State information-driven surrogate-assisted differential evolution for computationally expensive constrained optimization problems

Abstract

In this paper, a state information-driven surrogate-assisted differential evolution called SI-SADE is proposed for solving expensive constrained optimization problems, in which both the population state and adaptive search mechanism are respectively evaluated and designed based on the feasibility and state information. Firstly, the multiple subpopulations are obtained by comprehensively considering the three different population states, i.e., infeasible, partially feasible, and fully feasible, and the diversified indicators of population individuals. Secondly, different ensemble mutation and environmental selection operations are tailored specially for subpopulations where both an inner evolution-driven parent expansion and update rate-based surrogate switch strategies are designed to regulate the search ability of the algorithm. Furthermore, to bypass the hard obstacles caused by complex constraints, a pure objective-based search rectification is used to locate the possible feasible region in the direction of minimizing objective value. Therefore, the SI-SADE achieves an adaptive balance between feasibility and convergence. Systematic experimental results on both the IEEE CEC2010 and CEC2017 benchmark problems demonstrate the high competitiveness of SI-SADE. More importantly, the SI-SADE performs excellently in solving a real-world case.

Introduction

Expensive constrained optimization problems (ECOPs) refer to problems where the evaluation of the objective or constraint functions is computationally expensive, such as aerodynamic optimization [1] and structural optimization [2]. It is a significant challenge for algorithms to find the global optimal solution of ECOPs under the limited function evaluations (FEs). In general, ECOPs with inequality constraints can be formulated as follows [3, 4]:

1

$$\begin{array}{cc}\text{minimize}f(\text{x}),\hfill & \text{x}\in S\\ subject\text{to}{g}_j(\text{x})\le 0,\hfill & j=1,\dots ,q\end{array}$$where $\text{x}=(x_1,x_2,\dots ,x_n)$ denotes the decision vector, bounded by their minimum and maximum values such as $L=(l_1,l_2,\dots ,l_n)$ and $U=(u_1,u_2,\dots ,u_n)$, respectively. $f(\text{x})$ is objective function to be minimized.$g_j(\text{x})$ is the $j-\text{th}$ inequality constraint. The degree of constraint violation of x on the j-th constraint can be formulated as follows:

2

$$\begin{array}{cc}{G}_j(\text{x})=max\left\{g_j(\text{x}),0\right\},\hfill & j=1,\dots ,q\end{array}$$

Then, the value of the overall constraint violation function at a specific point x can be calculated as

3

$$G(\text{x})=\sum _{j=1}^q{G}_j(x)$$

It is evident that x is feasible when $G(\text{x})=0$.

Evolutionary algorithms (EAs) are effective at solving complex optimization problems, particularly those with large search spaces and complex functions. However, EAs typically require a substantial number of function evaluations to fully exert the search performance for the optimization problem, so using traditional EAs to solve ECOPs is usually inefficient [5]. To address this issue, surrogate-assisted evolutionary algorithms (SAEAs) have been introduced to solve ECOPs. A surrogate model is a mathematical model that uses expensive sample data to approximate the objective and constraints of the expensive problem [6]. In SAEAs, the predicted values from the surrogate model replace a portion of the real FEs in ECOPs, thereby conserving computational resources [7]. Moreover, the collaborative design of the evolutionary and selection strategies, the modeling mechanism of surrogate model and the utilization of objective information are the key factors that affect the performance of SAEAs. In terms of balancing between exploration and exploitation, the rational allocation of evolutionary and selection strategies can establish dynamic equilibrium in the search process. Surrogate model modeling strategies can be categorized into fixed modeling and dynamic adaptive modeling, where fixed modeling adopts the fixed execution or periodic alternating execution of global and local surrogate models. For constrained optimization problems, incorporating partial objective information during iterations facilitates the algorithm to break the search barriers formed by complex constraints. Notably, state information generated at the optimization process can serve as a guide for adaptive search adjustments. This feedback-driven dynamic regulation approach not only enhances algorithmic robustness but also represents a crucial research direction for improving optimization efficiency.

In general, the state information available in the optimization process can be categorized into two types. One type is the individual state information derived from real FEs or simple calculations, including constraint violation value, minimum distance to other individuals and so on. The other type refers to easily obtainable population state information, including feasibility rate, update rate, and diversity. Individuals possessing valuable information are typically utilized to guide the current population toward more promising regions. Wang et al. [8] selected two different individuals, such as the individual with the smallest constraint violation and the individual with the smallest objective value, as guiding individuals in DE/rand-to-best/1 and DE/current-to-rand/1, respectively. Similarly, Yang et al. [9] designed a comprehensive mutation operation that used beneficial information from high-quality solutions and adequately exploited potential information hidden in bad solutions. Furthermore, many algorithms [10, 11–12] have used the population’s state information to enhance the algorithm’s robustness. Feasibility rate is the most commonly used population state parameter. Wang et al. [13] designed an adaptive trade-off model (ATM) where the population is divided into three situations based on the feasibility rate. In the infeasible situation, non-dominated sorting (NDS) [14] is employed to select individuals with fewer violated constraints, guiding them toward the feasible region from various directions. In the semi-feasible situation, the fitness function is adjusted based on the feasibility rate to modify the survival probability of feasible and infeasible solutions. In the feasible situation, the method of individual selection is solely based on the objective value. Subsequently, Wang and Cai [15] designed the improved adaptive trade-off model (IATM), which incorporates diversity in the infeasible situation. Moreover, Gong et al. [16] and Jia et al. [17] have also improved ATM in various aspects. Different from the ATM's strategy which primarily relies on feasibility rate to regulate selection methods, Chu et al. [18] adaptively adjust the mutation operation through state information such as population feasibility rate and optimal solution update information. Similarly, Li et al. [19] incorporated fitness value improvement information from the optimal individual. If the improvement is insufficient, the algorithm is judged to be in a premature convergence or stagnation state according to the population distribution, and the population is dispersed or aggregated by designing an evolutionary strategy. Based on the above research, state information is predominantly used to guide the application of evolutionary or selection strategies, playing an indispensable role in enhancing the algorithm's performance. However, most studies do not make full use of state information, often relying on only a single type of individual or population state information.

The surrogate model can be constructed in a global or local way. For the global way, most algorithms [9, 20, 21] utilized all the real evaluated solutions in the archive to construct a global surrogate model. Such a global surrogate can capture the overall trend of the function and locate the promising regions [22, 23]. However, constructing an accurate global surrogate model presents challenges [24], and global surrogate models typically exhibit slow convergence [23]. In contrast, local surrogates can improve accuracy and convergence speed within local regions [22, 25]. Consequently, Li and Zhang [25] proposed a method characterized by multiple penalties and multiple local surrogates (MPMLS). In MPMLS, an adaptive penalty factor is designed, and each penalty factor corresponds to a subproblem and a subregion. For each subproblem, the individual with the optimal fitness value is designated as the subregion’s center, around which a local surrogate model is constructed. Experimental results indicate that MPMLS’s strategy of partitioning the space into subregions and constructing local surrogates mitigates inaccurate predictions of surrogate models. Nevertheless, MPMLS does not tackle the issue of local surrogates becoming trapped in local optima. To leverage the benefits of both global and local surrogates, Wang et al. [22] introduced a method in which differential evolution is integrated with global and local surrogates (GLoSADE). In the first stage of GLoSADE, the generalized regression neural network is employed to construct the global surrogate and explore the design space. During the second stage, a local surrogate is utilized to further refine the promising solutions identified in the first phase. The alternating construction of global and local surrogates promotes a balance between exploration and exploitation. Similar to GLoSADE, Liu et al. [7] adopted a two-phase approach: initially employing a global surrogate for design space exploration until the population becomes feasible and then switching to sequential execution of global and local surrogates to exploit the feasible region. Furthermore, studies in [18, 26, 27] have also incorporated a combination of global and local surrogates into their algorithmic frameworks.

In recent years, considerable attention has been focused on the dynamic adaptive surrogate model construction strategy, and some researchers have considered using state information to guide the construction of surrogate models. For instance, Wang et al. [28] utilized the feasibility rate to adjust the frequency of global and local search dynamically. When the population feasibility is low, the local search is performed more frequently to drive the population into the feasible region. Conversely, when feasibility is high, global searches are prioritized to enhance the overall population quality. However, the algorithm assumes that the population transitions from infeasible to feasible, limiting its applicability in situations with complex constraints or a large feasible region but intricate objective functions. Pan et al. [29] dynamically applied three search modes—global search, elite-based local search, and opposition-based search [30] based on the optimal solution’s update state and the population update rate after each search mode’s application. The above algorithm has efficiently enhanced the adaptability by applying a dynamic adaptive adjustment surrogate model. Different from the above studies, Wu et al. [31] did not use state information to guide the selection of individuals involved in the construction of the surrogate model, but used the k-fold cross-validation method to select the kernel with the highest prediction accuracy from the five widely used kernels to construct the model. However, most of the current research applies a fixed model construction mode, and there are still deficiencies in the systematic exploration of dynamic adaptive model modeling.

Furthermore, most algorithms prioritize feasible solutions, accelerating the population’s convergence to the feasible region while often overlooking the valuable objective information in infeasible solutions [32]. Infeasible solutions with favorable objective information have been shown to be beneficial in solving ECOPs [33]. Therefore, the importance of infeasible solutions is emphasized. For example, Jiao et al. [34] controlled the proportion of feasible and infeasible solutions in the first stage, approaching the global optimum from both the infeasible and feasible regions. In the second stage, the feasibility rule (FR) [35] was applied to push the whole population to the feasible region. Unlike the above study, to help the population overcome the infeasible obstacles, numerous multi-objective algorithms [36, 37–38] initially disregard constraints to reach the unconstrained Pareto front, subsequently incorporating constraints to guide the population toward the constrained Pareto front. Moreover, the correlation between constraints and objective optimization is a state information indicator that is often used to guide optimization recently. Song et al. [39] proposed a method that utilizes this correlation indicator to determine the necessity of constructing constraint surrogate models for enhancing the current evolutionary search process. Similarly, Wang et al. [40] used this correlation indicator to design a fitness function that incorporates objective information by balancing the weights of objective values and constraint violation values. Recent studies indicate a growing interest in approaches that navigate toward the feasible region while simultaneously optimizing objectives.

Based on the preceding review of ECOP research, existing research makes limited use of state information, and most studies focus on only one aspect among the evolutionary strategy, selection method, surrogate model construction, or utilization of objective information. Therefore, we propose a State Information-driven Surrogate-Assisted Differential Evolution (SI-SADE) for solving computationally expensive constrained optimization problems. Specifically, SI-SADE comprehensively utilizes both population state information, such as feasibility and update rate, and individual state information, including the individual update state and the minimum distance to other individuals. Furthermore, SI-SADE uses state information to guide the adjustment of strategies, including evolutionary and selection strategies, adaptive modeling of surrogate models, and utilization of objective information. In this way, SI-SADE achieves a balance between feasibility and convergence. The principal contributions of this study can be summarized as follows:

1. An adaptive sub-population search framework is designed to dynamically adjust evolutionary and selection strategies based on the population’s state. Specifically, based on the population’s feasibility rate, three distinct states are defined: infeasible, partially feasible, and fully feasible. This classification allows for the design of tailored optimization strategies for each state. To balance search resources, sub-populations are formed from the main population based on individual information. Then, targeted combinations of evolutionary strategies and environmental selection methods are assigned to these classified sub-populations to emphasize either exploration or exploitation.

2. A search mechanism based on population feasibility and update rate is designed to improve the search ability of the SI-SADE. Specifically, the parents will be expanded to adjust the distribution of offspring during the early stage or when the population is in the infeasible state. Moreover, for the population in the partially feasible or fully feasible state, a global surrogate is built when the population update rate falls below θ. For the population in the fully feasible state and at the later stage of optimization, the local and global surrogates are switched in real time based on their actual optimization outcomes.

3. In the infeasible state, to overcome the hard obstacles posed by complex constraints, the algorithm initially conducts a constraint-driven search. Subsequently, an objective-driven search is applied to a subset of individuals in the current population. This approach enhances the acquisition of objective information within the infeasible region.

Background methods and related techniques

This section outlines essential techniques employed in this study, including radial basis function, differential evolution, and method for handling constraints.

Radial basis function (RBF)

RBF [41] is a simple interpolation function. The basic principle is to approximate the complex function relationship by a linear combination of a set of basis functions.

