An Efficient Reconstructed Differential Evolution Variant by Some of the Current State-of-the-art Strategies for Solving Single Objective Bound Constrained Problems

In this section, we elaborate on one of the classic variants of DE and use it as the basis for our research in this paper. In the following sections, we will gradually develop based on this basic DE.

Like other evolutionary algorithms, DE typically generates an initial population of $N$ individuals based on the following formula.

where the function $rand(a,b)$ returns a random number within the range of $[a,b]$. $x_{L,j}$ and $x_{U,j}$ respectively represent the lower bound and upper bound of dimension $j$.

The core of differential evolution is the mutation operator. Since the original differential evolution (DE) was proposed in 1997 [14], various improved mutation operators have been introduced, including DE/rand/1, DE/rand/2, DE/best/1, DE/best/2, and DE/current-to-best/1. The most famous one is the DE/current-to-pbest/1 mutation strategy introduced by JADE in 2009 [15]. This strategy is also the most commonly used efficient strategy for LSHADE and its descendant variants. Therefore, we use it as the basis of our research in this paper. Its expression is as follows:

where $k$ represents the number of iterations. $x_{i,j}$ refers to the value of the $j$-th dimension of the $i$-th individual. $r1$ and $r2$ are two different random indices used for randomly selecting individuals from the population. $p$ is a randomly selected index value from the top $100p\%$ individuals in the population sorted by fitness values, with a range of $(0,1]$. $F$ is a scale factor used to control the size of differential vectors, with a range of $[0,1]$.

Another major part of LSHADE is crossover, which is controlled by a crossover probability $Cr$ ranging from 0 to 1. For each dimension of every new trial solution generated after mutation for each individual in the population, there is a probability of $(1-Cr)$ that it will be replaced by the value of the same dimension from its parent. This inheritance comes from earlier genetic algorithms in evolutionary computation [16]. The expression is as follows:

where $u_{i,j}$ is the new individual generated by executing crossover strategy between $v_{i,j}$ and $x_{i,j}$. $j_{rand}$ represents a random integer within the range of $[0,D]$ (where $D$ is the dimension of the optimization problem). The purpose of $j_{rand}$ is to ensure that at least one dimension adopts the value generated by mutation strategy.

Finally, there is the selection part. LSHADE compares the fitness values of offspring individuals with their corresponding parent individuals and keeps the one with better fitness value. The algorithm will perform this selection operation on each individual in the population separately. Its expression is as follows.

where $f(u_{i}^{(k)})$ and $f(x_{i}^{(k)})$ indicates the evaluation function of the individual $u_{i}^{(k)}$ and $x_{i}^{(k)}$.

In this section, we will combine different strategies of basic DE with various advanced DE variants proposed in recent years, and develop and propose a reconstructed DE (RDE).

The performance of the proposed algorithm RDE is tested on the CEC 2024 Competition for Single-Objective Real Parameter Numerical Optimization[20]. The benchmark suite, CEC2024, comprises 29 test functions (with $30D$) with diverse features, including unimodal functions ($f_{1}-f_{2}$), multimodal functions ($f3-f9$), hybrid funtions ($f10-f19$), and composition functions ($f20-f29$). The variables in hybrid functions are randomly divided into subcomponents, and different basic functions are used for each subcomponent. The composition functions utilize basic benchmark functions to create problems with a randomly located global optimum and several randomly located deep local optima, making them more challenging to solve.

In this study, the algorithm parameter settings are set as follows:

• 1)

  population maximum and minimum sizes $N_{min}=4$ and $N_{max}=18\cdot D$;

• 2)

  success history archive memory size $H=5$;

• 3)

  rank greediness factor $k_{r}=3$;

• 4)

  external archive size $Ar=1$;

• 5)

  differential mutation $p$ value: $p_{max}=0.25$;

• 6)

  initialized sub-population resource ratios $\gamma_{1}=\gamma_{2}=0.5$;

• 7)

  initialized scale factor and crossover rate in memory archive $\mu_{F}=0.3$ and $\mu_{Cr}=0.8$;

• 8)

  Cauthy pertubation strategy rate $p_{r}=0.2$;

According to the guidelines of CEC2024, 25 independent runs are implemented. The maximum evaluation number is set to $max\_n\!f\!es=10000\cdot D$, where the $D$ is the dimensionality of the optimization problems. The search range is set to $[-100,100]^{D}$. Fitness errors less than $1\times 10^{−8}$ are considered to 0. The experiments were performed using the following system: CPU: Intel^® Core™i7-12700kf 3.6GHz, 32GB DDR4-3600MHz RAM, Programming Language: C++, and Operator System: Windows 10.

The Wilcoxon rank-sum test is utilized to determine if there is a statistically significant difference between two algorithms in solving a problem, with a significance level of $\alpha=0.05$. Each problem is tested 51 times independently in a row. The symbol “$+$” denotes the superior algorithm when there is a significant difference between the pair, while “$-$” indicates the relatively inferior one. “$=$” signifies that no significant difference exists between the pair of algorithms.

The performance of the RDE was compared to the other compatitors which showed excellent performance on previous CEC benchmark suites, including:

• 1)

  LSHADE-RSP[11]: LSHADE Algorithm with Rank-Based Selective Pressure Strategy for Solving CEC 2017 Benchmark Problems;

• 2)

  iLSHADE-RSP[12]: An improved LSHADE-RSP algorithm with the Cauchy perturbation;

• 3)

  HSES[21]: Hybrid Sampling Evolution Strategy for Solving Single Objective Bound Constrained Problems;

• 4)

  EBOwithCMAR[22]: Effective Butterfly Optimizer using Covariance Matrix Adapted Retreat phase;

• 5)

  LSHADE[10]: Improving the search performance of SHADE using linear population size reduction.

As shown in Table I, RDE demonstrates superiority over competitive algorithms on CEC2024. In addition, RDE also performs well on relatively more complex composition functions ($f20-f29$), indicating the effectiveness of recombination research. Compared with LSHADE-RSP, iLSHADE-RSP, HSES, EBOwithCMAR and LSHADE, RDE achieves better solutions on 13, 15, 11, 14 and 17 problems respectively and fails to beat opponents on 4, 3, 7, 5 and 3 problems respectively. Although RDE shows significant advantages over advanced variants of the DE series, the performance gap is not significantly widened compared to HSES.

Table II shows that the algorithm complexity of the proposed RDE is 4.95. $T_{0}$ is the time result by running the assessing test program in [20]. $T_{1}$ is the time cost result by running the 18th function and $T_{2}$ is the time cost result of averaging 5 runs of the algorithm on $f_{18}$, respectively. Both of them are with 200000 evaluations.