Modified LSHADE-SPACMA with new mutation strategy and external archive mechanism for numerical optimization and point cloud registration

Abstract

Numerical optimization and point cloud registration are critical research topics in the field of artificial intelligence. The differential evolution algorithm is an effective approach to address these problems, and LSHADE-SPACMA, the winning algorithm of CEC2017, is a competitive differential evolution variant. However, LSHADE-SPACMA’s local exploitation capability can sometimes be insufficient when handling these challenges. Therefore, in this work, we propose a modified version of LSHADE-SPACMA (mLSHADE-SPACMA) for numerical optimization and point cloud registration. Compared to the original approach, this work presents three main innovations. First, we present a precise elimination and generation mechanism to enhance the algorithm’s local exploitation ability. Second, we introduce a mutation strategy based on a modified semi-parametric adaptive strategy and rank-based selective pressure, which improves the algorithm’s evolutionary direction. Third, we propose an elite-based external archiving mechanism, which ensures the diversity of the external population and can accelerate the algorithm’s convergence progress. Additionally, we utilize the CEC2014 (Dim = 10, 30, 50, 100) and CEC2017 (Dim = 10, 30, 50, 100) test suites for numerical optimization experiments, comparing our approach against: (1) 10 recent CEC winner algorithms, including LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig, L-SRTDE, and LSHADE-SPACMA; (2) 4 advanced variants: APSM-jSO, LensOBLDE, ACD-DE, and MIDE. The results of the Wilcoxon signed-rank test and Friedman mean rank test demonstrate that mLSHADE-SPACMA not only outperforms the original LSHADE-SPACMA but also surpasses other high-performance optimizers, except that it is inferior L-SRTDE on CEC2017. Finally, 25 point cloud registration cases from the Fast Global Registration dataset are applied for simulation analysis to demonstrate the potential of the developed mLSHADE-SPACMA technique for solving practical optimization problems. The code is available at https://github.com/ShengweiFu?tab=repositories and https://ww2.mathworks.cn/matlabcentral/fileexchange/my-file-exchange

1 Introduction

Optimization algorithms play a central role in many areas of modern science and engineering, particularly in solving complex optimization problems such as numerical optimization (Chauhan and Shivani 2024; Fu, et al. 2023; Li and Meng 2024), the capacitated vehicle routing problem (Souza et al. 2023), and point cloud registration (Zhao, 2021), parameter identification (Zhang, et al. 2024a, b), optimizing the parameters (Chauhan and Yadav 2023), scheduling problems (Zhou, et al. 2021), and others (Chauhan and Yadav 2024b; Fu, et al. 2024; Hong et al. 2024a, b). The Differential Evolution (DE) algorithm is widely used for these problems due to its simplicity, efficiency, and ease of implementation. Since the DE algorithm was first proposed by Storn and Price (Storn and Price 1997) in 1997, it has become one of the most popular evolutionary algorithms, demonstrating excellent global search capabilities across a variety of test problems.

Despite its many advantages, the DE algorithm still faces several challenges when dealing with optimization problems requiring complex adaptations. The main issues include maintaining population diversity, avoiding premature convergence, and effectively balancing global exploration and local exploitation (Li et al. 2023). To address these problems, researchers have developed various improved DE variants. These variants introduce parametric or semi-parametric adaptive mechanisms, propose new mutation operators, utilize historical successful parameters, modify external archiving mechanisms, or combine DE with other optimization techniques. These enhancements aim to improve the algorithm’s ability to solve complex problems while maintaining its simplicity and effectiveness (Chakraborty et al. 2023).

For example, in 2009, Zhang et al. (2009) proposed adaptive differential evolution based on an external archive (JADE), which was adopted by most later DE variants. JADE is seen as an effective method for solving optimization problems by combining historical data and adaptive parameter updates to provide optimization directions. In 2013, Tanabe et al. (2013) developed an adaptive differential evolution algorithm based on success history (SHADE) by using the history of successful parameter settings to guide the selection of control parameters, thereby enhancing the performance of JADE. In 2014, Piotrowski (2018) proposed LSHADE with a linear population reduction strategy to improve the robustness and performance of SHADE, making it one of the most popular DE variants. In 2021, Gao et al. (2021) introduced a new variant of JADE by incorporating the chaotic local search (CLS) mechanism. Exploiting the ergodic and non-repetitive properties of chaos, this variant is able to diversify the population and thus has the opportunity to explore a vast search space. In 2022, Li et al. (2022a, b) introduced the meta-knowledge transfer (MKT) based differential evolution (MKTDE) algorithm, which successfully addressed the knowledge transfer problem in multi-task optimization by combining a meta-knowledge transfer mechanism, a multi-task multi-population framework, and an elite solution transfer method. Additionally, Kumar et al. (2022) developed a new DE variant, OLSHADE-CS, by integrating orthogonal array-based initialization, a neighborhood search strategy, a conservative selection scheme, and a parameter adaptation strategy. In 2023, Wang et al. (2023) targeted specific genes in candidate solutions in differential evolution (DE), proposing a simple and efficient method called gene targeting based differential evolution (GTDE). Furthermore, Li et al. (2023) developed an improved differential evolution algorithm named MjSO, introducing a novel probabilistic selection mechanism and a directed mutation strategy, while eliminating the use of an archiving scheme. Meanwhile, Li et al. (2023a, b, c) introduced an improved framework based on population state evaluation (PSE), which can be freely embedded into various existing DE variants and population-based metaheuristics. Hong et al. (Hong et al. 2023a, b) introduced UMOEAs-III, incorporating innovations like an adaptive scaling factor, a crossover rate updating mechanism, sequential quadratic programming, and rank-based mutation strategy. In 2024, Zhang et al. (2024) presented the collective ensemble learning (CEL) paradigm, which combines the advantages of multiple strategies to generate offspring, effectively improving the performance of differential evolution algorithms. Additionally, Song et al. (2024) proposed a differential evolution algorithm (PCM-DE) with a disturbance mechanism and stagnation indicator based on the covariance matrix, which is competitive in terms of solution accuracy and convergence speed. Li et al. (2024) proposed a differential evolution framework (DEF-FM) based on a fluid model and used it for feature selection. Deng et al. (2024) proposed a quantum differential evolution algorithm based on quantum adaptive mutation strategy and population state evaluation framework (PSEQADE). Yuan et al. (2024) proposed a two-phase migration strategy (TMS) to improve the performance of multi-operator differential evolution (IMODE-TMS).

Based on the success of LSHADE, Mohamed et al. (2017) introduced the LSHADE-SPACMA algorithm, which incorporates a semi-parameter adaptive control mechanism into LSHADE and combines it with the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) to enhance population diversity and the global search ability of the algorithm. LSHADE-SPACMA achieved remarkable results in the global optimization competition 2017, performing well on many optimization problems. Despite its excellent optimization performance, LSHADE-SPACMA still encounters performance bottlenecks in some extreme optimization scenarios. Particularly in the algorithm’s parameter settings, improving its flexibility and adaptability without sacrificing robustness has become a key issue in current research. Although there have been many improved variants of differential evolution algorithm, there are few studies on parameter sensitivity analysis and excellent individual guidance. In addition, the No Free Lunch (NFL) theorem (Wolpert and Macready 1997) logically proves that no optimization algorithm can solve all optimization problems. As the winner of CEC2017, LSHADE-SPACMA sometimes exhibited the following shortcomings:

• (1) The iteration information of the current population is not fully utilized, resulting in a lack of local exploitation ability in some cases.

• (2) There is a lack of sensitivity analysis for the semi-parametric approach, and the advantages of the semi-parametric approach are not fully leveraged.

• (3) The mutation operator sometimes fails to effectively guide the evolution direction.

• (4) The external archive update mechanism exhibits randomness, which may remove some better individuals from the search and consequently affect the quality and diversity of the population.

To address the limitations of LSHADE-SPACMA, this study proposes mLSHADE-SPACMA, which significantly enhances algorithm performance across various optimization problems. Improvements include an advanced parameter control strategy, refined mutation operator, and a robust external archiving mechanism. The main contributions of this paper are as follows:

• (5) Addressing the inefficiency of LSHADE-SPACMA in utilizing current population iterative information, we introduce a precise elimination and generation strategy, enhancing the algorithm’s local exploitation capability.

• (6) We implement sensitivity analysis to identify parameter values superior to the original algorithm, effectively boosting performance.

• (7) A modified semi-parametric strategy and rank-based selective pressure mutation strategy are proposed to lead the evolutionary direction of the algorithm.