Given a set of $N$ distinct points ${\text{x}}_1,\dots ,{\text{x}}_N\in R^n$ and their corresponding function values $f({\text{x}}_1),\dots ,f({\text{x}}_N)\in R$, the RBF can be expressed by the following mathematical expression:

4

$$\widehat{f}\left(\text{x}\right)={\text{w}}^T\phi =\sum _{i=1}^N{\omega }_i\varphi \left(∥\text{x}-{\text{x}}_i∥\right)$$${\text{x}}_i$ represents the center of the i-th basis function, while x denotes the point being predicted. The notation $∥·∥$ signifies the Euclidean norm, which quantifies the distance between vectors. $\varphi \left(·\right)$ represents the basis function, defining the mapping from input space to feature space. Additionally, ${\omega }_i$ corresponds to the weight associated with each basis function. The cubic form $\varphi \left(r\right)=r^3$ is utilized in this paper due to its successful application in various SAEAs [9, 25, 26]. Furthermore, $\phi$ is an N × 1 vector containing the values of the basis function $\varphi \left(·\right)$, and w can be calculated as:

5

$$\text{w}={\left({\mathrm{\Phi }}^T\mathrm{\Phi }\right)}^{-1}{\mathrm{\Phi }}^T\text{y}$$

In the above equation, $y={\left(f\left({\text{x}}_1\right),...,f\left({\text{x}}_N\right)\right)}^T$, $\mathrm{\Phi }$ denotes the so-called Gram matrix, which is defined as follows:

6

$$\mathrm{\Phi }=\left[\begin{array}{ccc}\varphi \left(∥{\text{x}}_1-{\text{x}}_1∥\right)& \cdots & \varphi \left(∥{\text{x}}_1-{\text{x}}_N∥\right)\\ ⋮& \ddots & ⋮\\ \varphi \left(∥{\text{x}}_N-{\text{x}}_1∥\right)& \cdots & \varphi \left(∥{\text{x}}_N-{\text{x}}_N∥\right)\end{array}\right]$$

Differential evolution (DE)

As one of the important EAs, DE [8, 11] significantly improves its global search ability and convergence speed by introducing differential mutation operation. Its basic operations include mutation, crossover, and selection. The specific process is to generate new candidate solutions through the differences of individuals in the parent.

Given a n-dimensions target vector as follows:

7

$${\text{x}}_i=\left({\text{x}}_{i,1},\dots ,{\text{x}}_{i,n}\right)$$

The mutant vector ${\text{v}}_i=\left(v_{i,1},\dots ,v_{i,n}\right)$ is generated based on a mutation operator for each target vector ${\text{x}}_i$ in the current population. In this study, the four variants of DE are used to generate candidate offspring because of its powerful global search ability. The four mutation operators can be expressed as follows:

1. DE/rand/1.

   8

   $${\text{v}}_i={\text{x}}_{r1}+F({\text{x}}_{r2}-{\text{x}}_{r3})$$

2. DE/rand/2.

   9

   $${\text{v}}_i={\text{x}}_{r1}+F_1({\text{x}}_{r2}-x_{r3})+F_2({\text{x}}_{r4}-{\text{x}}_{r5})$$

3. DE/random-to-random/1.

   10

   $${\text{v}}_i={\text{x}}_{\mathit{rand}}+F_1({\text{x}}_{r1}-{\text{x}}_{\mathit{rand}})+F_2({\text{x}}_{r2}-{\text{x}}_{r3})$$

4. DE/current-to-rand/1.

   11

   $${\text{v}}_i={\text{x}}_{\mathit{cur}}+F_1({\text{x}}_{r1}-{\text{x}}_{\mathit{cur}})+F_2({\text{x}}_{r2}-{\text{x}}_{r3})$$

In the above equations, ${\text{x}}_{\mathit{rand}}$, ${\text{x}}_{r1}$, ${\text{x}}_{r2}$, ${\text{x}}_{r3}$, ${\text{x}}_{r4}$ and ${\text{x}}_{r5}$ are six randomly solutions chosen from parents, and ${\text{x}}_{\mathit{cur}}$ is the selected specific solution. $F_1$ and $F_2$ are scaling factors. Following the mutation operator, the crossover is applied to the target vector ${\text{x}}_i$ and mutant vector ${\text{v}}_i$ to generate the trial vector $u_i$. The binomial crossover is expressed as follows:

12

$$u_{i,j}=\left\{\begin{array}{cc}{v}_{i,j}\hfill & ifran{d}_j\le CRorj=j_{\mathit{rand}}\\ x_{i,j}\hfill & \mathit{otherwise}\end{array}\right),j=1,\dots ,n$$

In the above formula, $j_{\mathit{rand}}$ is a randomly selected integer from the set $\left\{1,\dots ,n\right\}$, $ran{d}_j$ represents a randomly generated number within the specified interval such as [0,1]. CR is the crossover probability parameter. Subsequently, the selection operator is applied to determine the superior solution between the parent vector x_i and the trial vector u_i for progression to the next generation.

Constraint handling

The ε-constraint (EC) method proposed in [42] is used to introduce constraint conditions of the original problem into a relaxation parameter ε to form a new relaxation constraint. This means that the original constraint $G(\text{x})\le 0$ is replaced by $G(\text{x})\le \epsilon$. The calculation of ε value is shown as follows:

13

$$\epsilon (0)=G({\text{x}}_{\theta })$$

14

$$\epsilon (gen)=\left\{\begin{array}{cc}\epsilon (0){\left(1-\frac{\mathit{gen}}{\mathit{Gen}}\right)}^{\mathit{cp}},\hfill & 0<gen<T_c,\\ 0,\hfill & \mathit{otherwise}\end{array}\right)$$

15

$$cp=-\frac{log(\epsilon (0))+\kappa }{log(1-T_c/Gen)}$$

The initial ε level, denoted as ε(0), is defined as the constraint violation value of the top θ-th individual at initialization. The ε level is updated after each iteration until the number of iterations reaches T_c. gen is the current iteration number, and Gen is the maximum number of iterations. Moreover, the value of κ directly affects the attenuation trend of ε level. In this study, EC was adjusted. This makes the downward trend of the ε value easier to control. The details are as follows:

16

$$\epsilon (0)=G({\text{x}}_{\theta })$$

17

$$\epsilon \left(\mathit{gen}\right)=\left\{\begin{array}{cc}\epsilon (0){\left(1-\frac{\mathit{gen}}{T_c}\right)}^{\mathit{cp}},\hfill & 0<gen<T_c\\ 0,\hfill & \mathit{otherwise}\end{array}\right)$$

18

$$cp=-\frac{log(\epsilon (0))+\kappa }{log(1-T_{\mathit{lambda}}/T_c)}$$

In this study, θ, T_c, T_lambda, and κ are set to 0.4*NP, 0.4*Gen, 0.95*T_c, and −log(e⁻⁵), respectively.

When x and y are compared, it is judged that x is better than y if and only if the following conditions are satisfied:

19

$$\left\{\begin{array}{cc}f(\text{x})<f(\text{y}),\hfill & ifG(\text{x})\le \epsilon \&G(\text{y})\le \epsilon \\ f(\text{x})<f(\text{y}),\hfill & ifG(\text{x})=G(\text{y})\\ G(\text{x})<G(\text{y}),\hfill & \text{otherwise}\end{array}\right)$$When ε is 0, the ε-constraint method is equivalent to FR.

In addition, there are many classical constraint handling techniques, such as stochastic ranking [43], penalty function methods [44], and so on. More constraint handling techniques can be found in [45].

Proposed algorithm

In this section, the motivation for this work is given first. Subsequently, “Main Framework of SI-SADE” offers the main framework of SI-SADE. The rest sections present the details of the proposed SI-SADE.

Motivation

Balancing exploration and exploitation capabilities is essential for an effective DE variant. Unbalanced parameters may lead to either premature convergence or stagnation [19]. Some evolutionary operators and control parameter settings of DE are conducive to global search, while others facilitate localized refinement [46]. In addition, it is observed that the local surrogate demonstrates higher prediction accuracy within its local region and offers greater computational efficiency [25]. Thus, to intensify the exploitation in the regions identified by high-potential solutions while simultaneously enhancing exploration efforts for less promising solutions, an adaptive subpopulation search framework (ASSF) is constructed. Specifically, the ASSF uses local surrogates within the main optimization process, and a feasibility rate-driven population division (FRPD) is designed to divide subpopulations. To further improve search performance under finite FEs, two challenges remain: (1) How can exploration abilities be enhanced in generating and selecting high-quality solutions? (2) how to locate the feasible region in a complex constraint environment?

The first challenge can be addressed by designing the state information-driven search mechanism (SISM) through the two aspects. For generating high-quality solutions, an inner evolution mechanism is employed in parent expansion (PE) to increase the diversity and quality of the parent subpopulation. Thus, the distribution of offspring solutions is improved. Subsequently, for selecting high-quality solutions, a suitable global surrogate switching mechanism should be designed to provide effective global prescreening capabilities. However, many SAEAs directly construct the global surrogates using all individuals in the archive. This approach can be computationally expensive when the archive contains numerous individuals, thereby decreasing the whole computational efficiency. Hence, the construction of global surrogates necessitates careful consideration. Furthermore, existing studies [12, 28] use the feasibility rate as the only indicator to determine whether the optimization is stagnant. This approach focuses primarily on constraints, while the challenges associated with objective optimization are overlooked. Hence, it is crucial to consider additional triggered indicators when switching to global surrogates. Considering these observations, an update rate-driven adaptive modeling (URAM) approach is proposed to address these challenges.

For the second, the existing studies predominantly favor feasible solutions, prioritizing constraints instead of objectives. This bias may lead the population into complex infeasible subregions, particularly when using local surrogate models. Hence, studies have shown a growing interest in utilizing objective information in recent years. However, most studies [39, 40] controlled the degree of utilizing objective information through the correlation between the objective and constraints. This approach only captures the overall trend within the occupied space of the population because it uses either the current population or the entire archive to measure the correlation. Notably, there is a lack of research focusing on optimizing objectives along the path of constraint optimization. Thus, a pure objective-driven search rectification (POSR) is proposed when the current parent population is infeasible.

Main framework of SI-SADE

To clearly explain the main framework of SI-SADE, Fig. 1 presents the flow chart of SI-SADE. SI-SADE can be roughly divided into the following eight steps:

1. Initialization: The Latin hypercube sampling (LHS) [47] is utilized to generate NP individuals in the design space. These individuals are then evaluated truly, and their corresponding objective and constraint values are stored in Data. Finally, the parameters are initialized.

2. Feasibility Ratio-driven Population Division: We first evaluate the feasibility rate of the entire population and divide it into three states: infeasible, partially feasible, and fully feasible. For different population states, diverse individual information is employed to divide the population. Details are in “Offspring generation and selection”.

3. Parent Expansion: We perform different operations for different population states. To be specific, when the population state is partially feasible or fully feasible and the FEs are less than the maximum number of points required for local modeling, the parent population is expanded. In addition, when the population state is infeasible, the parent population is expanded in each iteration.

4. Model Management: RBF is used for modeling here, and a local surrogate model is built when the population is in the infeasible state. In addition, when the population is in a partially feasible or fully feasible state, URAM is performed to achieve adaptive modeling, which is introduced in “Update rate-driven adaptive modeling (URAM)”.

5. Offspring Generation: different evolutionary strategies that emphasize either exploration or exploitation are conducted according to the characteristics of subpopulations derived from FRPD. Details are in “Feasibility ratio-driven population division (FRPD)”.

6. Offspring Selection: different selection ways are utilized for various subpopulations.

7. Population Update: FR is used to compare the selected individuals with the parent to update the population. In particular, when the population state is infeasible, POSR will be performed after updating the population, and then the population will be updated again. The specific details are in “Pure objective-driven search rectification (POSR)”. Finally, the parameters are updated.

8. Termination Criterion: Repeat steps (2)–(7) until the maximum number of function evaluations or other termination conditions in the real-world application is met.

