 2) − | ||
DTLZ3 | 3 | 9.9070e − 1 (2.67e − 1) = | 1.2035e + 0 (2.48e − 1) − | 9.1350e − 1 (1.14e − 1) = | 9.6831e − 1 (1.02e − 1) = | 9.5081e − 1 (9.91e − 2) = | 1.0395e + 0 (1.57e − 1) − | 8.7245e − 1 (2.58e − 1) | ||
DTLZ4 | 3 | 5.6974e − 1 (1.19e − 1) − | 1.0206e + 0 (8.45e − 2) − | 5.1083e − 1 (8.00e − 2) − | 4.6431e − 1 (1.13e − 1) − | 1.4167e + 0 (1.84e − 1) − | 3.9712e − 1 (1.69e − 1) − | 4.2756e − 1 (4.05e − 2) − | ||
DTLZ5 | 3 | 3.0078e − 1 (3.21e − 2) − | 1.1410e + 0 (1.49e − 1) − | 5.0231e − 1 (5.54e − 2) − | 7.8707e − 1 (1.34e − 1) − | 3.5182e − 1 (3.87e − 2) − | 5.9300e − 1 (2.99e − 2) − | 5.2722e − 1 (5.90e − 2) − | ||
DTLZ6 | 3 | 6.4761e − 1 (1.93e − 1) − | 1.3944e + 0 (9.18e − 2) − | 7.4365e − 1 (8.41e − 2) − | 1.5437e + 0 (8.84e − 2) − | 7.4778e − 1 (1.02e − 1) − | 6.0124e − 1 (5.57e − 2) − | 1.0561e + 0 (8.66e − 2) − | ||
ZDT1 | 2 | 7.8192e − 1 (3.10e − 2) − | 1.0383e + 0 (6.89e − 2) − | 3.6500e − 1 (3.83e − 2) − | 3.3249e − 1 (4.52e − 2) − | 9.0033e − 1 (5.48e − 2) − | 3.7093e − 1 (2.35e − 2) − | 1.2118e + 0 (1.23e − 1) − | ||
ZDT2 | 2 | 9.3303e − 1 (4.49e − 2) − | 1.0110e + 0 (1.98e − 2) − | 4.2942e − 1 (1.31e − 1) − | 5.0025e − 1 (3.44e − 1) − | 9.8143e − 1 (3.77e − 2) − | 4.4852e − 1 (6.64e − 2) − | 1.1787e + 0 (1.01e − 1) − | ||
ZDT3 | 2 | 8.8362e − 1 (4.10e − 2) − | 1.0833e + 0 (5.39e − 2) − | 4.2857e − 1 (7.50e − 2) − | 8.1090e − 1 (1.01e − 1) − | 9.0184e − 1 (7.42e − 2) − | 5.4774e − 1 (3.87e − 2) − | 1.2219e + 0 (9.31e − 2) − | ||
ZDT4 | 2 | 1.0061e + 0 (4.68e − 2) − | 1.7449e + 0 (2.14e − 1) − | 9.4711e − 1 (7.36e − 2) − | 9.5781e − 1 (1.41e − 1) − | 9.0914e − 1 (2.41e − 1) = | 9.3729e − 1 (8.61e − 2) − | 8.9735e − 1 (7.22e − 2) | ||
ZDT6 | 2 | 1.3583e + 0 (3.67e − 1) − | 1.1561e + 0 (6.25e − 1) − | 6.8469e − 1 (1.56e − 1) − | 5.5590e − 1 (4.27e − 1) − | 1.0641e + 0 (1.39e − 1) − | 6.9695e − 1 (1.12e − 1) − | 1.3267e + 0 (1.02e − 1) − | ||
+/-/= | 0/22/2 | 1/23/0 | 1/21/2 | 1/22/1 | 0/23/1 | 1/21/2 | 0/23/1 |
The bold font represents the best performance on this problem. Where the symbols ‘+’, ‘ −’ and ‘=’ indicate the results of compared algorithm is significantly better, significantly worse, and statistically similar to that obtained by SIACGDED at a 0.05 level by the Wilcoxon’s rank sum test, respectively.