• (8) We propose an elite-based external archive mechanism that ensures population diversity and accelerates algorithmic convergence and precision.

• (9) We conduct numerical optimization experiments using the CEC2014 and CEC2017 test suites, comparing the results with those of the 10 CEC-winning and 4 advanced variant algorithms in recent years.

• (10) A simulation analysis of 25 point cloud registration cases demonstrates mLSHADE-SPACMA’s potential for solving complex optimization problems through rapid global registration.

The subsequent sections of this paper are organized as follows: Sect. 2 reviews the related literature. Section 3 describes our proposed mLSHADE-SPACMA method in detail. Section 4 presents numerical experiments and their analysis. Section 5 describes the modeling and simulation analysis of point cloud registration. Finally, Sect. 6 concludes the paper and outlines future research directions.

2 Related work

2.1 DE

Differential Evolution (DE) is a population-based optimization method well-suited for common optimization problems (Storn and Price 1997). The algorithm leverages the principles of mutation, crossover, and selection from biological evolution to optimize solutions through repeated iterations. The essence of DE lies in the iterative search and update within the solution vector space, aiming to identify the optimal solution. The initial step of the algorithm involves randomly generating a series of candidate solutions using Eq. (1).

$$x_{i}^{j} = L^{j} + rand \cdot \left( {U^{j} - L^{j} } \right), i \in \left\{ {1,2, \ldots ,NP} \right\}, j \in \left\{ {1,2, \ldots ,D} \right\}$$

(1)

where ${L}^{j}$ and ${U}^{j}$ represent the lower and upper bounds, respectively, of the first $j$ dimensions of the optimization problem. The term $rand$ denotes a random number between 0 and 1, $NP$ represents the population size, and $D$ indicates the number of optimization variables, namely the dimension.

Next is the mutation process, where each individual generates new individuals by utilizing others in the current population, performing the mutation via the vector difference method. The five commonly used mutation strategies—DE/rand/1, DE/best/1, DE/current-to-best/1, DE/best/2, and DE/rand/2—are reported in Eq. (2).

$${\varvec{v}}_{i} = \left\{ {\begin{array}{*{20}c} {{\varvec{x}}_{r1} + F \cdot \left( {{\varvec{x}}_{r2} - {\varvec{x}}_{r3} } \right)} \\ {{\varvec{x}}_{best} + F \cdot \left( {{\varvec{x}}_{r1} - {\varvec{x}}_{r2} } \right)} \\ {{\varvec{x}}_{i} + F \cdot \left( {{\varvec{x}}_{r1} - {\varvec{x}}_{i} } \right) + F \cdot \left( {{\varvec{x}}_{r2} - {\varvec{x}}_{r3} } \right) } \\ {{\varvec{x}}_{i} + F \cdot \left( {{\varvec{x}}_{r1} - {\varvec{x}}_{r2} } \right) + F \cdot \left( {{\varvec{x}}_{r3} - {\varvec{x}}_{r4} } \right)} \\ {{\varvec{x}}_{r1} + F \cdot \left( {{\varvec{x}}_{r2} - {\varvec{x}}_{r3} } \right) + F \cdot \left( {{\varvec{x}}_{r4} - {\varvec{x}}_{r5} } \right)} \\ \end{array} } \right.$$

(2)

where ${{\varvec{x}}}_{r1}$, ${{\varvec{x}}}_{r2}$, ${{\varvec{x}}}_{r3}$, ${{\varvec{x}}}_{r4}$, and ${{\varvec{x}}}_{r5}$ denote five distinct individuals randomly selected from the population, ${{\varvec{x}}}_{best}$ represents the optimal individual in the entire population. ${\varvec{F}}$ demonstrates scaling factor, ${{\varvec{v}}}_{i}$ shows the variation of position. If the new position exceeds the boundary, adjustments are made according to Eq. (3).

$$\nu_{i}^{j} = \left\{ {\begin{array}{*{20}c} {\left( {L^{j} + x_{i}^{j} } \right)/2,\nu_{i}^{j} < L_{j} } \\ {\left( {U^{j} + x_{i}^{j} } \right)/2,\nu_{i}^{j} > U_{j} } \\ \end{array} } \right.$$

(3)

Then, a crossover operation is performed to combine the mutated individual with the original individual, generating a new candidate solution. This process involves mixing components of both individuals according to a predetermined crossover probability.

$$u_{i}^{j} = \left\{ {\begin{array}{*{20}c} {\nu_{i}^{j} ,rand < Cr or j = j_{rand} } \\ {x_{i}^{j} ,otherwis} \\ \end{array} } \right.$$

(4)

where $Cr$ denotes the crossover probability, ${j}_{rand}\in \left\{\text{1,2},\dots ,D\right\}$ represents a dimension index randomly selected from among dimensions.

Finally, the selection operation evaluates the newly generated offspring via the fitness function (objective function). Individuals from the offspring that exhibit higher fitness, along with the most excellent individuals from the parent generation, are selected to form the next generation population. This process is reported in Eq. (5).

$${\varvec{x}}_{i} = \left\{ {\begin{array}{*{20}c} {{\varvec{u}}_{i} ,} & {f\left( {{\varvec{u}}_{i} } \right) \le f\left( {{\varvec{x}}_{i} } \right)} \\ {{\varvec{x}}_{i} ,} & {otherwise} \\ \end{array} } \right.$$

(5)

where $f(\cdot )$ represents the objective function of the problem to be optimized.

The mutation, crossover, and selection steps are iteratively repeated until a stopping condition is met, such as reaching a maximum number of iterations or convergence of the objective function.

2.2 LSHADE

2.2.1 Parameter introduction

The LSHADE variant (Tanabe and Fukunaga 2014) introduces historical memory components ${{\varvec{M}}}_{CR}$, ${{\varvec{M}}}_{F}$, contain $H$ item ${\varvec{C}}{\varvec{R}}$ and ${\varvec{F}}$, and initializes all memory values at 0.5, in contrast to the original DE. It is important to note that both ${\varvec{C}}{\varvec{R}}$ and ${\varvec{F}}$ values range from 0 to 1, where $H$ represents the size of historical memory. Additionally, the external archive ${\varvec{A}}$ is initially empty, as no solutions have been generated yet.

$$\begin{array}{c}\left\{\begin{array}{c}{M}_{CR,k}=0.5\\ {M}_{F,k}=0.5\end{array}\right., k=\text{1,2},3\cdots ,H\end{array}$$

(6)

2.2.2 Mutation strategy

LSHADE utilizes the current-to-pbest/1 variation strategy, as delineated in Eq. (7), where the scaling factor ${F}_{i}$ is determined using Eq. (8).

$$\begin{array}{*{20}c} {\user2{v}_{i} = \user2{x}_{i} + F_{i} \cdot \left( {\user2{x}_{{pbest}} - \user2{x}_{i} } \right) + F_{i} \cdot \left( {\user2{x}_{{r1}} - \user2{x}_{{r2}} } \right)} \\ \end{array}$$

(7)

$$\begin{array}{c}{F}_{i}=randc\left({M}_{F,r3},0.1\right)\end{array}$$

(8)

In LSHADE, ${{\varvec{x}}}_{pbest}$ represents determined by selecting the top $NP\times p$ performers from the population, where $p\in (\text{0,1})$ denotes the algorithm’s degree of greediness—a smaller value indicates higher greed. ${{\varvec{x}}}_{r1}$ shows randomly chosen from the current population, while ${{\varvec{x}}}_{r2}$ demonstrates randomly selected from the external archive. In Eq. (8), ${F}_{i}$ represents generated from a Cauchy distribution with a mean of ${M}_{F,r3}$ (values randomly chosen from historical memory) and a standard deviation of 0.1. If ${F}_{i}$ falls outside the 0 to 1 range, it must be regenerated using Eq. (8).

2.2.3 Crossover probability

The crossover probability $C{R}_{i}$ for the search agent ${x}_{i}$ is determined by a Gaussian distribution, as described in Eq. (9). If C $C{R}_{i}$ generated by Eq. (9) falls outside the range of [0, 1], it is then replaced with the value closest to the boundary (0 or 1).

$$R_{i} = \left\{ {\begin{array}{*{20}c} 0 & {M_{CR,r3} = \bot } \\ {randn\left( {M_{CR,r3} ,0.1} \right),} & {{\text{otherwise}}} \\ \end{array} } \right.$$

(9)

where $\perp$ represents the terminal value, and ${M}_{CR,r3}$ denotes to the mean parameter selected from the historical memory for the crossover probability.