[See PDF for image]

Fig. 1

Flowchart of the proposed algorithm. Both the FRPD and specific strategic settings, such as the ways to generate and select offspring, constitute the adaptive sub-populations search framework (ASSF). The notations of other characters are shown as below: U, V, W: obtained sub-populations; Pop: the current population; NDS: non-dominated sorting; CV: constraint violation value; NC: number of constraints satisfied; Obj: exact objective values of individuals; Dist: negative distance from the nearest individual; FEs: the current consumed number of function evaluations; r: the scaling parameter for determining modeling points; min(120, r*n): maximum number of modeling points required for local surrogates; Global: the label for conducting global mutation operations; Local&Global: the label for performing both local and global mutation operations. The arrows of the three colors (red, green and blue) correspond to the algorithm execution flow under the three population states respectively, and the black arrow is the common path

Algorithm 1 gives the SI-SADE’s pseudo-code. The concise step-by-step representation of the pseudo-code is shown below.

[See PDF for image]

Algorithm 1:

SI-SADE

In lines 1–9, all key parameters are initialized. Gen counts the number of iterations in the optimization process, while TriggerGen indicates the generation at which the global elite search surrogate triggers. The individual update state vector is represented by R. URmean calculates the average update rate between two sampling periods. The globalflag indicates the switching of global surrogates with elite search, and URglobalGen marks the generation corresponding to the sampling update rate. θ defines a low update rate threshold, while URinterval specifies the update rate sampling interval. URflag acts as the global surrogate switching flag for optimization stagnation. Lastly, ω denotes the population size coefficient in POSR.

In lines 11–36, the strategy allocation of offspring generation and selection is presented. U, V, and W are divided from the parent population based on various state information of the population and individuals in FRPD. OffType and Offscreen are two labels that represent the ways to conduct corresponding strategies in SISM for these sub-populations. Specifically, Local&Global represents the hybrid evolutionary strategies including DE/rand/1, DE/rand/2, and DE/current-to-rand/1; Global means the hybrid evolutionary strategies including DE/rand/1, DE/rand/2, and DE/random-to-random/1. Thus, targeted exploitation and exploration strategies for local and global regions are arranged for the corresponding sub-populations.

In line 37, POSR is performed on the infeasible population after each iteration. In lines 38–41, the x_best is recorded and the parameters are updated, where the update rate is calculated by dividing the number of successfully updated individuals by NP.

Adaptive sub-populations search framework (ASSF)

ASSF consists of four parts: population feasibility evaluation, subpopulation division, offspring generation, and offspring selection. “Feasibility ratio-driven population division (FRPD)” gives the details of the population feasibility evaluation and subpopulation division. “Offspring generation and selection” provides details of offspring generation and offspring selection.

Feasibility ratio-driven population division (FRPD)

When addressing ECOPs, the feasibility state of the population does not always transition from completely infeasible to fully feasible, as it depends on the size of the feasible region and the complexity of constraints. If the feasible region is large and the objective function is complex, then the existing algorithms for complex constraint problems may exhibit limited performance on such issues. This suggests that a tailored compromise strategy is required for each population feasibility state, enabling dynamic guidance of the search direction in different states.

As shown in Algorithm 2 lines 1–8, The feasible state of the population is divided into:

1. infeasible: when all individuals of the population are infeasible.

2. partially feasible: when some individuals of the population are feasible.

3. fully feasible: when all individuals are of the population feasible.

Remark:

While the ASSF method categorizes the population into three states based on feasibility rates, similar to ATM, the algorithm strategies for each state differ significantly. As the state transitions from infeasible to fully feasible, ATM shifts the environmental selection pressure from constraints to objectives by designing specific selection strategies. Unlike ATM, ASSF allocates search resources more efficiently by integrating various evolutionary and selection strategies.

Generally, the overall characteristics of the population can be grasped by dividing the feasibility state, while the features of local areas can be captured by utilizing valuable individual information. Thus, SI-SADE fully uses individual state information based on the population's feasibility state.

In the fully feasible state, to prevent the population from converging to the local optimum of the objective function due to over-exploitation, the updated state information of individuals is considered. As presented in Algorithm 2 lines 11–17, Individuals that are successfully updated in the last generation are stored in U; otherwise, they are stored in V. Successfully updated individuals will be reinforced on exploitation, motivating them to find the optimal solution in their regions until they stop updating; For individuals not successfully updated, the focus shifts to exploration, considering two aspects: (1) enabling full exploration of the feasible region, and (2) helping individuals trapped in local optima escape from their dilemmas.

[See PDF for image]

Algorithm 2:

Feasibility Rate-driven Population Division

The partially feasible state plays a pivotal role in the optimization process, but many algorithms tend to push the entire population into the feasible region quickly while ignoring the search around the feasible region boundary. In Algorithm 2 lines 18–21, the feasible individuals are stored in U, while the remaining infeasible individuals are ranked based on NDS, where three metrics—objective value, constraint violation, and minimum distance, are treated as three objectives. The top half of the individuals with better ranking values are stored in V, and the last half is stored in W. For the subpopulation U, we emphasize exploitation because predictions within the feasible region may not be sufficiently accurate at this stage. Therefore, we aim for individuals in U first search regions where they are located. In addition, for subpopulation V, the information contained in these individuals may be more valuable than that of feasible solutions. There are three purposes to strengthen the exploitation of V: (1) CV: to promote individuals with small constraint violation values to enter the feasible region; (2) Obj: to optimize the objective value around the feasible region boundary; (3) Dist: to explore sparse regions to prevent sub-feasible regions from being ignored.

In the infeasible state, like most algorithms, SI-SADE aims to locate the feasible region. As shown in Algorithm 2 lines 22–24, the solutions are ranked by sequentially comparing two indicators, such as the constraint violation and the number of satisfied constraints. The top half of individuals are stored in V, while the rest are stored in W.

Offspring generation and selection

For offspring generation, exploration and exploitation should be integrated into the evolutionary strategy to improve the algorithm’s robustness. Hence, this paper introduces hybrid methods to fully utilize different DE mutation operations. Specifically, we employ three operations, DE/rand/1, DE/rand/2, and DE/random-to-random/1, to explore global space comprehensively. These approaches ensure promising regions are not overlooked and help to prevent getting trapped in local optima. Conversely, employing DE/current-to-rand/1 enables algorithm to shift focus to a more localized search space. This strategy allows SI-SADE to concentrate computational efforts on a specific region of interest, typically around a promising solution.

For offspring selection, FR and EC are used to select offspring on feasible and infeasible individuals, respectively. FR is a relatively greedy selection rule. Although it does not introduce parameters, it emphasizes constraints and does not incorporate objective information. Hence, FR is used here to encourage feasible individuals to explore within the feasible region. Moreover, EC can partially utilize objective information, facilitating exploration around the feasible region, especially when the global optimal solution is located at the boundary of the feasible region. Therefore, EC is used to increase the survival probability of infeasible solutions with excellent objective values.

State information-driven search mechanism (SISM)

[See PDF for image]

Algorithm 3:

State Information-driven Search Mechanism

The pseudo-code of SISM is presented in Algorithm 3. In line 1, Neighbor is used to construct a local surrogate model and to serve as an evolutionary parent population. In lines 3–9, when the population state is fully feasible or partially feasible, and if FEs is less than the maximum number of points required for local surrogate modeling, PE will expand the current population Pop; otherwise, URAM will be performed. Additionally, in lines 10–12, if there is no feasible individual in the population, then PE will be performed to expand the Neighbor. The details of PE and URAM can be found in “Parent expansion (PE)” and “Update rate-driven adaptive modeling (URAM)”, respectively. In lines 13–19, mutation operations and selection rules defined in the ASSF are used to generate candidate offspring and select the exact offspring, respectively. Subsequently, the exact offspring X_best is evaluated, and the population is updated.

Parent expansion (PE)

As illustrated in Fig. 2, the original parent population is expanded using DE to create an additional population that is three times its size. This expanded population is then combined with the original parent population to form a new, larger parent population. Subsequently, this combined population is used to guide the generation of offspring.

[See PDF for image]

Fig. 2

Schematic diagram of parent expansion. The green circles represent the individuals in the original parent population, and the blue circles are the newly generated individuals

Specifically, a modified version of DE is employed to generate new individuals near the original parent population. This approach ensures that the expanded parent population retains a significant portion of the genetic information from the original population, thereby preventing excessive loss of valuable traits. Our adjusted DE can be express as follow:

20

$${\text{v}}_i={\text{x}}_{\mathit{cur}}+F_{\mathit{min}}({\text{x}}_{r1}-{\text{x}}_{\mathit{cur}})$$x_cur is a selected individual, and x_r1 is an individual randomly selected from the current parent population. The F_min value is the minimum value in the F value interval in this study.

Figures 3 and 4 fully demonstrates the performance of PE. The design space is designed to be 2-dimensional, with the upper and lower bounds of both dimensions set to 10 and 0, respectively. Five parent individuals are initialized by LHS, represented as green points in the diagram. The number of offspring solutions is 96; the blue points represent the offspring generated by the expanded population after PE, while the red points correspond to the offspring generated by the original population without PE. To more clearly illustrate the area covered by the offspring population, the following method is used: first, a central individual is obtained by calculating the mean values of all the dimensions of the offspring solutions. Then, the main offspring population is identified around this central individual, comprising approximately 80% of the total population size to which the central individual belongs. Finally, the area occupied by this offspring population is colored, with the region's color set to match that of the offspring population. We conducted experiments on three global mutation operations applied in this study. This section shows the results of DE/rand/1, and the rest are in the appendix. Furthermore, the Fig. 5 is the convergence curve of the method with and without PE. The test function is IEEE D10 CEC2010 [48] F8.

[See PDF for image]

Fig. 3

Population distribution map in the early stage of optimization

[See PDF for image]

Fig. 4

Population distribution map in the later stage of optimization

[See PDF for image]

Fig. 5

Convergence curve with and without PE

Figure 3 illustrates the distribution of the generated offspring in the early stage when the number of truly evaluated solutions is insufficient to form the local sub-region. It can be observed that the distribution of blue individuals is better than that of red individuals. This can be attributed to the evolutionary path being more diversified during mutation after PE. By implementing PE in the early optimization stage, the diversity of the offspring is fully enhanced, allowing for a more comprehensive exploration of the entire design space.

In Fig. 4, individuals outside the boundary are no longer repaired, which makes the graph reflect the distribution of the offspring population in the later stage of optimization. By this time, local sub-regions have formed, and the upper and lower boundaries of the local evolution are set to coincide with the boundaries of the entire design space, reducing the frequency of repair strategy activation. The blue area remains primarily within the design space bounded by [0,10]², while the red area extends beyond the boundary. At this stage, the application of PE fully enhances the exploitation capabilities of local surrogates, which is very effective for the exploration of small and challenging regions. However, excessive exploitation risks premature convergence. In contrast, without PE, a broader search is conducted, increasing the survival probability of diverse global individuals, which aids global exploration.

Figure 5 supports the previous discussion of PE performance. Although implementing PE results in faster convergence, premature convergence may occur. Therefore, the application of PE should be approached with caution, considering its specific characteristics. PE has different effects on the algorithm during various stages of optimization. When the population is in the partially feasible or fully feasible state, PE is executed when the current FEs are less than the maximum modeling points required by the local surrogate, thereby enhancing the diversity of the offspring. In addition, PE is always performed in the infeasible state, as it is assumed that when the population remains in the infeasible state for an extended period, the feasible region may be small and challenging to reach.

Update rate-driven adaptive modeling (URAM)

The local surrogate model is switched from local to global under two conditions: (1) when the population exhibits a low overall update rate, (2) during later stage of optimization to intensify search near the optimum solution. The first condition addresses the problem of the population falling into the local optimum and not being able to escape due to the limitation of the local surrogate, while the second enhances convergence toward optimal regions.

The URAM comprises four principal components:

1. Global surrogate switching judgment when the update rate of the population is low: when the population exists in partially feasible or fully feasible states with iteration count below TriggerGen, then the update rate of the population in the previous generations will be judged. As specified in Algorithm 4 lines 3–6, initial threshold violation (update rate < θ) triggers three sequential operations: That is, URflag is set to 1, the current generation is recorded to URglobalGen, and the update rate of the last population is recorded in URmean. After that, as shown in lines 7–13, when the generation reaches URglobalGen + URinterval, the URglobalGen is updated and the mean update rate within the URinterval is calculated. If the average update rate is lower than the threshold θ, URflag is set to 1 and the average update rate is saved to URmean.

2. Global surrogates with elite search switching judgment: as in Algorithm 4 lines 14–17, when the population state is fully feasible and the number of iterations reaches TriggerGen, globalflag is set to 1. In lines 18–19, in the TriggerGen + 1 generation, if the improvement on the update rate of the TriggerGen generation is less than 0.1, the globalflag will be set to 0. After that, in lines 20–21, if the population update rate is less than the threshold θ, globalflag will also be set to 0. This conservative approach prevents excessive commitment to the elite-search strategy.