Problem | GDE3 | MOEADDE | NSGAII | LCSA | LMEA | SPEA2SDE | ACGDENSGAII | SIACGDE | ||
---|---|---|---|---|---|---|---|---|---|---|
VNT1 | 3 | 4.1061e − 1 (3.03e − 2) − | 3.4987e − 1 (2.02e − 2) − | 3.6818e − 1 (3.41e − 2) − | 3.9298e − 1 (4.91e − 2) − | 5.2968e − 1 (3.00e − 2) − | 4.4238e − 1 (2.85e − 2) − | 3.4988e − 1 (3.93e − 2) − | ||
VNT2 | 3 | 1.4062e − 1 (1.99e − 2) − | 1.2871e − 1 (1.27e − 2) − | 1.2878e − 1 (1.85e − 2) − | 1.3418e − 1 (2.91e − 2) − | 1.6965e − 1 (1.52e − 2) − | 1.4616e − 1 (1.72e − 2) − | 1.0872e − 1 (1.74e − 2) − | ||
VNT3 | 3 | 1.4576e − 1 (1.12e − 2) − | 4.7590e − 1 (3.59e − 2) = | 1.3851e − 1 (3.13e − 2) − | 4.4576e − 1 (2.69e − 2) − | 2.2039e − 1 (4.02e − 2) − | 3.3171e − 1 (4.04e − 2) − | 4.8823e − 1 (3.72e − 2) | ||
VNT4 | 3 | 4.4409e − 1 (3.32e − 2) − | 3.1403e − 1 (4.03e − 2) − | 3.8689e − 1 (3.03e − 2) − | 3.1331e − 1 (4.15e − 2) − | 4.7496e − 1 (4.26e − 2) − | 4.8042e − 1 (3.42e − 2) − | 3.1891e − 1 (3.39e − 2) − | ||
WFG1 | 3 | 1.1993e − 2 (3.24e − 3) − | 6.0675e − 2 (2.11e − 2) + | 1.4661e − 1 (2.49e − 2) + | 5.5414e − 2 (1.39e − 2) + | 8.1154e − 3 (6.33e − 3) − | 1.3821e − 2 (4.54e − 3) − | 2.0609e − 2 (8.75e − 3) | ||
WFG2 | 3 | 4.2776e − 1 (3.56e − 2) − | 1.9787e − 1 (3.14e − 2) − | 4.2725e − 1 (4.15e − 2) − | 6.4917e − 1 (6.34e − 2) − | 2.9965e − 1 (5.09e − 2) − | 4.1425e − 1 (3.14e − 2) − | 4.3531e − 1 (4.03e − 2) − | ||
WFG3 | 3 | 5.1338e − 1 (3.53e − 2) − | 4.8023e − 1 (3.49e − 2) − | 5.3908e − 1 (4.32e − 2) = | 5.1037e − 1 (4.33e − 2) − | 4.7693e − 1 (2.89e − 2) − | 5.3531e − 1 (4.42e − 2) = | 5.3158e − 1 (2.54e − 2) | ||
WFG4 | 3 | 3.8559e − 1 (3.04e − 2) − | 3.7957e − 1 (4.75e − 2) − | 3.7184e − 1 (2.94e − 2) − | 4.9452e − 1 (5.01e − 2) − | 4.1885e − 1 (2.24e − 2) − | 3.9661e − 1 (3.18e − 2) − | 6.4911e − 1 (2.84e − 2) | ||
WFG5 | 3 | 3.9421e − 1 (3.16e − 2) − | 3.5281e − 1 (1.31e − 2) − | 3.2584e − 1 (3.29e − 2) − | 6.1957e − 1 (2.11e − 2) − | 5.3951e − 1 (3.22e − 2) − | 4.1632e − 1 (2.40e − 2) − | 3.6193e − 1 (3.72e − 2) − | ||