2.2.4 External archiving and historical memory updating

In the LSHADE algorithm, an external archive ${\varvec{A}}$ is utilized to enhance population diversity and mitigate premature convergence. If a parent vector outperforms its offspring, it is retained for the next generation; otherwise, it is stored in ${\varvec{A}}$. Once ${\varvec{A}}$ reaches its preset size, elements are randomly replaced by new insertions. Values of ${F}_{i}$ and $C{R}_{i}$ that successfully produce offspring are recorded as success values, denoted ${S}_{CR}$ and ${S}_{F}$, respectively. These are used to update the historical memories ${{\varvec{M}}}_{CR}$ and ${{\varvec{M}}}_{F}$ according to Eq. (10) and Eq. (11) at each generation’s conclusion. The cell index $k$ used in updating ${{\varvec{M}}}_{CR}$ and ${{\varvec{M}}}_{F}$ starts from 1 and increments by 1 each generation, resetting to 1 when the memory size $H$ is exceeded. It is important to note that if both ${S}_{CR}$ and ${S}_{F}$ are empty, indicating no subvector outperformed the original vector, ${{\varvec{M}}}_{CR}$ and ${{\varvec{M}}}_{F}$ remain unchanged.

$$M_{F,k} = \left\{ {\begin{array}{*{20}c} {M_{F,k} ,} & {S_{F} = \emptyset } \\ {{\text{mean}}_{WL} \left( {S_{F} } \right),} & {{\text{otherwise}}} \\ \end{array} } \right.$$

(10)

$$M_{CR,k} = \left\{ {\begin{array}{*{20}c} {M_{CR,k} ,} & {S_{CR} = \emptyset } \\ { \bot ,} & {max\left( {{\mathcal{S}}_{CR} } \right) = 0} \\ {{\text{mean}}_{WL} \left( {{\mathcal{S}}_{CR} } \right),} & {otherwise} \\ \end{array} } \right.$$

(11)

where the mean ${\text{mean}}_{WL}$ is calculated as the weighted average according to Eq. (12), where ${S}_{n}$ represents either ${S}_{CR}$ and ${S}_{F}$.

$$\begin{array}{c}{\text{mean}}_{WL}\left(S\right)=\frac{\sum_{n=1}^{\left|\mathbf{S}\right|} {\omega }_{n}\cdot {S}_{n}^{2}}{\sum_{n=1}^{\left|\mathbf{S}\right|} {\omega }_{n}\cdot {S}_{n}}\end{array}$$

(12)

$$\begin{array}{c}{\omega }_{n}=\frac{\left|f\left({{\varvec{u}}}_{n}\right)-f\left({{\varvec{x}}}_{n}\right)\right|}{\sum_{n=1}^{\left|{\varvec{s}}\right|} \left|f\left({{\varvec{u}}}_{n}\right)-f\left({{\varvec{x}}}_{n}\right)\right|}\end{array}$$

(13)

2.2.5 Linear population size reduction

In LSHADE, linear population size reduction (LPSR) is utilized to dynamically adjust the population size according to a linear function. Initially, the population sizes at the first generation and at the final generation $G$ (the number of terminated iterations) are set as ${NP}_{init}$ and ${NP}_{min}$, respectively. The population size for generation $g+1$ is updated using the equation provided below:

$$NP_{{g + 1}} = {\text{round}}\left[ {\left( {NP_{{min}} - NP_{{init}} } \right) \cdot FEs/MaxFEs + NP_{{init}} } \right]$$

(14)

where round indicates rounding to the nearest integer, $FEs$ represents to the current number of evaluations, MaxFES denotes the maximum allowable number of evaluations.

2.3 iL-SHADE

iL-SHADE (Brest, Maučec, and Bošković, 2016) is an improved version of LSHADE with the following improvements:

First, the historical memory parameters, ${M}_{CR}$ and ${M}_{F}$, are updated using methods detailed in Eq. (15) and Eq. (16). Initially, all values of ${M}_{CR}$ are set to 0.8, which differs from the original L-SHADE setting of 0.5. Throughout the population evolution process, the values of ${M}_{CR,H}$ and ${M}_{F,H}$ are consistently maintained at 0.9.

$$M_{F,k} = \left\{ {\begin{array}{*{20}c} {M_{F,k} ,} & {{\mathbf{S}}_{F} = \emptyset } \\ {\frac{{\left( {{\text{mean}}_{WL} \left( {{\mathbf{S}}_{F} } \right) + M_{F,k} } \right),}}{2}} & {otherwise} \\ \end{array} } \right.$$

(15)

$$M_{CR,k} = \left\{ {\begin{array}{*{20}c} { \bot ,} & {M_{CR,k} = \bot or\,max\left( {S_{CR} = 0} \right)} \\ {\frac{{\left( {{\text{mean}}_{WL} \left( {S_{CR} } \right) + M_{CR,k} } \right)}}{2},} & {{\text{otherwise}}} \\ \end{array} } \right.$$

(16)

Secondly, at the start of the iteration, low $Cr$ values and high $F$ values are not conducive to effective population evolution. Therefore, adjustments in iL-SHADE are made using Eq. (17) and Eq. (18).

$$Cr_{i} = \left\{ {\begin{array}{*{20}c} {\max \left( {Cr_{i} ,0.5} \right),} & {FEs < 0.25 \cdot {{MaxFEs}}} \\ {\max \left( {Cr_{i} ,0.25} \right),} & {0.25 \cdot {{MaxFEs}} \le FEs < 0.5 \cdot {{MaxFEs}}} \\ \end{array} } \right.$$

(17)

$$F_{i} = \left\{ {\begin{array}{*{20}c} {\min \left( {F_{i} ,0.7} \right),FEs < 0.25 \cdot {{MaxFEs}}} \\ {\min \left( {F_{i} ,0.8} \right),0.25 \cdot MaxFEs \le FEs < 0.5 \cdot {{MaxFEs}}} \\ {\min \left( {F_{i} ,0.9} \right),0.5 \cdot MaxFEs \le FEs < 0.75 \cdot {{MaxFEs}}} \\ \end{array} } \right.$$

(18)

Finally, an adaptive greedy control parameter strategy is implemented, as detailed in Eq. (19).

$$p = p_{{min}} + \left( {p_{{max}} - p_{{min}} } \right) \cdot FEs/MaxFEs$$

(19)

where ${p}_{min}$ and ${p}_{max}$ represent the lower and upper limits of the greedy factor, respectively, set at 0.1 and 0.2 in iL-SHADE.

2.4 jSO

jSO (Brest, Maučec, and Bošković, 2017) represents an enhanced version of iL-SHADE, introducing a novel weighted mutation strategy, “DE/current-to-pbest-w/1”. This strategy is described in detail as outlined in Eq. (20).

$$\begin{array}{c}{v}_{i}={x}_{i}+F{w}_{i}\cdot \left({x}_{pbest}-{x}_{i}\right)+{F}_{i}\cdot \left({x}_{{r}_{1}}-{x}_{{r}_{2}}\right)\end{array}$$

(20)

In the evolution process, an initially smaller factor $F{w}_{i}$ is adopted for the early stages to promote exploration, while a larger $F{w}_{i}$ is used in later stages to enhance exploitation. Additionally, during evolution, the maximum greed factor ${p}_{max}$, is set at 0.25, with ${p}_{min}$ being half of ${p}_{max}$. Concurrently, the initial value of ${M}_{F}$ in jSO is set to 0.3.

$$Fw_{i} = \left\{ {\begin{array}{*{20}c} {0.7 \cdot F_{i} , \cdot FEs < 0.2 \cdot Max{{F}}s} \\ {0.8 \cdot F_{i} , \cdot 0.2 \cdot Max{{F}}s \le FEs < 0.4 \cdot Max{{F}}s} \\ {1.2 \cdot F_{i} , \cdot FEs \ge 0.4 \cdot Max{{F}}s} \\ \end{array} } \right.$$

(21)

2.5 LSHADE-RSP

LSHADE-RSP (Stanovov, Akhmedova, and Semenkin, 2018) is the first version to improve jSO by introducing rank-based selective pressure (RSP) and adjusting $Cr$ and $F$ parameters. The description of the RSP-based mutation strategy “DE/current-to-pbest-w/r” is presented in Eq. (22).