3. Construction of global surrogate: as present in Algorithm 4 lines 22–29, when the value of URflag or globalflag is 1, a global surrogate model is constructed; otherwise the local surrogate model is built. It should be noted that the two flags are not simultaneously 1. In addition, when the globalflag value is 1, the mutation mode of the offspring is modified to elite search. In particular, the mutation operations include DE /random-to-random/1, DE/best/1, and DE/rand-to-pbest/1 (inspired by the study of [49], DE/rand-to-best/1 was improved. p is set to 0.1).

4. parameter updating: as shown in Algorithm 4 lines 30–35, If the update rate is improved after switching the global surrogate, the value of URinterval is reduced by 1, but the minimum is 2, that is, the global and local surrogates are executed in turn. Conversely, the value of URinterval is doubled. The above operation ensures that the interval constructed by the global surrogate is dynamically adjusted when the global surrogate cannot handle the algorithm dilemma.

[See PDF for image]

Algorithm 4:

Update Rate-driven Adaptive Modeling

Pure objective-driven search rectification (POSR)

The prioritization of objectives over constraints in optimization problems has garnered significant attention, especially when dealing with complex constraints. Due to an excessive preference for feasible solutions, the population is unable to traverse the complex infeasible region and reach all feasible sub-regions [50]. The existing studies [39, 40, 51] focus on leveraging the correlation between the objective and constraints to balance them dynamically. However, designing a compromise indicator to exploit this correlation and establishing a reasonable balance remains a significant challenge. Our study incorporates POSR into the infeasible state of the population, introducing only one easily controllable parameter.

The specific operation of POSR is shown in Algorithm 5. Three mutation operations are employed in line 3 following the label of Local&Global. Notably, although individuals with an excellent objective value may not be able to replace the parent based on FR successfully, they are still used for surrogate modeling, which enhances the model’s prediction accuracy for the objective.

[See PDF for image]

Algorithm 5:

Algorithm 5: Pure Objective-driven Search Rectification

Experimental studies

A series of comparative experiments was performed to fully demonstrate the performance of SI-SADE. The ablation comparison experiments and parameter analysis are presented in “Effect of components in SI-SADE” and “Parameter sensitivity analysis” of this paper, respectively. The performance of SI-SADE in solving a real-world case is demonstrated in “Real-world application”. Additionally, the visualized figures of other evolutionary strategies of PE, along with the convergence curves, are included in the “Appendix”.

Parameter settings

The SI-SADE employs the following parameter settings:

1. population size: NP is set to 40 for each test problem.

2. the coefficient of the number of solutions to construct the local surrogate: r is set to 7, and the number of solutions to build the local surrogate is set to K = min(FEs, min(120, r * n)).

3. the proportion coefficient of POSR population size: ω is set to 0.5.

4. the coefficient of the corresponding iteration number when the global surrogate of the elite search is triggered: Trigger is set to 0.8, then TriggerGen is set to 0.8*maxFEs/NP.

5. the update rate threshold: θ is set to 0.1.

6. related parameters of DE: F = 0.2:0.2:1; CR = 0.4:0.2:1; the number of offspring for each mutation operation is 500, and the total number of offspring is 1500.

7. Maximum number of function evaluations: The value of maxFEs is consistent with studies [7, 22, 26], with a value of 3000. In addition, the statistical results obtained by each comparison method for each test problem with maxFEs = 1000 are given in the “Appendix”. The experimental results show that SI-SADE is competitive on ECOPs.

Comparative algorithms

The comparative algorithms are enumerated as follows:

1. GLoSADE [22]: The optimization process is divided into exploration and exploitation. In the exploration phase, all individuals in the archive are used to construct a global surrogate. In the exploitation stage, a local search is performed on the new population formed in the exploration stage.

2. MPMLS [25]: An adaptive penalty coefficient is proposed, and each penalty coefficient corresponds to a subproblem. The individual with the best performance of (the pair of sub-problems is selected from the population as the center of the sub-region to build a local surrogate. Compared with FR, it improves the emphasis on the objective value and can approach the feasible region from different directions.

3. SA-TSDE [7]: Before the population finds the feasible region, a global surrogate is built and its proposed gradient repair strategy is executed. When some individuals of the population are feasible, the feasible individuals are clustered, and a local surrogate is constructed for the clustering region to find the predicted optimal solution.

4. DSI [52]: Two sampling strategies are proposed. When the predicted optimal solution is close to the archived individual, the solution with the greatest uncertainty is selected. Additionally, when the RBF prediction error occurs, the individuals used in the modeling will be chosen when constructing the objective surrogate and constraint surrogates. The algorithm autonomously modulates its search intensity in response to the optimization trajectories observed in both the surrogate model and the actual fitness landscape within the promising region.

5. GF-SAEAs [26]: Constraints are stratified into multiple levels based on the differential computational costs between the objective function and constraint evaluations. The potential sub-feasible regions of each constraint level are explored and utilized by three search mechanisms, and an adaptive population regeneration mechanism is implemented to restart the algorithm and mitigate premature convergence.

Comparison with the state-of-the-art algorithms on two test suites

In this work, the benchmark problems of IEEE CEC2010 (F1, F7, F8, F13, F14 and F15) and IEEE CEC2017 [53] (F1, F2, F4, F5, F13, F20 and F22), including 10-dimensional and 30-dimensional, are selected to verify the performance of SI-SADE.

Each comparative algorithm is executed independently for 25 iterations on each benchmark problem, terminating upon the exhaustion of FEs. Statistical significance of the performance differentials between the proposed method and the comparative algorithms (GF-SAEAs, DSI, SA-TSDE, MPMLS, and GLoSADE) is assessed utilizing the Wilcoxon rank sum test (WRST) with a significance level of α = 0.05. If an algorithm fails to obtain feasible solutions in all 25 runs, the success rate, defined as the percentage of runs yielding feasible solutions, is reported. The statistical results are listed in Table 1, in which the ‘+’, ‘~’, and ‘−’ represent that the SI-SADE is better than, comparable to, or worse than the compared algorithm in terms of WRST. Therefore, in Table 1, there may be a larger mean value but it is highlighted with a gray background. The “Average Ranking” of each algorithm is the sum of its rankings on each problem divided by the number of problems.

Table 1. Experimental results (maxFEs = 3000) obtained by each compared method on the benchmark problems

Problem	D	GLoSADE	MPMLS	SA-TSDE	DSI	GF-SAEAs	SI-SADE
CEC2010F1	10	−4.77E−01(3.39E−02)~	−5.83E−01(1.01E−01)−	−5.29E−01(1.03E−01)−	−4.08E−01(1.40E−01) +	−3.49E−01(3.79E−02) +	−4.74E−01(9.14E−02)
	30	−3.05E−01(2.11E−02)+	−3.88E−01(6.24E−02)~	−3.05E−01(6.12E−02)+	−3.12E−01(6.94E−02)+	−3.20E−01(4.02E−02)+	−3.77E−01(5.07E−02)
CEC2010F7	10	4.29E+03(3.05E+03)+	1.05E+02(4.70E+02)+	5.32E+02(1.97E+03)+	2.47E+01(8.21E+01)~	3.49E+03(4.75E+03)+	1.05E+01(2.50E+01)
	30	2.43E+05(2.01E+05)+	1.37E+03(1.87E+03)~	7.52E+03(1.15E+04)+	3.06E+03(4.67E+03)+	1.04E+04(2.10E+04)+	1.31E+03(2.10E+03)
CEC2010F8	10	4.68E+04(4.16E+04)+	1.57E+02(3.26E+02)~	8.81E+04(1.83E+05)+	3.02E+03(4.31E+03)+	1.62E+04(2.23E+04)+	5.43E+01(7.76E+01)
	30	5.11E+06(1.63E+06)+	5.55E+06(1.65E+07)+	2.79E+06(1.47E+06)+	1.13E+05(4.26E+05)+	1.13E+04(2.57E+04)~	4.56E+03(1.08E+04)
CEC2010F13	10	−6.23E+01(1.63E+00)−	−6.00E+01(3.13E+00)−	−5.75E+01(4.35E+00)~	−5.72E+01(4.18E+00)~	−5.59E+01(5.29E+00)~	−5.50E+01(6.44E+00)
	30	−4.29E+01(8.12E+00)+	−5.65E+01(3.27E+00)−	−5.37E+01(5.26E+00)~	−5.21E+01(3.90E+00)~	−5.53E+01(1.55E+00)~	−5.33E+01(4.47E+00)
CEC2010F14	10	1.20E+09(3.12E+09)+	1.57E+05(7.78E+05)+	1.04E+13(1.69E+13)+	3.24E+13(6.43E+13)+	5.89E+07(1.07E+08)+	5.86E+00(1.22E+00)
	30	1.380E+10(1.36E+10)+	1.16E+08(3.33E+08)+	5.47E+13(3.18E+13)+	8.18E+13(1.54E+14)+	7.63E+05(1.90E+06)+	4.55E+01(3.84E+01)
CEC2010F15	10	52%+	4.33E+11(5.83E+11)+	2.65E+14(2.38E+14)+	4.87E+13(5.95E+13)+	48%+	1.74E+01(2.24E+01)
	30	7.33E+11(1.12E+12)+	4.96E+13(1.29E+14)+	1.38E+15(9.40E+14)+	2.12E+14(1.63E+14)+	1.58E+12(1.27E+12)+	2.87E+05(5.22E+05)
CEC2017F1	10	2.31E+02(6.72E+01)+	5.20E+02(2.96E+02)+	6.32E−01(2.35E+00)+	8.56E−08(2.25E−07)−	4.05E+00(1.03E+01)+	6.41E−05(3.20E−04)
	30	9.85E+03(1.47E+03)+	3.76E+04(6.43E+03)+	9.16E+03(4.86E+03)+	5.24E+02(7.21E+02)−	4.70E+04(8.13E+03)+	1.00E+03(6.48E+02)
CEC2017F2	10	1.13E+02(4.40E+01)+	1.48E+02(1.29E+02)+	2.99E+00(9.41E+00)+	9.59E−07(3.93E−06)−	6.06E−01(2.03E+00)+	2.43E−07(1.86E−07)
	30	4.70E+03(7.98E+02)+	1.01E+04(1.99E+03)+	2.94E+03(1.21E+03)+	2.90E+02(1.86E+02)−	1.57E+04(3.13E+03)+	7.13E+02(3.59E+02)
CEC2017F4	10	8.54E+01(1.56E+01)+	5.31E+01(1.61E+01)+	6.57E+01(1.18E+01)+	5.97E+01(9.13E+00)+	4.67E+01(1.35E+01)~	4.46E+01(1.16E+01)
	30	3.82E+02(2.04E+01)+	2.67E+02(8.31E+01)+	2.05E+02(4.34E+01)~	2.19E+02(4.95E+01)+	3.32E+02(7.42E+01)+	1.85E+02(2.09E+01)
CEC2017F5	10	6.71E+00(6.56E−01)+	5.09E+00(1.12E+00)+	8.49E+00(9.48E−01)+	3.46E+00(1.60E+00)+	5.19E+00(7.59E−01)+	2.85E+00(4.34E−01)
	30	3.72E+01(7.71E+00)+	3.37E+01(2.24E+01)~	7.14E+01(3.84E+01)+	3.35E+01(2.92E+01)−	3.36E+01(2.17E+01)~	3.05E+01(1.27E+01)
CEC2017F13	10	1.34E+01(1.71E+01)+	1.56E+01(2.57E+01)+	1.18E+02(1.99E+02)+	9.32E+02(3.24E+03)~	5.56E+00(1.34E+01)~	8.31E+00(2.03E+01)
	30	0%+	0%+	36%−	0%+	0%+	16%
CEC2017F20	10	2.18E+00(3.31E−01)~	2.13E+00(3.18E−01)~	1.70E+00(5.21E−01)−	3.15E+00(5.75E−01)+	1.85E+00(1.98E−01)−	1.98E+00(4.91E−01)
	30	9.74E+00(6.82E−01)~	9.84E+00(4.29E−01)~	1.01E+01(6.57E−01)+	1.19E+01(9.45E−01)+	9.83E+00(5.21E−01)~	9.86E+00(6.09E−01)
CEC2017F22	10	7.12E+01(4.68E+01)+	6.31E+00(1.69E+01)~	2.86E+03(4.32E+03)+	68%+	9.86E+00(2.17E+01)~	9.37E+00(2.00E+01)
	30	0%~	0%~	0%~	0%~	0%~	0%
Win/Tie/Loss		21/4/1	14/9/3	18/5/3	16/5/5	16/9/1	NA
Average Ranking		4.58	3.14	3.28	4.50	3.62	1.88