WFG6 | 3 | 3.9129e − 1 (3.82e − 2) − | 3.1384e − 1 (3.21e − 2) − | 3.2998e − 1 (2.92e − 2) − | 5.6264e − 1 (3.97e − 2) − | 5.2581e − 1 (3.24e − 2) − | 4.0052e − 1 (2.27e − 2) − | 3.4274e − 1 (4.09e − 2) − | ||
WFG7 | 3 | 3.9701e − 1 (3.16e − 2) − | 3.8370e − 1 (3.61e − 2) − | 3.1329e − 1 (3.94e − 2) − | 6.1965e − 1 (3.55e − 2) = | 4.1572e − 1 (1.98e − 2) − | 3.3888e − 1 (3.66e − 2) − | 6.2638e − 1 (2.83e − 2) | ||
WFG8 | 3 | 3.6368e − 1 (2.90e − 2) − | 3.4912e − 1 (2.34e − 2) − | 3.1269e − 1 (3.13e − 2) − | 5.0110e − 1 (3.33e − 2) − | 4.4169e − 1 (3.83e − 2) − | 4.1418e − 1 (2.46e − 2) − | 3.4876e − 1 (3.15e − 2) − | ||
WFG9 | 3 | 3.7370e − 1 (3.01e − 2) − | 3.4841e − 1 (2.38e − 2) − | 3.1711e − 1 (3.23e − 2) − | 5.6093e − 1 (3.28e − 2) − | 4.1388e − 1 (3.17e − 2) − | 3.3280e − 1 (3.37e − 2) − | 5.9597e − 1 (2.45e − 2) | ||
DTLZ1 | 3 | 1.7844e − 2 (2.11e − 2) − | 1.7073e − 1 (1.93e − 1) = | 1.3992e − 1 (7.72e − 2) + | 4.5853e − 2 (2.54e − 2) − | 2.7596e − 1 (1.74e − 1) + | 8.5965e − 2 (7.37e − 2) = | 8.6788e − 2 (4.81e − 2) | ||
DTLZ2 | 3 | 4.1335e − 1 (3.94e − 2) − | 4.9404e − 1 (3.14e − 2) − | 3.2196e − 1 (3.19e − 2) − | 6.5682e − 1 (2.10e − 2) − | 5.4981e − 1 (3.55e − 2) − | 4.3623e − 1 (1.82e − 2) − | 3.7830e − 1 (2.89e − 2) − | ||
DTLZ3 | 3 | 2.4167e − 1 (6.42e − 2) = | 1.5296e − 1 (1.00e − 1) − | 1.3322e − 1 (6.27e − 2) − | 1.1064e − 1 (6.36e − 2) − | 1.1361e − 1 (5.61e − 2) − | 2.3046e − 1 (7.72e − 2) − | 2.8561e − 1 (1.04e − 1) | ||
DTLZ4 | 3 | 2.6485e − 1 (8.55e − 2) − | 1.6441e − 1 (1.24e − 1) − | 3.2718e − 1 (6.79e − 2) − | 6.3973e − 1 (3.35e − 2) − | 5.8842e − 2 (2.84e − 2) − | 3.7805e − 1 (1.43e − 1) − | 3.6914e − 1 (3.37e − 2) − | ||
DTLZ5 | 3 | 9.2035e − 1 (1.89e − 2) − | 4.3657e − 1 (7.07e − 2) − | 7.8871e − 1 (3.89e − 2) − | 6.3602e − 1 (6.51e − 2) − | 8.7198e − 1 (2.66e − 2) − | 7.6854e − 1 (4.37e − 2) − | 9.4030e − 1 (1.58e − 2) | ||
DTLZ6 | 3 | 7.8487e − 1 (9.86e − 2) − | 2.1561e − 1 (5.15e − 2) − | 6.5308e − 1 (4.74e − 2) − | 2.3081e − 1 (4.83e − 2) − | 7.1408e − 1 (8.14e − 2) − | 4.5902e − 1 (5.54e − 2) − | 9.2944e − 1 (9.98e − 3) | ||