$$\begin{array}{c}{{\varvec{v}}}_{i}={{\varvec{x}}}_{i}+F{w}_{i}\cdot \left(\begin{array}{c}{{\varvec{x}}}_{pbest}-{{\varvec{x}}}_{i}\end{array}\right)+{F}_{i}\cdot \left(\begin{array}{c}{{\varvec{x}}}_{p{r}_{1}}-{{\varvec{x}}}_{p{r}_{2}}\end{array}\right)\end{array}$$

(22)

where ${{\varvec{x}}}_{p{r}_{1}}$ denotes the vector selected from the population P based on rank-based probability, and ${{\varvec{x}}}_{p{r}_{2}}$ represents the vector chosen from P either according to rank-based probability or randomly selected from the archive A. The selection probability of individual $i$ is calculated using Eq. (23).

$$\begin{array}{c}P{r}_{i}=Ran{k}_{i}/(Ran{k}_{1}+Ran{k}_{2}+\cdot \cdot \cdot +Ran{k}_{NP})\end{array}$$

(23)

$$\begin{array}{c}Ran{k}_{i}=k\left(NP-i\right)+1\end{array}$$

(24)

In this context, the highest and lowest ranks are assigned to the best and worst individuals, respectively. The index $i$ from the set $i\in \{\text{1,2},3,...,NP\}$ denotes the position of individuals in the fitness-sorted population array. The parameter $k$ serves as the control factor that regulates the greediness of the rank-based selection process.

Secondly, $Cr$ and $F$ are adjusted by Eq. (25) and Eq. (26) during evolution.

$$Cr_{i} = \left\{ {\begin{array}{*{20}c} {{\text{max}}\left( {Cr_{i} ,0.7} \right),\,FEs < 0.25 \cdot MaxFEs} \\ {{\text{max}}\left( {Cr_{i} ,0.6} \right),\,0.25 \cdot Max{\text{F}}s \le FEs < 0.5 \cdot MaxFEs} \\ {Cr_{i} ,\,0.5 \cdot Max{\text{F}}s \le FEs} \\ \end{array} } \right.$$

(25)

$$Cr_{i} = \left\{ {\begin{array}{*{20}c} {max\left( {Cr_{i} ,0.7} \right),\,FEs < 0.25 \cdot {{MaxFE}}s} \\ {} \\ {max\left( {Cr_{i} ,0.6} \right),\,0.25 \cdot {Max{FE}}s \le FEs < 0.5 \cdot {Max{FE}}s} \\ {} \\ {Cr_{i} ,\,0.5 \cdot {{MaxFE}}s \le FEs} \\ \end{array} } \right.$$

(26)

Finally, different parameter settings in LSHADE-RSP are as follows: the greed factor ${p}_{max}$ in jSO is set at 0.17, with ${p}_{min}$ being half of ${p}_{max}$. Additionally, the control factor $k$ for the rank-based selection’s greediness is adjusted to 3.

2.6 CMA-ES

CMA-ES is an effective evolutionary strategy for addressing various optimization problems (Hansen et al. 2003). This strategy models the search space using a multivariate normal distribution and generates new individuals via a Gaussian distribution, thereby tracking the evolutionary trajectory of the population. The algorithm automatically adjusts several parameters, including the mean vector $m$, the covariance matrix $C$, and the step size $\sigma$. The specific operational steps are as follows:

• (1) Initialization: The population is initialized and evaluated using the fitness function.

• (2) Generation of New Individuals: New individuals are generated according to a Gaussian distribution:

  $$\begin{array}{c}{x}_{i}=N\left(m,{\sigma }^{2}C\right)\end{array}$$

  (27)

• (3) Updating the Mean Vector m: The best μ individuals are used to update the mean vector m as follows:

  $$\begin{array}{c}\begin{array}{c}m=\sum_{i=1}^{\mu } {w}_{i}{{\varvec{x}}}_{i}\end{array}\end{array}$$

  (28)

  where weights ${w}_{i}$ satisfy:

  $$\begin{array}{c}\sum_{i=1}^{\mu } {w}_{i}=1\end{array}$$

  (29)
  $$\begin{array}{c}\begin{array}{c}{w}_{1}\ge {w}_{2}\ge \dots \ge {w}_{\mu }\end{array}\end{array}$$

  (30)

• (4) Updating σ and C: The step size σ and the covariance matrix C are updated based on the selection outcomes.

• (5) Termination: These steps are repeated until a termination condition is met.

2.7 LSHADE-SPACMA

2.7.1 Semi-parametric adaptive strategy

Parameter configuration plays a crucial role in the performance of differential evolution algorithms, with adaptive parameter settings demonstrating that algorithm performance is intimately connected to the specific parameters chosen for given problems. Different problems may require tailored parameter values to optimize algorithm performance (Das et al. 2016). To facilitate semi-parametric self-adaptation of ${F}_{i}$ and $C{R}_{i}$, LSHADE-SPA modifies the calculation method for ${F}_{i}$ during the first half of the evaluation times in the LSHADE algorithm. The specific formula is outlined below:

$$F_{i} = 0.45 + 0.1 \cdot rand,~~FEs < \max FEs/2$$

(31)

2.7.2 The LSHADE-SPACMA hybrid framework

In order to improve the efficiency of the algorithm, a hybrid framework combining LSHADE-SPA and an improved CMA-ES algorithm was developed (Mohamed et al. 2017). The crossover operation is introduced into the CMA-ES algorithm to enhance the exploration function of the framework. The crossover operation is performed according to Eq. (4) after the CMA-ES step to generate offspring.

The whole framework starts with a common population $P$, where each individual ${\varvec{x}}$ in the population can generate offspring individuals ${\varvec{u}}$ via LSHADE or CMA-ES. The assignment of such individuals is based on a class probability variable ($FCP$), whose value is randomly selected from slot ${M}_{FCP}$. At the end of each generation, the memory slot ${M}_{FCP}$ is updated according to the performance of each algorithm, thus gradually increasing the proportion of the population allocated to the better-performing algorithm. The update process is performed only for cases where new individuals are successfully generated. The specific ${M}_{FCP}$ memory slot update mechanism is as follows:

$${M}_{FCP,g+1}=\left(1-c\right){\cdot M}_{FCP,g}+c\cdot {\Delta }_{Alg1}$$

(32)

In the update mechanism, the variable vector $c$ is used to adjust the size of each update. The improvement rate of different mutation operators, ${\Delta }_{Alg1}$ represents the algorithm’s performance improvement from one generation to the next and is calculated using Eq. (33).

$${\Delta }_{Alg1}=min\left(0.8,max\left(0.2,{\omega }_{Alg1}/({\omega }_{Alg1}+{\omega }_{Alg2})\right)\right)$$

(33)

where 0.2 and 0.8 represent the minimum and maximum probability values assigned to each mutation operator, respectively. This is done to ensure that both algorithms can run simultaneously, maintaining algorithmic diversity and the ability to balance exploration with exploitation. Therefore, the value of $FCP$ is always kept between 0.2 and 0.8. In addition, ${\omega }_{Alg1}$ represents the accumulation of differences between the new and old fitness values of algorithm $Alg1$ and serves as an indicator to evaluate the change in performance of each individual under this algorithm, as shown in Eq. (34).

$${\omega }_{Alg1}=\sum_{i=1}^{n} f\left({{\varvec{x}}}_{i}\right)-f\left({{\varvec{u}}}_{i}\right)$$

(34)

3 The proposed mLSHADE-SPACMA method

3.1 Motivation

The motivation to improve the LSHADE-SPACMA algorithm mainly comes from its performance bottlenecks when dealing with different optimization problems, which mainly include failing to make full use of the current population information during the iteration process, resulting in insufficient local search ability. At the same time, the lack of sensitivity analysis of the semi-parameter adaptive strategy weakens the adaptability of the algorithm in different problem scenarios. In addition, the mutation strategy sometimes fails to guide the evolution direction effectively, and the randomness of the external archive update mechanism may remove excellent individuals, affecting population diversity and overall quality of the algorithm. Therefore, by introducing a more accurate elimination and generation mechanism, improving the mutation strategy, and optimizing the external archive mechanism, the overall performance and convergence speed of the algorithm are improved to better address various optimization problems.