In the CEC2010 test suite, we find that MPMLS achieves the best overall performance on 10D CEC2010F1, while SI-SADE maintains a similar overall performance to MPMLS on 30D CEC2010F1 with a smaller standard deviation. Since the constraint value of this benchmark problem changes significantly at the boundary of the feasible region, causing RBF to predict infeasible solutions as feasible ones. The worse the results on CEC2010F1, the more serious the influence of the surrogate model prediction inaccuracy. MPMLS’s strategy of dividing subregions effectively reduces the possibility of guiding the search into inaccurate regions, so it performs well on CEC2010F1. SI-SADE obtains the best overall performance on both CEC2010F7 and CEC2010F8, and the standard deviation is smaller than that of other algorithms, indicating that SI-SADE guarantees population diversity while maintaining high convergence speed. Furthermore, GLoSADE achieves the best overall performance on 10D CEC2010F13, because FR performs better than EC and other constraint handling techniques that introduce objective information on this problem. For the 30D CEC2010F13, the subregions strategy of MPMLS enables it to achieve better overall performance because most algorithms converge prematurely on this problem. The overall performance of SI-SADE on CEC2010F14 and CEC2010F15 is significantly better than those of other algorithms. This is due to the fact that proposed ASSF provides a variety of offspring for the algorithm and allocates search resources efficiently, and the collaboration of POSR and PE enhances the algorithm’s ability to find feasible region. Moreover, the different penalty factors set by MPMLS allow the population to approach the feasible region from various directions, resulting in the second-best overall performance. However, GF-SAEAs and GLoSADE found only 12 and 13 feasible solutions in 25 independent runs on 10D CEC2010F15, respectively. In summary, among the 12 test problems of CEC2010, SI-SADE demonstrates the best overall performance on 8 of them. Additionally, MPMLS and GLoSADE achieve the best performance on 3 and 1 of the test problems, respectively.

In the CEC2017 test suite, DSI demonstrates superior overall performance on CEC2017F1 and CEC2017F2 compared to other algorithms. This advantage stems from its proposed adaptive adjustment search intensity strategy, which ensures high convergence rates on these test problems. SI-SADE ranks second in performance, trailing only DSI, whereas MPMLS exhibits the poorest performance due to its use of local surrogates, which leads the population to become trapped in local optima. Moreover, SI-SADE excels in both dimensions of CEC2017F4, owing to the enhanced exploration capabilities provided by the proposed ASSF. While SI-SADE outperforms others on 10D CEC2017F5, DSI takes the lead on 30D CEC2017F5, with SI-SADE securing the second position. Additionally, the standard deviations of SI-SADE on both 10D and 30D of CEC2017F4 and CEC2017F5 are lower than those of the algorithms with no significant difference from SI-SADE, which confirms SI-SADE’s stability. The overall performance of SI-SADE on 10D CEC2017F13 is the best, but the stability is not as good as that of GF-SAEAs, so the mean value is larger than that of GF-SAEAs. On 30D CEC2017F13, SA-TSDE obtained the best overall result, which means that its proposed gradient repair strategy played a full role. In the case of CEC2017F20, SA-TSDE, and GLoSADE showcase optimal performance in 10 and 30 dimensions, respectively. MPMLS reasserts its formidable search capabilities on 10D CEC2017F22, claiming the top spot in overall performance. In conclusion, among the 12 selected test problems of CEC2017, DSI achieves the best overall performance on 5 of them, followed by SI-SADE which obtains the best overall performance on 4 of them. Rounding out the results, SA-TSDE, MPMLS, and GLoSADE each clinch the top position in one test problem.

Based on the preceding discussions and the statistical results presented in Table 1, SI-SADE and MPMLS emerge as the top two performers, securing the first and second average ranking, respectively. This success can be attributed to the accelerated convergence rates facilitated by local surrogates. However, SI-SADE distinguishes itself by effectively addressing a key limitation of MPMLS: the tendency to become trapped in local optima when using local surrogates. Furthermore, SI-SADE exhibits superior overall performance in 12 out of the 24 test problems. This impressive feat underscores the algorithm’s robust and promising capabilities across various optimization challengess.

Effect of components in SI-SADE

To intuitively and verify the effectiveness of each component, these problems of IEEE CEC2010 (F1, F7, F8, F13, F14, and F15) and IEEE CEC2017 (F1, F2, F4, F5, F13, F20 and F22), including 10-dimensional and 30-dimensional, were used to compare the performance of SI-SADE and its variants with different component combinations. All experimental results are obtained over 25 independent runs in Matlab R2023a and are listed in Tables 3, 4 and 5, respectively. The “AR” in all tables represents the average rank of the corresponding algorithm based on “Mean” values. The description of the variants is shown in the following in the Table 2.

Table 2. The description of variants for components

Variant name	Description
RemoveAll	All components are removed from SI-SADE. Additionally, RemoveAll uses local surrogates, with the feasibility rule as the selection rule
IF	The population state is fixed to be infeasible, and only the strategies in this state are executed
PF	The population state is fixed to be partially feasible, and only the strategies in this state are executed
FF	The population state is fixed to be fully feasible, and only the strategies in this state are executed
ASSF	ASSF component is added to RemoveAll
NoPE	PE component is removed from SI-SADE
NoURAM	URAM component is removed from SI-SADE
NoPOSR	POSR component is removed from SI-SADE
LocalRepair	The upper and lower bounds of the boundary repair method are changed from global to local
NoAdEC	The adjusted EC is replaced the traditional EC

s

Table 3. Experimental results of five variants of SI-SADE

Problem	D	RemoveAll	IF	HF	FF
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−4.90E−01(1.26E−01)	−4.87E−01(1.11E−01)	−4.95E−01(1.12E−01)	−5.24E−01(1.07E−01)
	30	−3.64E−01(7.05E−02)	−3.76E−01(6.22E−02)	−3.74E−01(3.11E−02)	−3.56E−01(3.60E−02)
CEC2010F7	10	3.15E+01(6.36E+01)	6.49E+01(1.35E+02)	1.09E+01(2.60E+01)	6.11E+01(2.14E+02)
	30	2.91E+03(4.09E+03)	1.64E+03(2.07E+03)	1.70E+03(2.19E+03)	1.49E+03(2.19E+03)
CEC2010F8	10	1.41E+02(1.99E+02)	1.84E+02(4.79E+02)	2.13E+02(3.97E+02)	2.40E+02(6.68E+02)
	30	8.64E+04(3.09E+05)	2.53E+03(2.73E+03)	7.33E+03(1.31E+04)	8.38E+03(1.38E+04)
CEC2010F13	10	−5.91E+01 (3.86E+00)	−5.98E+01(4.72E+00)	−5.49E+01(6.24E+00)	−6.08E+01(2.57E+00)
	30	−5.60E+01(2.45E+00)	−5.43E+01(4.09E+00)	−4.96E+01(7.57E+00)	−5.60E+01(3.05E+00)
CEC2010F14	10	2.36E+09(1.15E+10)	6.64E+00(1.49E+00)	5.62E+00(1.17E+00)	7.97E+05(3.76E+06)
	30	3.80E+10(9.69E+10)	4.80E+01(3.17E+01)	5.20E+01(4.02E+01)	6.76E+02(1.66E+03)
CEC2010F15	10	28%	3.18E+03(6.67E+03)	7.80E+07(3.82E+08)	52%
	30	60%	5.82E+05(1.27E+06)	2.61E+06(7.94E+06)	80%
CEC2017F1	10	2.90E+02(3.04E+02)	4.37E−01(7.75E−01)	4.31E−07(1.08E−06)	1.22E−07(8.91E−08)
	30	4.38E+04(9.10E+03)	9.41E+03(2.19E+03)	1.37E+03(4.90E+02)	1.18E+03(8.19E+02)
CEC2017F2	10	7.02E+01(1.08E+02)	3.58E−01(4.89E−01)	8.12E−07(1.82E−06)	2.75E−07(2.18E−07)
	30	1.08E+04(2.29E+03)	5.27E+03(1.62E+03)	1.49E+03(4.31E+02)	7.63E+02(3.94E+02)
CEC2017F4	10	5.77E+01(2.19E+01)	5.33E+01(1.73E+01)	4.51E+01(1.16E+01)	4.67E+01(1.21E+01)
	30	1.79E+02(3.14E+01)	1.81E+02(2.76E+01)	1.80E+02(2.04E+01)	1.85E+02(2.10E+01)
CEC2017F5	10	3.04E+00(9.13E−01)	4.88E+00(8.13E−01)	3.12E+00(1.01E+00)	3.08E+00(9.37E−01)
	30	2.90E+01(1.03E+01)	3.65E+01(2.08E+01)	3.52E+01(2.13E+01)	3.54E+01(2.11E+01)
CEC2017F13	10	5.56E+00(1.68E+01)	4.05E+00(1.15E+01)	1.20E+01(2.22E+01)	6.25E+01(1.34E+02)
	30	0%	16%	12%	0%
CEC2017F20	10	2.31E+00(2.62E−01)	2.14E+00(2.77E−01)	2.08E+00(2.10E−01)	2.09E+00(2.43E−01)
	30	9.92E+00(4.94E−01)	9.95E+00(5.96E−01)	9.94E+00(4.68E−01)	9.68E+00(5.61E−01)
CEC2017F22	10	1.32E+01(2.38E+01)	7.02E+01(6.55E+01)	2.19E+01(2.81E+01)	48%
	30	0%	0%	0%	0%
AR		7.84	6.14	6.26	6.6

Table 4. Experimental results of four variants of SI-SADE

Problem	D	ASSF	NoPE	NoURAM	NoPOSR
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−5.46E−01(1.13E−01)	−4.84E−01(1.12E−01)	−5.84E−01(1.08E−01)	−4.72E−01(9.15E−02)
	30	−3.52E−01(4.83E−02)	−3.54E−01(5.82E−02)	−3.70E−01(6.25E−02)	−3.77E−01(5.35E−02)
CEC2010F7	10	5.29E+01(2.33E+02)	4.54E+01(1.95E+02)	1.24E+01(3.34E+01)	1.06E+01(2.45E+01)
	30	2.31E+03(2.68E+03)	1.74E+03(2.10E+03)	1.30E+03(2.05E+03)	1.31E+03(2.06E+03)
CEC2010F8	10	1.12E+02(1.67E+02)	1.13E+02(1.63E+02)	5.66E+01(8.33E+01)	5.43E+01(7.60E+01)
	30	5.99E+03(1.02E+04)	7.97E+03(1.89E+04)	4.26E+03(9.49E+03)	4.56E+03(1.06E+04)
CEC2010F13	10	−5.72E+01(4.21E+00)	−5.66E+01(4.74E+00)	−5.62E+01(5.59E+00)	−5.51E+01(6.37E+00)
	30	−5.40E+01(5.70E+00)	−5.33E+01(4.59E+00)	−5.41E+01(3.19E+00)	−5.50E+01(4.14E+00)
CEC2010F14	10	6.23E+02(2.80E+03)	5.75E+01(1.41E+02)	5.59E+00(1.21E+00)	1.55E+05(7.61E+05)
	30	6.79E+05(3.32E+06)	5.82E+01(4.48E+01)	4.92E+01(3.96E+01)	6.18E+01(7.41E+01)
CEC2010F15	10	36%	80%	4.17E+01(8.95E+01)	6.30E+06(2.46E+07)
	30	1.98E+14(2.00E+14)	96%	1.10E+05(1.54E+05)	1.25E+11(6.14E+11)
CEC2017F1	10	2.94E+02(3.60E+02)	1.35E−07(9.04E−08)	1.61E+02(2.66E+02)	2.84E−07(8.82E−07)
	30	4.00E+04(9.67E+03)	2.58E+03(1.83E+03)	1.91E+04(7.54E+03)	1.00E+03(6.33E+02)
CEC2017F2	10	1.35E+02(1.44E+02)	1.69E−07(1.54E−07)	4.72E+01(7.51E+01)	2.30E−07(1.78E−07)
	30	8.99E+03(2.26E+03)	1.44E+03(6.80E+02)	7.38E+03(1.39E+03)	5.62E+02(2.36E+02)
CEC2017F4	10	4.58E+01(9.47E+00)	4.43E+01(8.12E+00)	5.20E+01(1.71E+01)	4.49E+01(1.17E+01)
	30	1.69E+02(3.95E+01)	1.76E+02(2.86E+01)	1.89E+02(3.82E+01)	1.85E+02(2.05E+01)
CEC2017F5	10	2.94E+00(3.64E−01)	3.02E+00(1.08E+00)	2.81E+00(4.43E−01)	2.82E+00(3.60E−01)
	30	3.55E+01(1.90E+01)	2.94E+01(1.26E+01)	3.04E+01(1.25E+01)	2.70E+01(1.60E+00)
CEC2017F13	10	6.79E+00(1.66E+01)	1.62E+01(2.59E+01)	8.51E+00(1.99E+01)	4.28E+00(1.19E+01)
	30	0%	32%	16%	0%
CEC2017F20	10	2.06E+00(2.68E−01)	1.85E+00(4.37E−01)	2.18E+00(2.87E−01)	1.97E+00(5.48E−01)
	30	9.76E+00(4.71E−01)	9.82E+00(6.50E−01)	9.71E+00(7.12E−01)	9.86E+00(5.97E−01)
CEC2017F22	10	2.06E+01(2.67E+01)	1.40E+01(2.34E+01)	4.86E+00(1.19E+01)	84%
	30	0%	0%	0%	0%
AR		7.06	5.40	5.06	4.58