ZDT1 | 2 | 8.5357e − 2 (5.01e − 2) − | 1.8170e − 1 (8.47e − 2) − | 6.8342e − 1 (2.54e − 2) − | 7.9718e − 1 (4.02e − 2) − | 0.0000e + 0 (0.00e + 0) − | 7.0805e − 1 (1.78e − 2) − | 3.2313e − 1 (5.48e − 2) − | ||
ZDT2 | 2 | 1.0558e − 1 (2.33e − 2) − | 9.4901e − 3 (2.20e − 2) − | 6.3183e − 1 (1.20e − 1) − | 6.2579e − 1 (2.53e − 1) − | 4.5989e − 2 (2.17e − 2) − | 6.3722e − 1 (4.41e − 2) − | 3.3576e − 1 (4.42e − 2) − | ||
ZDT3 | 2 | 9.8039e − 4 (5.37e − 3) − | 4.9689e − 3 (8.14e − 3) − | 6.5800e − 1 (4.05e − 2) − | 4.4583e − 1 (8.72e − 2) − | 4.0421e − 2 (5.36e − 2) − | 6.4515e − 1 (2.95e − 2) − | 3.2265e − 1 (3.87e − 2) − | ||
ZDT4 | 2 | 5.0000e − 2 (1.53e − 1) = | 0.0000e + 0 (0.00e + 0) − | 2.5385e − 1 (8.75e − 2) + | 0.0000e + 0 (0.00e + 0) − | 2.4095e − 1 (1.22e − 1) + | 2.7614e − 2 (8.47e − 2) = | 3.4883e − 2 (7.86e − 2) | ||
ZDT6 | 2 | 5.5017e − 1 (3.29e − 1) − | 6.2013e − 1 (3.42e − 1) − | 5.0975e − 1 (5.63e − 2) − | 7.5684e − 1 (9.69e − 2) − | 2.3176e − 1 (5.89e − 2) − | 5.0119e − 1 (6.09e − 2) − | 2.8401e − 1 (4.19e − 2) − | ||
+/-/= | 1/21/2 | 1/22/1 | 3/19/2 | 2/21/1 | 5/18/1 | 5/19/0 | 0/21/3 |
Problem | GDE3 | MOEADDE | NSGAII | LCSA | LMEA | SPEA2SDE | ACGDENSGAII | SIACGDE | ||
---|---|---|---|---|---|---|---|---|---|---|
VNT1 | 3 | 4.3520e − 1 (1.28e − 2) | 1.4927e + 0 (2.11e − 2) | 1.9050e − 1 (9.64e − 3) | 5.4482e − 1 (2.09e − 2) | 8.0590e − 1 (1.75e − 2) | 4.5761e + 0 (4.27e − 2) | 1.9391e + 0 (2.14e − 2) | 1.9545e + 0 (1.39e − 2) | |
VNT2 | 3 | 2.7341e − 1 (5.38e − 3) | 1.4914e + 0 (1.96e − 2) | 1.8379e − 1 (3.15e − 3) | 5.5694e − 1 (2.14e − 2) | 7.4865e − 1 (7.66e − 3) | 3.3687e + 0 (9.50e − 2) | 1.9318e + 0 (8.72e − 3) | 1.9358e + 0 (8.67e − 3) | |
VNT3 | 3 | 2.7389e − 1 (4.49e − 3) | 1.5005e + 0 (2.09e − 2) | 1.8355e − 1 (4.85e − 3) | 5.5637e − 1 (2.33e − 2) | 6.8591e − 1 (5.41e − 3) | 2.3958e + 0 (1.96e − 2) | 1.9467e + 0 (1.52e − 2) | 1.9579e + 0 (3.16e − 3) | |
VNT4 | 3 | 3.4775e − 1 (9.33e − 3) | 1.5355e + 0 (1.58e − 2) | 1.9118e − 1 (3.73e − 3) | 5.6945e − 1 (2.36e − 2) | 8.1982e − 1 (1.53e − 2) | 4.4627e + 0 (1.78e − 1) | 1.9897e + 0 (2.42e − 2) | 2.1074e + 0 (2.35e − 2) | |
Rank | 2 | 5 | 1 | 3 | 4 | 8 | 6 | 7 |