3.2 Precise elimination and generation mechanisms

Eliminating the poorest of the population and elite guidance have been widely used to improve the performance of algorithms (Chauhan and Shivani 2024; Chauhan and Yadav 2024a), most of which are imprecise. In mLSHADE-SPACMA, we propose a precise elimination and generation mechanism to ensure the high quality of solutions. Notably, this mechanism is operational only during the first half of the evaluations. Inspired by the “survival of the fittest” observed in natural selection, this mechanism is implemented through the following specific steps:

After a round of evaluations, individuals with poor fitness are identified for elimination. The selection mechanism for this process is detailed below:

$$PE_{m} = ceil\left( {\rho \cdot NP} \right),~~FEs < \max {{F}}s/2$$

(35)

where ${PE}_{m}$ denotes the quantity of individuals to be eliminated, with $ceil$ indicating the smallest integer greater than or equal to itself, and $\rho$ representing the percentage of the population to be eliminated. Individuals are selected for elimination based on fitness rankings, ensuring that those with the lowest performance are removed. Subsequently, an equivalent number of new individuals, summed up as ${PE}_{m}$, are generated through the following generation mechanism:

$$x_{i} = x_{{best1}} + rand \cdot \left( {x_{{best1}} - x_{{best2}} } \right),~~i \in \left\{ {1,2, \ldots ,PE_{m} } \right\},~~FEs < Max{{F}}s/2$$

(36)

where ${x}_{best1}$ and ${x}_{best2}$ represent the best and second-best ranked individuals in the current population, respectively. This design ensures that newly generated individuals not only inherit characteristics from the strongest members of the population but also enhance population diversity through the introduction of a random mutation factor, denoted as $rand$. To more clearly illustrate the precise elimination and generation mechanism, Fig. 1 is provided, detailing the entire process from elimination to generation. Evidently, the proposed mechanism not only increases the efficiency of the algorithm but also ensures the diversity and quality of the solutions. Moreover, in Fig. 2, we provide the visualization results of the proposed mechanism in this paper. Obviously, eliminating poor individuals and generating individuals guided by excellent individuals is beneficial to the algorithm in finding the global optimal solution.

Fig. 1

Schematic of the precise elimination and generation mechanisms

Fig. 2

Visualization of the precise elimination and generation mechanisms

3.3 Mutation strategy based on modified semi-parameter adaptation strategy and RSP

In this section, we propose new mutation strategies aimed at ensuring population diversity while improving the evolution direction of individuals. Firstly, we conducted parameter sensitivity analysis and modified the semi-parameter adaptive strategy of the original LSHADE-SPACMA scaling factor ${F}_{i}$, as shown in Eq. (37). The detailed experiment will be given in Sect. 4.3.

$$F_{i} = 0.5 + 0.1 \cdot rand,~~FEs < Max{{F}}s/2$$

(37)

To enhance the local exploitation capability of LSHADE-SPACMA, the RSP strategy from LSHADE-RSP is integrated. In mLSHADE-SPACMA, new mutation strategies are defined using Eq. (7), Eq. (22), and Eq. (37), culminating in a strategy detailed in Eq. (38) and named mLSHADE. Unlike LSHADE-RSP, the scaling factor ${F}_{i}$ employs an improved semi-parametric adaptive strategy to maintain population diversity. Furthermore, this study applies the RSP strategy only for ${r}_{1}$, enhancing the likelihood that ${{\varvec{x}}}_{{pr}_{1}}$ outperforms ${{\varvec{x}}}_{{r}_{2}}$, thus increasing the probability that the population’s evolutionary direction will improve. Additionally, the external archiving mechanism is fully utilized, particularly in the later stages of evolution, where random selection of ${{\varvec{x}}}_{{r}_{2}}$ can significantly boost population diversity.

$$\user2{v}_{i} = \user2{x}_{i} + F_{i} \cdot \left( {\user2{x}_{{pbest}} - \user2{x}_{i} } \right) + F_{i} \cdot \left( {\user2{x}_{{pr_{1} }} - \user2{x}_{{r_{2} }} } \right)$$

(38)

3.4 Elite archiving mechanism

In LSHADE-SPACMA, when the archive size exceeds a predetermined threshold, a subset of individuals is randomly removed, which may inadvertently eliminate high-quality solutions and reduce both the quality and diversity of the archive. To address these shortcomings, the proposed mLSHADE-SPACMA adopts an elite archiving mechanism designed to optimize algorithm performance by ensuring that only the best individuals are retained in the external archive. This approach aims to maintain both the quality and diversity of the archive and prevent the persistence of poor-quality solutions during the optimization process.

Specifically, when the archive size |A| exceeds the preset limit, the random removal method employed by LSHADE-SPACMA is replaced. Instead, individuals are screened based on their fitness values, and those with inferior fitness are selectively eliminated. This elite archive mechanism guarantees that the archive remains populated with highly fit individuals, continuously steering the search towards a more optimal solution space throughout the optimization process. This enhancement not only improves the convergence accuracy of the algorithm but also boosts the diversity of the solutions and the robustness of the algorithm.

In summary, the pseudocode of mLSHADE-SPACMA is presented in Algorithm 1.

Algorithm 1

mLSHADE-SPACMA

4 Numerical experiment

4.1 Review of CEC2014 and CEC2017

In this section, we conduct numerical experiments on the algorithm using a total of 59 test functions, which include 30 from the CEC2014 (Liang 2013) and 29 from the CEC2017 (Awad 2016) suites, detailed descriptions of these functions are available in the Supplementary materials, Tables S1, S2. Notably, Function F2 from CEC2017 has been excluded due to its instability at high dimensions. These test suites are categorized into four types of functions: Unimodal, Multimodal, Hybrid, and Composition. These categories are designed to comprehensively evaluate the exploration and exploitation capabilities of the algorithm:

• (1) Unimodal functions (F1 to F3): These have a unique global optimal solution and are used to assess the algorithm’s exploitation efficiency.

• (2) Multimodal functions (F4 to F10): These include multiple local optima, testing the algorithm’s ability to escape local minima during global searches.

• (3) Hybrid functions (F11 to F20) and Composition functions (F21 to F30): These are employed to evaluate the algorithm’s overall performance in balancing exploration and exploitation.

The mLSHADE-SPACMA algorithm was developed using Matlab 2023a, and all comparative experiments were conducted on a desktop computer equipped with an Intel Core i7-12700F, 2.10 GHz CPU, and 16GB RAM, except for L-STRDE.

4.2 Competitor algorithm parameter setting

Maintaining the original parameter settings of the algorithms is essential for ensuring a fair comparison. In the following experiments, mLSHADE-SPACMA was compared with 14 other algorithms, including LSHADE (Tanabe and Fukunaga 2014), EBOwithCMAR (Kumar 2017), jSO (Brest et al. 2017), LSHADE-cnEpSin (Mohamed et al. 2017), HSES (Zhang and Shi 2018), LSHADE-RSP (Stanovov et al. 2018), ELSHADE-SPACMA (Hadi 2018), EA4eig (Bujok 2022), L-SRTDE (Stanovov and Semenkin 2024), APSM-jSO (Li et al. 2023), LensOBLDE (Yu et al. 2024), ACD-DE (Meng et al. 2024), MIDE (Yang et al. 2024), and LSHADE-SPACMA (Mohamed et al. 2017). Detailed parameter settings for these competitor algorithms can be found in Table 1.

Table 1 Detailed parameter settings for the selected competitor algorithm

For the experiments, the remaining experimental parameters were set according to references (Liang et al. 2013) and (Wu et al. 2016), and the number of evaluations was determined to be 10,000 times the problem dimension (D). Specifically, dimensions of 10, 30, 50, and 100 for CEC2014 and CEC2017 test functions were selected for numerical experiments, with corresponding MaxFES set at 100,000, 300,000, 500,000, and 1,000,000, respectively. Additionally, each algorithm was run 51 times to ensure robust statistical validity. The results of each run were recorded and statistically analyzed using the Wilcoxon signed-rank test (Derrac et al. 2011) and the Friedman mean rank test (Friedman 1940). It is important to note that the statistical analysis in this study is based on the error, defined as the difference between the optimal value obtained by the algorithm and the global optimum of the test function.