Table 5. Experimental results of SI-SADE and two variants of SI-SADE

Problem	D	LocalRepair	NoAdEC	SI-SADE
		Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−6.00E−01(1.13E−01)	−4.71E−01(9.23E−02)	−4.74E−01(8.95E−02)
	30	−3.74E−01(6.16E−02)	−3.75E−01(4.59E−02)	−3.77E−01(4.96E−02)
CEC2010F7	10	1.69E+02(4.86E+02)	1.38E+01(2.48E+01)	1.05E+01(2.45E+01)
	30	2.54E+06(9.20E+06)	2.16E+03(2.28E+03)	1.31E+03(2.06E+03)
CEC2010F8	10	2.52E+03(8.67E+03)	5.48E+01(7.82E+01)	5.43E+01(7.60E+01)
	30	2.74E+06(3.51E+06)	1.48E+04(3.45E+04)	4.56E+03(1.06E+04)
CEC2010F13	10	−5.65E+01(5.92E+00)	−5.86E+01(4.53E+00)	−5.50E+01(6.31E+00)
	30	−5.27E+01(4.05E+00)	−5.59E+01(3.28E+00)	−5.33E+01(4.38E+00)
CEC2010F14	10	1.28E+05(6.29E+05)	9.22E+04(4.52E+05)	5.86E+00(1.20E+00)
	30	1.32E+08(6.02E+08)	4.46E+01(2.68E+01)	4.55E+01(3.77E+01)
CEC2010F15	10	2.05E+02(4.42E+02)	7.90E+06(3.40E+07)	1.74E+01(2.20E+01)
	30	2.53E+05(5.23E+05)	3.65E+07(5.10E+07)	2.87E+05(5.11E+05)
CEC2017F1	10	1.25E+01(3.00E+01)	6.41E−05(3.14E−04)	6.41E−05(3.14E−04)
	30	2.21E+03(9.43E+02)	1.60E+03(8.96E+02)	1.00E+03(6.35E+02)
CEC2017F2	10	6.15E+01(1.41E+02)	2.66E−07(2.24E−07)	2.43E−07(1.82E−07)
	30	1.59E+03(6.56E+02)	1.17E+03(5.96E+02)	7.13E+02(3.52E+02)
CEC2017F4	10	4.56E+01(7.45E+00)	4.90E+01(8.88E+00)	4.46E+01(1.14E+01)
	30	1.86E+02(1.86E+01)	1.83E+02(2.43E+01)	1.85E+02(2.05E+01)
CEC2017F5	10	8.63E+00(1.34E+00)	3.22E+00(1.16E+00)	2.85E+00(4.25E−01)
	30	1.23E+02(8.88E+01)	3.21E+01(1.57E+01)	3.05E+01(1.25E+01)
CEC2017F13	10	4.17E+01(5.14E+01)	8.54E+00(1.99E+01)	8.30E+00(1.99E+01)
	30	28%	16%	16%
CEC2017F20	10	1.95E+00(4.30E−01)	2.16E+00(2.92E−01)	1.98E+00(4.81E−01)
	30	9.82E+00(6.12E−01)	9.70E+00(6.63E−01)	9.86E+00(5.97E−01)
CEC2017F22	10	56%	6.64E+00(1.89E+01)	9.37E+00(1.96E+01)
	30	0%	0%	0%
AR		7.48	5.56	4.02

As shown in Tables 3, 4 and 5, SI-SADE has the lowest AR value, indicating that all components of SI-SADE contribute significantly to its overall performance.

The comparison between RemoveAll and ASSF in Tables 3 and 4 demonstrates that the proposed ASSF significantly outperforms RemoveAll on complex constraint problems, particularly F14 and F15 of CEC2010. This benefits from reasonable allocation of search resources by ASSF. More importantly, the ASSF framework facilitates the easy integration of corresponding strategies based on different population states, thereby enhancing the algorithm’s robustness. Furthermore, the AR values of IF, HF, and FF are significantly higher than SI-SADE, demonstrating that the strategy of any single state in SI-SADE cannot outperform the overall performance of SI-SADE. This further validates the effectiveness of the proposed ASSF.

PE improves the overall performance of SI-SADE in CEC2010, including F7, F8, F14, and F15. On the one hand, PE increases the diversity of offspring; on the other hand, it aids the algorithm in locating the narrow feasible region in the infeasible state and has a positive effect on complex constraints such as CEC2010F14 and F15. However, the proportion of feasible solutions found in 25 independent runs on the 30D CEC2017F13 problem decreased. This decrease can be attributed to PE’s enhancement of the exploitation ability of the local surrogate, which improves the local surrogate’s search capability but may also accelerate convergence.

URAM significantly enhances the performance of SI-SADE on CEC2017F1 and F2. The algorithm using local surrogates can search for solutions near the global optimum on CEC2017F1 and F2. However, most individuals become trapped in local optima, hindering the transfer of search resources to more promising areas. Consequently, switching to global surrogates significantly improves the overall performance. However, it inevitably reduces the overall performance of SI-SADE on the 10D CEC2010F1. When switching to a global surrogate, the population gains access to a broader landscape, enabling it to escape local optima. However, this switch may also amplify any inaccuracies in the surrogate model. This phenomenon is also observed in algorithms utilizing the global surrogate model. Overall, URAM contributes positively to the robustness of the algorithm.

POSR enhances the algorithm’s ability to locate the feasible region. When POSR is removed, the quality of feasible solutions for CEC2010F15 decreases, and fewer feasible solutions are found for 30D CEC2017F13 and 10D CEC2017F22. This indicates that the proposed POSR enables the population to overcome complex constraints, which is one of the main reasons why SI-SADE outperforms other algorithms on F14 and F15 of CEC2010. Table 6 lists the mean objective value of the best solution and mean FEs when SI-SADE and NoPOSR firstly search for a feasible solution in 25 independent runs. The comparison between NoPOSR and SI-SADE in Table 6 reveals that SI-SADE achieves similar or better values with comparable FEs. This confirms that POSR does not waste valuable FEs and can lay a good foundation for subsequent optimization.

Table 6. Experimental results of mean objective value of the first feasible solution and the average FEs are found

Problem	2010F14D10	2010F15D10	2010F14D30	2010F15D30	2017F13D10
	Mean(FEs)	Mean(FEs)	Mean(FEs)	Mean(FEs)	Mean(FEs)
NoPOSR	2.16E+13(95)	3.33E+11(555)	1.26E+14(95)	1.76E+13(450)	1.57E+04(283)
SI-SADE	2.11E+13(102)	6.26E+09(570)	1.27E+14(97)	1.72E+13(410)	1.09E+04(293)

The ablation experiments comparing LocalRepair and SI-SADE in Table 5 demonstrate that while the LocalRepair demonstrates the best performance on D10 CEC2010F1 with statistically significant improvements, it shows performance weaknesses in most other benchmark functions. These results confirm the validity of SI-SADE’s setting of the upper and lower bounds of the boundary repair method to align with the upper and lower bounds of the design domain.

The results of NoAdEC and SI-SADE show that the adjusted EC performs well in dealing with benchmark functions with complex constraints such as CEC2010F15. When the cut-off generation is set the same, the adjusted EC will not converge prematurely, that is, compared with the traditional EC, it can introduce more objective information.

Parameter sensitivity analysis

Problems of IEEE CEC2010 (F1, F7, F8, F13, F14, and F15) and IEEE CEC2017 (F1, F2, F4, F5, F13, and F20), including 10-dimensional and 30-dimensional, are chosen to discuss the effectiveness of five critical parameters in SI-SADE. All experimental results are obtained over 30 independent runs in Matlab R2023a.

NP varies from 20 to 90 with increments of 10 (20, 30, 40, 50, 60, 70, 80, and 90) for comparison. Tables 7 and 8 list the results of SI-SADE with different population sizes on these selected problems. NP represents the static population size in the iterative process. A larger NP can alleviate premature convergence. However, it increases unnecessary computational costs by evaluating similar or poor solutions, potentially reducing the algorithm's exploration capability given a fixed number of function evaluations. Conversely, a smaller NP might lead to premature convergence but can facilitate the population in overcoming local optima and allow for more iterations. Therefore, selecting the appropriate population size NP for the algorithm is vital. Based on the Average Ranking (AR) values in Tables 7 and 8, SI-SADE's performance varies significantly with different NP values, indicating the algorithm's sensitivity to this parameter. The optimal performance, shown by the lowest AR value, was achieved when NP = 40. Consequently, we recommend setting NP to 40 for SI-SADE.

Table 7. Experimental results of SI-SADE with different NP (20, 30, 40, 50)

Problem	D	NP = 20	NP = 30	NP = 40	NP = 50
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−5.13E−01(1.28E−01)	−5.03E−01(1.33E−01)	−4.65E−01(9.13E−02)	−5.12E−01(9.13E−02)
	30	−3.33E−01(3.38E−02)	−3.46E−01(4.18E−02)	−3.52E−01(2.92E−02)	−3.63E−01(4.98E−02)
CEC2010F7	10	2.63E+02(1.13E+03)	3.20E+01(7.73E+01)	9.60E+00(2.29E+01)	8.90E+00(1.76E+01)
	30	8.21E+02(1.18E+03)	7.24E+02(9.14E+02)	1.51E+03(1.83E+03)	1.51E+03(1.65E+03)
CEC2010F8	10	3.04E+02(8.45E+02)	4.08E+02(8.33E+02)	5.38E+01(7.45E+01)	1.21E+02(3.70E+02)
	30	1.15E+06(5.25E+06)	9.61E+04(1.84E+05)	8.12E+04(1.85E+05)	2.07E+05(5.00E+05)
CEC2010F13	10	−5.49E+01(6.51E+00)	−5.62E+01(5.82E+00)	−5.61E+01(6.54E+00)	−5.83E+01(4.78E+00)
	30	−5.37E+01(5.19E+00)	−5.52E+01(4.77E+00)	−5.69E+01(4.12E+00)	−5.38E+01(4.40E+00)
CEC2010F14	10	4.47E+00(1.96E+00)	1.11E+04(6.08E+04)	5.81E+00(1.24E+00)	6.81E+00(2.83E+00)
	30	4.54E+01(3.15E+01)	4.59E+01(2.93E+01)	8.05E+01(9.09E+01)	1.09E+02(2.53E+02)
CEC2010F15	10	6.53E+01(1.38E+02)	3.14E+01(1.41E+02)	1.78E+01(2.23E+01)	1.34E+05(5.72E+05)
	30	6.20E+06(1.73E+07)	6.18E+06(8.99E+06)	6.53E+06(1.03E+07)	9.03E+06(1.96E+07)
CEC2017F1	10	6.34E−02(3.47E−01)	1.86E−07(1.42E−07)	2.51E−07(8.23E−07)	1.81E−07(3.04E−07)
	30	1.49E+03(1.27E+03)	1.67E+03(8.26E+02)	1.36E+03(9.37E+02)	1.46E+03(5.90E+02)
CEC2017F2	10	3.13E−07(2.74E−07)	2.96E−02(1.62E−01)	2.50E−07(1.92E−07)	1.96E−07(2.64E−07)
	30	6.07E+02(3.58E+02)	7.27E+02(3.89E+02)	8.20E+02(5.71E+02)	7.81E+02(3.01E+02)
CEC2017F4	10	5.14E+01(9.37E+00)	4.55E+01(8.92E+00)	4.41E+01(1.12E+01)	4.34E+01(1.10E+01)
	30	1.96E+02(3.49E+01)	1.86E+02(1.98E+01)	1.81E+02(2.01E+01)	1.88E+02(1.94E+01)
CEC2017F5	10	1.37E+00(1.31E+00)	1.86E+00(4.85E−01)	2.89E+00(4.59E−01)	3.86E+00(3.10E−01)
	30	2.97E+01(1.44E+01)	3.39E+01(1.82E+01)	2.88E+01(5.74E+00)	3.25E+01(1.61E+01)
CEC2017F13	10	4.84E+00(1.57E+01)	3.24E+00(1.13E+01)	1.12E+01(2.32E+01)	1.11E+01(2.25E+01)
	30	/	/	/	/
CEC2017F20	10	2.05E+00(4.18E−01)	2.00E+00(4.74E−01)	1.97E+00(5.44E−01)	1.88E+00(5.52E−01)
	30	9.84E+00(5.24E−01)	9.67E+00(6.83E−01)	9.77E+00(6.47E−01)	9.79E+00(5.72E−01)
AR		4.83	4.37	3.41	3.50