4.3 Semi-parametric sensitivity analysis

In this section, we conduct sensitivity analysis on the semi-parameter to find more robust and superior parameter values, and change the semi-parameter adaptive strategy into the following formula:

$$F_{i}$$

(39)

where ${F}_{min}$ represents the lower limit of the scaling factor, while ${F}_{a}$ denotes the amplitude scaling factor. For the purposes of this study, we have established 10 different cases to explore various parameter settings. The detailed configurations for these cases are outlined in Table 2

Table 2 Detailed parameter settings for different cases

The Friedman mean rank test results for different cases are depicted in Fig. 3. The analysis indicates that Case 5 performs slightly worse than Cases 6 and 7 on the CEC2014 (Dim = 10) and the CEC2017 (Dim = 30). It also shows worse performance compared to Cases 6 through 9 on the CEC2014 test (Dim = 30), yet it outperforms all other cases in the remaining scenarios. Furthermore, the original algorithm, LSHADE-SPACMA (Case 3), does not yield satisfactory results, whereas Cases 5 through 8 achieve better outcomes. Notably, Case 5 records the smallest mean value in the last analysis, indicating the best overall effect. Consequently, for the mLSHADE-SPACMA configuration in this study, we adopt the semi-parametric strategy set forth in Case 5.

Fig. 3

Semi-parametric sensitivity analysis results Friedman mean rank test

4.4 Ablation experiment

In this work, three modifications are made to LSHADE-SPACMA to enhance its optimization performance. Three variants of mLSHADE-SPACMA, named mLSHADE-SPACMA1, mLSHADE-SPACMA2, and mLSHADE-SPACMA3, are designed to study the effects of the precise elimination and generation mechanism, the mutation strategy based on rank selection pressure, and the elite archiving mechanism on the performance of the proposed algorithm. In addition, the strategies are analyzed in combination, including mLSHADE-SPACMA12, mLSHADE-SPACMA13, mLSHADE-SPACMA23. The parameters of LSHADE-SPACMA and mLSHADE-SPACMA are consistent with Table 1. Note that we perform all ablation experiments based on modified semi-parameters. Detailed experimental results are presented in Tables S3-S10 in the Supplementary materials.

The Friedman mean rank test of the experimental results of different strategies is shown in Fig. 4. The results show that mLSHADE-SPACMA are worse than mLSHADE-SPACMA1 on CEC2014 (Dim = 30), mLSHADE-SPACMA3 and mLSHADE-SPACMA23 on CEC2017 (Dim = 10), and better than the six compared cases and the original algorithm in all other cases. Meanwhile, we find that mLSHADE-SPACMA1 and mLSHADE-SPACMA3 achieve good results, indicating that the third mechanism is the most effective. According to the Mean value, it can be concluded that the proposed strategy is effective and it makes sense to develop mLSHADE-SPACMA.

Fig. 4

Impact analysis of different improvement strategies

4.5 Numerical experiments based on CEC2014

The detailed experimental results of CEC2014 are presented in Tables S11–S18 in the Supplementary materials.

4.5.1 CEC2014 Wilcoxon signed-rank test

In this section, we conduct a statistical analysis of the CEC2014 run results using the non-parametric Wilcoxon signed-rank test. Detailed experimental results from the Wilcoxon signed-rank test for CEC2014 are presented in Tables S15–S18 of the Supplementary materials. Table 3 summarizes the results, comparing mLSHADE-SPACMA against 9 CEC competition winners and 4 advanced variants, including LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig, L-SRTDE, APSM-jSO, LensOBLDE, ACD-DE, and MIDE. The results are marked with ‘ + ’, ‘−’, and ‘ = ’, indicating whether mLSHADE-SPACMA performed better, worse, or similarly compared to its competitors, respectively. The detailed analysis across different dimensions is as follows:

• (1) For 10 dimension, mLSHADE-SPACMA outperforms (and is inferior to) LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig,L-SRTDE,APSM-jSO, LensOBLDE, ACD-DE, and MIDE are 11 (7), 4 (12), 13 (5), 9 (5), 18 (6), 12 (7), 7 (6), 7 (11), 12 (12), 11 (7), 20 (5), 11 (7), and 13 (5), respectively. This shows that mLSHADE-SPACMA is better than the other 9 competitor algorithms.

• (2) For 30 dimension, mLSHADE-SPACMA outperforms (and is inferior to) LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig,L-SRTDE,APSM-jSO, LensOBLDE, ACD-DE, and MIDE are 17 (2), 15 (6), 27 (0), 19 (3), 21 (7), 24 (0), 14 (4), 19 (5), 12 (17), 16 (7), 25 (4), 16 (6), and 19 (5), respectively. This shows that the developed mLSHADE-SPACMA is worse than L-SRTDE, better than the 12 competitor algorithms, and significantly outperforms jSO, and LensOBLDE (in at least 25 out of 30 cases, it performs better than the competitors).

• (3) For 50 dimension, mLSHADE-SPACMA outperforms (and is inferior to) LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig,L-SRTDE,APSM-jSO, LensOBLDE, ACD-DE, and MIDE are 22 (3), 19 (7), 28 (1), 18 (4), 19 (9), 27 (1), 15 (7), 20 (8), 13 (14), 20 (5), 23 (7), 19 (5), and 22 (3), respectively. We show that mLSHADE-SPACMA is worse than L-SRTDE, better than the other 12 competitor algorithms, and significantly outperforms jSOand LSHADE-RSP.

• (4) For 100 dimension, mLSHADE-SPACMA outperforms (and is inferior to) LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig, L-SRTDE, APSM-jSO, LensOBLDE, ACD-DE, and MIDE are 21 (3), 21 (7), 28 (2), 17 (11), 15 (14), 27 (2), 15 (6), 18 (10), 13 (15), 22 (4), 22 (8), 24 (3), and 27 (2), respectively. This shows that the developed mLSHADE-SPACMA is worse than L-SRTDE, better than the other 12 competitor algorithms, and significantly outperforms jSO, LSHADE-RSP, and MIDE.

Table 3 Wilcoxon signed-rank test results for different competitor algorithms (CEC2014)

From “Total 1” in Table 3, we observe that mLSHADE-SPACMA secures more wins than the 12 other competing algorithms, slightly worse than L-SRTDE. The “Total 2” reveals that its advantages become increasingly evident as dimensions’ increase, suggesting enhanced efficacy in handling complex problems. This trend is visually confirmed by Fig. 5, which shows mLSHADE-SPACMA’s superior performance across higher dimensions in the CEC2014 test suite. Overall, the algorithm wins 925 tests, loses 322, and draws 313, demonstrating its robustness and effectiveness.

Fig. 5

Wilcoxon signed-rank test results for CEC2014

4.5.2 CEC2014 Friedman mean rank test

In this section, we utilize the Friedman average rank test, a robust non-parametric method, to statistically analyze the performance of various algorithms on the CEC2014 test suite. Table 4 presents the results from 51 runs of multiple algorithms, including LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig,L-SRTDE,APSM-jSO, LensOBLDE, ACD-DE, and mLSHADE-SPACMA. The results, particularly evident in the last two columns of the table, highlight that our algorithm, mLSHADE-SPACMA, consistently achieved the best average ranking across all dimensions. It was closely followed by ELSHADE-SPACMA and EBOwithCMAR in performance. Additionally, Fig. 6 provides a visual representation of these rankings, illustrating the superior performance of our proposed algorithm compared to its competitors. A detailed analysis of the results is as follows:

• (1) For 10 dimension, mLSHADE-SPACMA is ranked sixth, worse than EA4eig, EBOwithCMAR, L-SRTDE, ELSHADE-SPACMA, APSM-jSO, but performing better than the other 8 competitor algorithms.

• (2) For dimensions 30, 50, and 100, mLSHADE-SPACMA ranks the best, surpassing the other 13 competing algorithms, demonstrating the superior capability of the algorithm presented in this paper in handling complex problems.

• (3) As indicated by the last line, the Friedman-p-value for all four dimensions are less than α, signifying statistically significant differences between mLSHADE-SPACMA and the other 13 competing algorithms on CEC2014.

• (4) Combining the results from Table 4 and Fig. 6, it is evident that as the dimensionality increases, the superiority of mLSHADE-SPACMA becomes pronounced. Overall, the algorithm we propose outperforms the comparative competitor algorithms on the CEC2014 test suite.

Table 4 CEC2014 Friedman mean rank test results (α=0.05)

Fig. 6

CEC2014 Friedman mean rank test results

4.6 Numerical experiments based on CEC2017

The detailed experimental results of CEC2017 are reported in Tables S19–S26 in the Supplementary materials.

4.6.1 CEC2017 Wilcoxon signed-rank test

In this section, we utilize the non-parametric Wilcoxon signed-rank test to perform a statistical analysis of the CEC2017 results. Detailed experimental results of the Wilcoxon signed-rank test for CEC2017 are reported in the Tables S23-S26 from Supplementary materials. Additionally, Table 5 summarizes the Wilcoxon signed-rank test results (α = 0.05) from 51 runs of LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig, L-SRTDE, APSM-jSO, LensOBLDE, ACD-DE, and mLSHADE-SPACMA. The following sections provide a detailed analysis of the statistical results across different competing algorithms and dimensions, as outlined below:

• (1) For 10 dimension, mLSHADE-SPACMA outperforms (and is inferior to) LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig,L-SRTDE,APSM-jSO, LensOBLDE, ACD-DE, and MIDE are 9 (6), 5 (14), 12 (3), 6 (5), 18 (6), 9 (4), 5 (6), 3 (15), 5 (13), 5 (8), 17 (7), 7 (10), and 12 (8), respectively. This shows that mLSHADE-SPACMA is worse than EBOwithCMAR, ELSHADE-SPACMA, EA4eig, L-SRTDE, APSM-jSO, and ACD-DE, better than the other 7 competitor algorithms.

• (2) For 30 dimension, mLSHADE-SPACMA outperforms (and is inferior to) LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig,L-SRTDE,APSM-jSO, LensOBLDE, ACD-DE, and MIDE are 19 (2), 12 (8 ), 27 (0), 9 (5), 15 (8), 23 (0), 12 (5), 17 (8), 6 (21), 11 (10), 26 (2), 12 (5), and 22 (4), respectively. This shows that the developed mLSHADE-SPACMA is worse than L-SRTDE, better than the 12 competitor algorithms, and significantly outperforms jSO and LensOBLDE (in at least 25 out of 30 cases, it is better than the competitors).

• (3) For 50 dimension, mLSHADE-SPACMA outperforms (and is inferior to) LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig,L-SRTDE, APSM-jSO, LensOBLDE, ACD-DE, and MIDE are 21 (5), 17 (8), 26 (0), 16 (9), 5 (16), 25 (0), 18 (6), 23 (4), 6 (23), 13 (7), 28 (1), 17 (5), and 25 (2), respectively. We show that mLSHADE-SPACMA is worse than HSES and L-SRTDE, better than the other 11 competitor algorithms, and significantly outperforms jSO, LSHADE-RSP, LensOBLDE, and MIDE.

• (4) For 100 dimension, mLSHADE-SPACMA outperforms (and is inferior to) LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig,L-SRTDE,APSM-jSO, LensOBLDE, ACD-DE, and MIDE are 24 (1), 22 (1), 27 (0), 19 (3), 8 (18), 26 (0), 21 (3), 25 (2), 7 (20), 23 (2), 27 (1), 23 (0), and 26 (2), respectively. This shows that the developed mLSHADE-SPACMA is worse than HSES and L-SRTDE, better than the other 11 competitor algorithms, and significantly outperforms jSO, LSHADE-RSP, EA4eig, LensOBLDE, ACD-DE, and MIDE.

Table 5 Wilcoxon signed-rank test results for different competitor algorithms (CEC2017)

The data from Table 5’s “Total 1” clearly shows that mLSHADE-SPACMA outperforms the other 13 CEC algorithms in terms of total wins. The “Total 2” column further reveals that while mLSHADE-SPACMA has moderate performance at 10 dimensions, its technical strengths are more apparent at higher dimensions. This trend is visually supported by Fig. 7, which illustrates mLSHADE-SPACMA’s increasing effectiveness with complex problems. On the CEC2017 test suite, the algorithm achieved a compelling record of 845 wins, 322 losses, and 341 ties out of 1508 tests, demonstrating its superior overall performance.

Fig. 7

Wilcoxon signed-rank test results for CEC2017

4.6.2 CEC2017 Friedman mean rank test

In this section, we apply the Friedman mean rank test, a robust non-parametric statistical test, to evaluate the results from the CEC2017 competition. Table 6 details the outcomes of this test for multiple algorithms, including LSHADE, EBOwithCMAR, jSO, LSHADE-cnEpSin, HSES, LSHADE-RSP, ELSHADE-SPACMA, EA4eig, L-SRTDE, APSM-jSO, LensOBLDE, ACD-DE, and mLSHADE-SPACMA, each tested across 51 runs. The data in the final two columns clearly show that our algorithm, mLSHADE-SPACMA, secures the best average rank across all tested dimensions, outperforming notable competitors such as ELSHADE-SPACMA and EBOwithCMAR. Figure 8 visually presents the ranking comparison between our proposed algorithm and the other competing algorithms. The detailed analysis results are as follows:

• (1) For the 10 dimension, mLSHADE-SPACMA ranks eighth, performing worse than EBOwithCMAR, LSHADE-cnEpSin, ELSHADE-SPACMA, EA4eig, L-SRTDE, APSM-jSO, ACD-DE, but better than other 6 competing algorithms.

• (2) For the 30 and 50 dimension, mLSHADE-SPACMA ranks second, performing worse than L-SRTDE, but better than other 12 competing algorithms.

• (3) For the 100 dimension, mLSHADE-SPACMA ranks the third, slightly worse than L-SRTDE and HSES, outperforming the other 11 competing algorithms, demonstrating the superior capability of the proposed algorithm in handling complex problems.

• (4) From the last column, it can be observed that the Friedman-p-value for all four dimensions are less than a, indicating a significant difference between mLSHADE-SPACMA and the other 13 competing algorithms on CEC2017.

• (5) Combining Table 6 and Fig. 8, it is clear that the advantages of mLSHADE-SPACMA become more and more obvious as the number of dimensions increases. Overall, on the CEC2017 test set, the performance of the proposed algorithm is worse than that of the CEC2024 champion algorithm L-SRTDE, and better than that of other comparison algorithms.

Table 6 CEC2017 Friedman mean rank test results (α=0.05)

Fig. 8

CEC2017 Friedman mean rank test results

4.7 Computational complexity analysis

In this section, we perform the complexity analysis of different algorithms using CEC2017. The computational complexity algorithm of the developed mLSHADE-SPACMA technique was determined following the CEC2017 benchmark process proposed by (Wu et al. 2016) and referred to (Hong 2024; Hong, 2023). All experiments were performed using the following platforms: (1) CPU: Intel Core i9-12900KS @ 3.40 GHz; (2) RAM: 64 GB; (3) Operating system: Windows 11; (4) Software: MATLAB 2023b.

Table 7 shows the computational complexity of the mLSHADE-SPACMA algorithm when the dimensions are 10, 30, 50 and 100, respectively. In this table ${T}_{0}$ represents the time calculated using the following statement:

Table 7 Computational complexity of mLSHADE-SPACMA and its competitors

${T}_{1}$ denotes the single run time of the benchmark function CEC2017-F18 for $2\times {10}^{5}$ evaluations in $D$ dimension space. ${T}_{2}$ represents the execution time of mLSHADE-SPACMA and its competitor algorithms on the function CEC2017-F18 for $2\times {10}^{5}$ evaluation in the $D$ dimension space. The time, ${\widehat{T}}_{2}$, is the average of ${T}_{2}$ over five runs. ${T}_{0}$, ${T}_{1}$, ${\widehat{T}}_{2}$, and $(({\widehat{T}}_{2}-{T}_{1})/{T}_{0})$. For the corresponding experimental results, see Table 7. The computational complexity of all competitor algorithms is calculated based on the same hardware and software environment. In Table 7, mLSHADE-SPACMA ranks 6th in computational complexity, lower than LSHADE, LSHADE-cnEpSin, HSES, ELSHADE-SPACMA and LSHADE-SPACMA, but better than EBOwithCMAR, jSO, LSHADE-RSP, EA4eig, APSM-jSO, LensOBLDE, ACD-DE, and MIDE. Considering the excellent performance of the proposed algorithm, the computational cost is worth it, and mLSHADE-SPACMA is still competitive among the comparison algorithms based on DE and CMA-ES.

4.8 mLSHADE-SPACMA strengths and limitations

Advantages of mLSHADE-SPACMA: It performs well in high-dimensional optimization problems, especially in the function test with dimension 100 in the CEC2014 and CEC2017 test sets, and the overall performance is better than the selected competing algorithms. At the same time, the proposed algorithm has good local exploitation capability. This is due to the introduction of the precise elimination and generation mechanism, which can better balance the global search and local exploitation capability, making its search efficiency in simple functions more efficient. In addition, the application of the elite file storage mechanism ensures the diversity of understanding, avoids the random deletion of high-quality solutions, and helps the algorithm maintain efficient convergence in the iterative process.