Table 8. Experimental results (mean + rank) of SI-SADE with different NP (60, 70, 80, 90)

Problem	D	NP = 60	NP = 70	NP = 80	NP = 90
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−5.19E−01(1.01E−01)	−4.97E−01(1.08E−01)	−4.97E−01(9.58E−02)	−4.88E−01(8.78E−02)
	30	−3.60E−01(3.56E−02)	−3.71E−01(2.68E−02)	−3.70E−01(2.79E−02)	−3.79E−01(3.90E−02)
CEC2010F7	10	2.81E+01(1.03E+02)	4.93E+01(1.45E+02)	2.06E+01(4.24E+01)	1.16E+01(1.60E+01)
	30	1.21E+03(1.51E+03)	1.63E+03(1.93E+03)	2.40E+03(2.34E+03)	2.96E+03(2.14E+03)
CEC2010F8	10	1.31E+02(1.87E+02)	1.92E+02(4.47E+02)	1.24E+02(2.00E+02)	1.16E+02(1.77E+02)
	30	1.34E+05(2.19E+05)	4.65E+05(9.18E+05)	2.22E+05(3.42E+05)	2.88E+05(3.79E+05)
CEC2010F13	10	−6.00E+01(3.34E+00)	−5.75E+01(5.57E+00)	−5.80E+01(5.07E+00)	−5.80E+01(4.14E+00)
	30	−5.42E+01(4.38E+00)	−5.51E+01(2.72E+00)	−5.70E+01(3.24E+00)	−5.59E+01(3.93E+00)
CEC2010F14	10	1.34E+01(3.50E+01)	1.10E+01(2.02E+01)	8.07E+00(1.61E+00)	1.13E+01(1.41E+01)
	30	4.56E+02(1.13E+03)	1.50E+03(5.69E+03)	1.29E+03(6.01E+03)	1.35E+04(5.20E+04)
CEC2010F15	10	2.81E+05(1.53E+06)	1.98E+05(1.05E+06)	3.06E+04(1.27E+05)	1.50E+05(4.14E+05)
	30	6.19E+06(1.02E+07)	1.83E+07(3.78E+07)	1.59E+07(1.96E+07)	2.06E+07(2.62E+07)
CEC2017F1	10	2.04E−08(4.33E−08)	4.33E−05(1.65E−04)	2.21E−05(1.14E−04)	3.06E−01(1.67E+00)
	30	1.55E+03(5.59E+02)	1.65E+03(6.31E+02)	1.84E+03(1.49E+03)	1.65E+03(6.31E+02)
CEC2017F2	10	7.38E−08(1.99E−07)	8.68E−01(4.76E+00)	2.85E−05(7.72E−05)	3.19E−04(1.13E−03)
	30	1.02E+03(4.40E+02)	1.23E+03(4.34E+02)	1.36E+03(5.78E+02)	1.22E+03(4.35E+02)
CEC2017F4	10	4.40E+01(7.83E+00)	4.53E+01(8.51E+00)	4.84E+01(9.11E+00)	4.40E+01(8.91E+00)
	30	1.78E+02(1.82E+01)	1.77E+02(1.93E+01)	1.83E+02(2.10E+01)	1.77E+02(1.93E+01)
CEC2017F5	10	5.14E+00(7.44E−01)	5.85E+00(4.08E−01)	6.07E+00(5.79E−01)	6.60E+00(7.37E−01)
	30	3.28E+01(1.55E+01)	2.86E+01(7.49E+00)	3.08E+01(1.31E+01)	2.86E+01(7.49E+00)
CEC2017F13	10	1.51E+01(2.45E+01)	1.17E+01(2.23E+01)	2.12E+01(2.61E+01)	1.44E+01(2.23E+01)
	30	/	/	/	/
CEC2017F20	10	1.95E+00(4.18E−01)	1.81E+00(5.46E−01)	2.00E+00(4.21E−01)	1.94E+00(4.57E−01)
	30	9.88E+00(6.04E−01)	9.83E+00(5.86E−01)	9.96E+00(6.80E−01)	9.83E+00(5.86E−01)
AR		4.24	5.24	5.46	4.96

The coefficient of the number of solutions to construct the local surrogate r is varied as 2, 3, 4, 5, 6, 7, 8, and 9 for comparison. K is the number of solutions involved in constructing the local surrogate. A suitably small K can provide higher prediction accuracy for the local space and faster convergence speed, but it may also increase the risk of falling into local optima. Conversely, a larger K allows the population to explore a wider range of the solution space, but at the cost of slower convergence. Therefore, choosing an appropriate r is crucial to balance between convergence speed and exploration capability. Tables 9 and 10 show that Average Ranking (AR) values exhibit a non-monotonic trend as r varies. Therefore, the AR value is calculated for different dimensions. Table 11 demonstrates that the AR values show distinct trends across two dimensions. Based on these results, the recommended value of r is 7, and K is recommended to be set to min (FEs, min(120, r * n)). This is because when the dimension is 30, the overall performance of SI-SADE is best when r is 4 and the corresponding K value is 120.

Table 9. Experimental results of SI-SADE with different r (2, 3, 4, 5)

Problem	D	r = 2	r = 3	r = 4	r = 5
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−5.93E−01(7.61E−02)	−5.84E−01(9.54E−02)	−5.29E−01(1.05E−01)	−5.35E−01(1.13E−01)
	30	−3.57E−01(3.83E−02)	−3.66E−01(5.08E−02)	−3.76E−01(5.09E−02)	−3.67E−01(3.94E−02)
CEC2010F7	10	1.49E+03(2.06E+03)	2.13E+02(7.07E+02)	8.44E+01(1.79E+02)	3.16E+01(5.99E+01)
	30	5.83E+03(5.26E+03)	1.79E+03(2.78E+03)	1.32E+03(2.00E+03)	1.56E+03(2.02E+03)
CEC2010F8	10	2.84E+03(6.17E+03)	5.36E+02(1.12E+03)	6.42E+02(1.47E+03)	1.69E+02(3.38E+02)
	30	6.78E+04(9.47E+04)	5.18E+03(5.64E+03)	1.28E+04(4.46E+04)	7.53E+04(3.07E+05)
CEC2010F13	10	−5.95E+01(3.91E+00)	−5.78E+01(5.06E+00)	−5.55E+01(5.63E+00)	−5.87E+01(4.39E+00)
	30	−5.56E+01(3.21E+00)	−5.55E+01(3.01E+00)	−5.38E+01(4.41E+00)	−5.40E+01(3.34E+00)
CEC2010F14	10	8.58E+03(1.54E+04)	1.13E+04(6.09E+04)	2.20E+04(8.35E+04)	1.39E+01(3.31E+01)
	30	2.74E+04(5.09E+04)	3.15E+02(1.42E+03)	4.35E+01(3.66E+01)	5.21E+01(5.22E+01)
CEC2010F15	10	8.90E+04(1.95E+05)	1.02E+03(4.26E+03)	4.84E+01(8.55E+01)	3.72E+02(1.91E+03)
	30	9.23E+06(1.16E+07)	6.75E+06(1.87E+07)	3.26E+05(5.52E+05)	1.27E+06(3.06E+06)
CEC2017F1	10	2.28E−08(5.12E−08)	9.81E−02(5.37E−01)	7.39E−08(9.34E−08)	1.22E−07(1.04E−07)
	30	3.14E+03(2.38E+03)	1.81E+03(2.91E+03)	1.09E+03(6.90E+02)	1.15E+03(7.04E+02)
CEC2017F2	10	6.71E−07(2.22E−06)	8.28E−03(4.53E−02)	1.20E−07(1.14E−07)	4.79E−01(2.62E+00)
	30	8.50E+02(5.16E+02)	7.11E+02(2.52E+02)	7.26E+02(3.56E+02)	8.60E+02(4.71E+02)
CEC2017F4	10	4.49E+01(9.78E+00)	4.14E+01(8.39E+00)	4.61E+01(1.11E+01)	4.48E+01(1.02E+01)
	30	1.65E+02(2.10E+01)	1.78E+02(2.34E+01)	1.85E+02(2.25E+01)	1.77E+02(1.83E+01)
CEC2017F5	10	7.66E+00(1.08E+00)	8.18E+00(1.27E+01)	4.58E+00(5.77E−01)	4.11E+00(9.98E−01)
	30	4.10E+01(3.09E+01)	3.18E+01(1.80E+01)	3.15E+01(1.60E+01)	3.44E+01(1.99E+01)
CEC2017F13	10	3.03E+01(4.10E+01)	1.45E+01(2.57E+01)	1.12E+01(2.22E+01)	1.32E+01(2.39E+01)
	30	/	/	/	/
CEC2017F20	10	1.75E+00(5.25E−01)	1.90E+00(5.23E−01)	1.90E+00(4.96E−01)	1.78E+00(5.44E−01)
	30	9.66E+00(6.08E−01)	9.87E+00(4.47E−01)	9.86E+00(6.03E−01)	9.78E+00(7.40E−01)
AR		4.98	4.93	4.04	4.00

Table 10. Experimental results of SI-SADE with different r (6, 7, 8, 9)