Limitations of mLSHADE-SPACMA: From Tables S3–S29, we can see that the algorithm developed in this paper performs well in most tests, but its global search ability is insufficient in some hybrid and composition functions, resulting in the exploration ability of the algorithm slightly worse than HSES, L-SRTDE, and LensOBLDE. In addition, the execution time is slightly higher than that of the original algorithm because the algorithm introduces the exact elimination and generation mechanism. From the Wilcoxon signed-rank test and Friedman mean rank test, we can find that our algorithm is inferior to the latest variant L-SRTDE on the CEC2017 test suite. In addition, we can also find that L-SRTDE has strong exploration performance, but the local exploitation ability can be further improved. Therefore, it is necessary to develop new algorithms with higher scalability and stronger robustness.

5 Simulation analysis of point cloud registration

5.1 Motivation

Point clouds have been utilized across numerous fields. These applications include rockfall monitoring (Bolkas et al. 2021), self-driving (Fernandes et al. 2021), three-dimensional reconstruction (Qin, 2023; Shi and Wang 2022), LiDAR odometry and mapping challenges (Guo et al. 2023; Lee et al. 2024; Lu, et al. 2023), and automatic robotic trajectory planning (Zhao et al. 2022). Among these applications, point cloud registration stands out as a crucial step. Point cloud registration involves integrating point cloud data from various temporal, spatial, and angular contexts into a unified reference coordinate system through rigid transformation (Yun et al. 2015). This process enables seamless fusion and stitching between point clouds, thereby constructing comprehensive 3D target object point cloud data. The Iterative Closest Point (ICP) algorithm, introduced by Besl et al. (1992), is widely employed in point cloud matching. It primarily accomplishes this task by establishing correspondences between two sets of points and computing the transformation matrix. However, the ICP algorithm suffers from several drawbacks, including unstable convergence, high computational complexity, sensitivity to noise and outliers, and heavy reliance on the initial position (Pottmann et al. 2006). In this section, addressing the ICP algorithm’s sensitivity to the initial position, we employ various evolutionary algorithms to optimize its initial position and conduct a comparative analysis.

5.2 Random sampling and optimization process

Point cloud random sampling is a common technique in computer graphics, computer vision, and 3D data processing. Its purpose is to randomly select a subset of points from a 3D point cloud dataset to facilitate more efficient processing, analysis, or visualization (Cheng et al. 2023). These point cloud data, widely utilized in laser scanning, 3D modeling, and remote sensing. Random sampling not only reduces computation and optimizes memory usage but also enhances data quality by eliminating some points for initial noise filtering.

In this study, random sampling is integrated with various evolutionary algorithms to develop a point cloud coarse registration algorithm. This algorithm provides an initial transformation matrix for ICP fine registration, thereby enhancing the accuracy and robustness of the ICP registration process. As depicted in Fig. 9, we utilize rabbits from the Stanford 3D Scanning Repository to visually illustrate the role of random sampling and evolutionary algorithms in the ICP registration process.

Fig. 9

ICP registration process combining random sampling and evolutionary algorithm

5.3 Fitness function design

In the coarse registration of point clouds using different evolutionary algorithms, the initial step involves randomly sampling the input point clouds. Subsequently, an objective function, also referred to as a fitness function, needs to be constructed. Here, the target point cloud is denoted as $q$, the point cloud to be transformed as $p$, and $n$ represents the number of moving point cloud points. Following random sampling, a KD-tree is employed to identify the nearest neighbor point set.

The primary objective of the evolutionary algorithm is to determine a rigid transformation matrix that optimally aligns the two-point clouds, p and q. Typically, six parameters are optimized, encompassing three translation distances and three rotation angles. Let the rotation transform be denoted as ${\varvec{R}}$, and the translation vector as ${\varvec{t}}$. The optimization objective of the evolutionary algorithm in coarse registration is to minimize the root mean square error between the corresponding points of the two sets of point clouds, as formulated in Eq. (40). The goal is to minimize the fitness value.

$$fitness = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left\| {\left. {q_{i} - \left( {p_{i} \cdot {\varvec{R}} + {\varvec{t}}} \right)_{2}^{2} } \right\|} \right.}}{n}}$$

(40)

where, ||∙|| represents the Euclidean distance between two points.

5.4 The simulation is analyzed in the fast global registration dataset

In this section, we conduct experiments using Bimba, Children, Drago, Angle, and Bunny models from the fast global registration (FGR) dataset (Zhou 2016). The dataset comprises a total of 25 cases, with each model featuring five pairs of point clouds exhibiting varying overlap rates. Similarly, for each algorithm and different point cloud dataset, we executed 51 trials and recorded the optimal value, mean value, and standard deviation of the root mean square error (RMSE) after ICP registration. The RMSE was calculated as shown in (Li et al. 2022a, b; Zhang et al. 2022). We compare these results with the original ICP (Besl and McKay 1992) and with the evolutionary algorithms LSHADE (Tanabe and Fukunaga 2014), EBOwithCMAR (Kumar, et al. 2017), jSO (Brest, et al. 2017), LSHADE-cnEpSin (Mohamed, et al. 2017), HSES (G. Zhang and Shi 2018), LSHADE-RSP (Stanovov, et al. 2018), ELSHADE-SPACMA (Hadi, et al. 2018), EA4eig (Bujok, et al. 2022), APSM-jSO (Li et al. 2023), LensOBLDE (Yu et al. 2024), ACD-DE (Meng, et al. 2024), MIDE (Yang, et al. 2024), LSHADE-SPACMA (Mohamed et al. 2017), and mLSHADE-SPACMA.

$$rmse = \sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left\| {R^{*} qi} \right.} } \left. { + t^{*} - Rq_{i} - t} \right\|_{2}^{2}$$

(41)

where $({{\varvec{R}}}^{*},{{\varvec{t}}}^{*})$ represents the true transformation matrix between point clouds, and $({\varvec{R}},{\varvec{t}})$ denotes the registration results of different algorithms.

Table 8 presents the mean (Ave) and standard deviation (Std) of the registration results after the initial ICP transformation, optimized by various competitor algorithms. The best results are highlighted in bold, illustrating that our algorithm consistently achieves the lowest values in most cases. To visually demonstrate the optimization results, Figs. 9, 10, 11, 12, 13 depict the registration outcomes for the Bimba, Children, Drago, Angle, and Bunny point clouds. The first column provides the input point cloud data, which varies in point number, overlap rate, and positioning. The last column uses log-scale color coding to highlight deviations between the transformed source point cloud, as aligned by the computed transformations, and the ground truth alignment. The intervening images show the registration results from different algorithms, with each figure displaying the best registration run for each algorithm. While the optimized results from each algorithm show improvement over the unoptimized ICP, the differences in registration quality among the 14 optimized algorithms are subtle. However, when considered alongside the data from Table 8, it is clear that our algorithm generally exhibits superior robustness (Fig. 14).

Table 8 Results of different algorithms for optimizing ICP registration

Fig. 10

Registration results of different algorithms on Bimba

Fig. 11

Registration results of different algorithms on Children

Fig. 12

Registration results of different algorithms on Dragon

Fig. 13

Registration results of different algorithms on Angle

Fig. 14

Registration results of different algorithms on Bunny

6 Summary and prospect

In this paper, we proposed an improved variant of the differential evolution algorithm, namely, modified LSHADE-SPACMA (mLSHADE-SPACMA), aimed at enhancing the global exploration and local exploitation capabilities of numerical optimization and point cloud registration. By introducing a precise elimination and generation strategy, a new mutation strategy based on modified semi-parametric and RSP, and an elitist external archive mechanism, mLSHADE-SPACMA improves both global exploration and local exploitation capabilities. Experiments on CEC2014 and CEC2017 test suites confirm the superiority of the proposed algorithm over the original LSHADE-SPACMA and other high-performance optimizers, except that it is inferior L-SRTDE on CEC2017. Additionally, by optimizing the initial transformation matrix of the classical point cloud registration algorithm ICP, improved point cloud alignment technology is demonstrated. Given the excellent performance of mLSHADE-SPACMA, future research could explore the following directions:

• (1) Apply mLSHADE-SPACMA to other optimization fields, such as human–computer collaboration, image segmentation, parameter identification, and others.

• (2) Combine the proposed strategy with other advanced algorithms or introduce other advanced mechanisms to develop a more powerful optimizer.

• (3) Develop multi-objective and binary versions of mLSHADE-SPACMA to solve other complex optimization problems.

• (4) Integrate with other point cloud registration technologies to develop a more efficient and stable point cloud registration technology.