Problem	D	r = 6	r = 7	r = 8	r = 9
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−5.10E−01(1.27E−01)	−4.65E−01(9.13E−02)	−4.80E−01(1.09E−01)	−4.87E−01(1.13E−01)
	30	−3.57E−01(2.98E−02)	−3.52E−01(2.92E−02)	−3.41E−01(3.88E−02)	−3.52E−01(3.55E−02)
CEC2010F7	10	2.02E+01(6.23E+01)	9.60E+00(2.29E+01)	1.47E+01(4.97E+01)	1.71E+01(6.53E+01)
	30	1.29E+03(1.50E+03)	1.51E+03(1.83E+03)	1.24E+03(1.65E+03)	9.85E+02(1.27E+03)
CEC2010F8	10	4.34E+02(1.36E+03)	5.38E+01(7.45E+01)	9.33E+01(1.58E+02)	3.30E+02(1.49E+03)
	30	3.70E+04(8.88E+04)	8.12E+04(1.85E+05)	1.63E+05(3.57E+05)	3.53E+05(5.31E+05)
CEC2010F13	10	−5.70E+01(4.95E+00)	−5.61E+01(6.54E+00)	−5.69E+01(5.47E+00)	−5.64E+01(5.90E+00)
	30	−5.36E+01(4.47E+00)	−5.69E+01(4.12E+00)	−5.49E+01(4.26E+00)	−5.36E+01(5.36E+00)
CEC2010F14	10	1.10E+04(6.04E+04)	5.81E+00(1.24E+00)	5.86E+00(1.05E+00)	6.02E+00(1.46E+00)
	30	5.49E+01(7.94E+01)	8.05E+01(9.09E+01)	5.07E+01(3.07E+01)	1.11E+02(2.91E+02)
CEC2010F15	10	4.18E+03(2.26E+04)	1.78E+01(2.23E+01)	1.50E+02(3.78E+02)	1.49E+03(5.60E+03)
	30	3.22E+06(6.59E+06)	6.53E+06(1.03E+07)	1.49E+07(2.24E+07)	1.43E+07(1.18E+07)
CEC2017F1	10	1.24E−07(1.28E−07)	2.51E−07(8.23E−07)	1.40E−07(1.38E−07)	1.45E−07(1.66E−07)
	30	1.24E+03(7.02E+02)	1.36E+03(9.37E+02)	1.33E+03(7.72E+02)	1.56E+03(1.09E+03)
CEC2017F2	10	2.94E−07(2.93E−07)	2.50E−07(1.92E−07)	1.75E−07(3.06E−07)	1.70E−07(1.56E−07)
	30	7.99E+02(5.45E+02)	8.20E+02(5.71E+02)	8.61E+02(4.28E+02)	9.96E+02(5.19E+02)
CEC2017F4	10	4.63E+01(1.02E+01)	4.41E+01(1.12E+01)	4.64E+01(8.50E+00)	4.76E+01(8.24E+00)
	30	1.81E+02(1.59E+01)	1.81E+02(2.01E+01)	1.81E+02(2.12E+01)	1.81E+02(1.93E+01)
CEC2017F5	10	3.47E+00(1.01E+00)	2.89E+00(4.59E−01)	2.95E+00(1.61E+00)	2.51E+00(5.66E−01)
	30	3.53E+01(2.08E+01)	2.88E+01(5.74E+00)	2.74E+01(2.71E+00)	3.23E+01(1.73E+01)
CEC2017F13	10	5.16E+00(1.41E+01)	1.12E+01(2.32E+01)	1.11E+01(2.33E+01)	3.55E+00(1.12E+01)
	30	/	/	/	/
CEC2017F20	10	1.97E+00(4.28E−01)	1.97E+00(5.44E−01)	1.97E+00(3.82E−01)	2.09E+00(4.05E−01)
	30	9.76E+00(6.88E−01)	9.77E+00(6.47E−01)	9.80E+00(6.10E−01)	9.55E+00(6.87E−01)
AR		4.54	4.02	4.46	5.02

Table 11. Experimental results of AR under different r values in different dimensions

D	r = 2	r = 3	r = 4	r = 5	r = 6	r = 7	r = 8	r = 9
10	4.79	5.38	4.75	4.04	4.83	3.71	4.00	4.50
30	5.14	4.45	3.27	4.00	4.23	4.36	4.95	5.59

The proportion coefficient ω, which determines the population size of POSR, is varied from 0.1 to 0.8 with increments of 0.1 for comparison. As ω increases, the role of POSR becomes more significant, placing greater emphasis on objective optimization. However, if ω is too large, it will consume more FEs but may not achieve better performance. Tables 12 and 13 show that the AR value rises when ω is larger than 0.5. The AR value is the smallest when ω = 0.5, so ω is recommended to take 0.5.

Table 12. Experimental results of SI-SADE with different ω (0.1, 0.2, 0.3, 0.4)

Problem	D	ω = 0.1	ω = 0.2	ω = 0.3	ω = 0.4
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−4.65E−01(9.13E−02)	−4.67E−01(8.98E−02)	−4.67E−01(8.98E−02)	−4.65E−01(9.13E−02)
	30	−3.42E−01(3.93E−02)	−3.60E−01(2.88E−02)	−3.42E−01(3.43E−02)	−3.52E−01(2.92E−02)
CEC2010F7	10	9.60E+00(2.29E+01)	9.49E+00(2.29E+01)	9.49E+00(2.29E+01)	9.60E+00(2.29E+01)
	30	1.12E+03(1.70E+03)	1.51E+03(1.83E+03)	1.12E+03(1.70E+03)	1.51E+03(1.83E+03)
CEC2010F8	10	5.38E+01(7.45E+01)	5.38E+01(7.45E+01)	5.38E+01(7.45E+01)	5.38E+01(7.45E+01)
	30	1.26E+05(2.59E+05)	8.12E+04(1.85E+05)	1.26E+05(2.59E+05)	8.12E+04(1.85E+05)
CEC2010F13	10	−5.61E+01(6.54E+00)	−5.59E+01(6.40E+00)	−5.59E+01(6.40E+00)	−5.61E+01(6.54E+00)
	30	−5.46E+01(3.75E+00)	−5.46E+01(5.11E+00)	−5.36E+01(3.80E+00)	−5.45E+01(5.09E+00)
CEC2010F14	10	1.27E+01(3.77E+01)	8.44E+00(1.51E+01)	2.21E+01(9.06E+01)	8.40E+00(1.51E+01)
	30	4.83E+01(2.90E+01)	7.09E+01(7.55E+01)	5.27E+01(4.38E+01)	7.08E+01(7.58E+01)
CEC2010F15	10	1.37E+05(4.87E+05)	3.45E+05(1.30E+06)	3.13E+05(1.67E+06)	3.45E+05(1.30E+06)
	30	2.70E+12(1.48E+13)	5.41E+07(1.99E+08)	1.50E+07(3.23E+07)	5.41E+07(1.99E+08)
CEC2017F1	10	2.51E−07(8.23E−07)	5.34E−05(2.92E−04)	5.34E−05(2.92E−04)	2.51E−07(8.23E−07)
	30	1.36E+03(9.37E+02)	1.39E+03(8.19E+02)	1.36E+03(9.37E+02)	1.39E+03(8.19E+02)
CEC2017F2	10	2.50E−07(1.92E−07)	2.53E−07(1.94E−07)	2.53E−07(1.94E−07)	2.50E−07(1.92E−07)
	30	7.93E+02(3.76E+02)	9.25E+02(5.40E+02)	7.78E+02(3.81E+02)	9.25E+02(5.40E+02)
CEC2017F4	10	4.41E+01(1.12E+01)	4.38E+01(1.09E+01)	4.38E+01(1.09E+01)	4.41E+01(1.12E+01)
	30	1.81E+02(2.01E+01)	1.81E+02(2.26E+01)	1.81E+02(2.01E+01)	1.82E+02(2.55E+01)
CEC2017F5	10	2.94E+00(4.47E−01)	2.91E+00(4.58E−01)	3.01E+00(5.36E−01)	2.93E+00(4.35E−01)
	30	3.70E+01(2.39E+01)	2.90E+01(1.07E+01)	3.47E+01(2.40E+01)	2.90E+01(1.07E+01)
CEC2017F13	10	1.19E+01(2.37E+01)	3.37E+00(1.11E+01)	1.33E+01(2.41E+01)	3.35E+00(1.10E+01)
	30	/	/	/	/
CEC2017F20	10	1.97E+00(5.44E−01)	1.97E+00(4.87E−01)	1.97E+00(4.87E−01)	1.97E+00(5.44E−01)
	30	9.77E+00(6.47E−01)	9.93E+00(6.43E−01)	9.76E+00(6.44E−01)	9.93E+00(6.43E−01)
AR		4.76	4.63	4.80	4.96

Table 13. Experimental results of SI-SADE with different ω (0.5, 0.6, 0.7, 0.8)

Problem	D	ω = 0.5	ω = 0.6	ω = 0.7	ω = 0.8
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−4.65E−01(9.13E−02)	−4.65E−01(9.13E−02)	−4.65E−01(9.13E−02)	−4.65E−01(9.13E−02)
	30	−3.52E−01(2.92E−02)	−3.52E−01(2.92E−02)	−3.42E−01(3.93E−02)	−3.52E−01(2.92E−02)
CEC2010F7	10	9.60E+00(2.29E+01)	9.60E+00(2.29E+01)	9.60E+00(2.29E+01)	9.60E+00(2.29E+01)
	30	1.51E+03(1.83E+03)	1.51E+03(1.83E+03)	1.12E+03(1.70E+03)	1.51E+03(1.83E+03)
CEC2010F8	10	5.38E+01(7.45E+01)	5.38E+01(7.45E+01)	5.38E+01(7.45E+01)	5.38E+01(7.45E+01)
	30	8.12E+04(1.85E+05)	8.12E+04(1.85E+05)	1.26E+05(2.59E+05)	8.12E+04(1.85E+05)
CEC2010F13	10	−5.61E+01(6.54E+00)	−5.61E+01(6.54E+00)	−5.61E+01(6.54E+00)	−5.61E+01(6.54E+00)
	30	−5.69E+01(4.12E+00)	−5.46E+01(3.62E+00)	−5.49E+01(5.30E+00)	−5.48E+01(3.82E+00)
CEC2010F14	10	5.81E+00(1.24E+00)	5.70E+00(1.21E+00)	5.92E+00(1.39E+00)	1.11E+04(6.09E+04)
	30	8.05E+01(9.09E+01)	7.09E+01(5.99E+01)	6.21E+01(6.52E+01)	9.65E+01(2.17E+02)
CEC2010F15	10	1.78E+01(2.23E+01)	4.64E+01(7.53E+01)	1.63E+04(8.78E+04)	9.40E+02(4.38E+03)
	30	6.53E+06(1.03E+07)	4.10E+06(6.26E+06)	3.70E+06(6.63E+06)	3.72E+06(4.29E+06)
CEC2017F1	10	2.51E−07(8.23E−07)	2.51E−07(8.23E−07)	2.51E−07(8.23E−07)	2.51E−07(8.23E−07)
	30	1.36E+03(9.37E+02)	1.39E+03(8.19E+02)	1.36E+03(9.37E+02)	1.39E+03(8.19E+02)
CEC2017F2	10	2.50E−07(1.92E−07)	2.50E−07(1.92E−07)	2.50E−07(1.92E−07)	2.50E−07(1.92E−07)
	30	8.20E+02(5.71E+02)	8.83E+02(5.10E+02)	9.43E+02(5.42E+02)	8.95E+02(5.46E+02)
CEC2017F4	10	4.41E+01(1.12E+01)	4.41E+01(1.12E+01)	4.41E+01(1.12E+01)	4.41E+01(1.12E+01)
	30	1.81E+02(2.01E+01)	1.82E+02(2.55E+01)	1.81E+02(2.01E+01)	1.82E+02(2.55E+01)
CEC2017F5	10	2.89E+00(4.59E−01)	2.88E+00(4.47E−01)	2.92E+00(4.83E−01)	2.93E+00(4.18E−01)
	30	2.88E+01(5.74E+00)	3.30E+01(1.74E+01)	3.15E+01(1.48E+01)	2.69E+01(2.02E+00)
CEC2017F13	10	1.12E+01(2.32E+01)	9.14E+00(2.07E+01)	8.79E+00(2.01E+01)	8.57E+00(1.84E+01)
	30	/	/	/	/
CEC2017F20	10	1.97E+00(5.44E−01)	1.97E+00(5.44E−01)	1.97E+00(5.44E−01)	1.97E+00(5.44E−01)
	30	9.77E+00(6.47E−01)	9.93E+00(6.43E−01)	9.77E+00(6.47E−01)	9.93E+00(6.43E−01)
AR		3.70	4.39	4.11	4.65

Table 14. Experimental results of SI-SADE with different Trigger (0.6, 0.7, 0.8, 0.9)

Problem	D	Trigger = 0.6	Trigger = 0.7	Trigger = 0.8	Trigger = 0.9
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−4.78E−01(1.01E−01)	−4.71E−01(8.74E−02)	−4.65E−01(9.13E−02)	−4.73E−01(8.56E−02)
	30	−3.43E−01(4.13E−02)	−3.43E−01(3.49E−02)	−3.52E−01(2.92E−02)	−3.53E−01(3.08E−02)
CEC2010F7	10	1.16E+01(3.07E+01)	8.10E+00(1.28E+01)	9.60E+00(2.29E+01)	1.09E+01(3.04E+01)
	30	1.21E+03(1.98E+03)	1.24E+03(1.96E+03)	1.51E+03(1.83E+03)	1.49E+03(1.84E+03)
CEC2010F8	10	5.20E+01(6.94E+01)	5.62E+01(7.82E+01)	5.38E+01(7.45E+01)	5.52E+01(7.52E+01)
	30	7.63E+04(1.30E+05)	1.54E+05(2.56E+05)	8.12E+04(1.85E+05)	7.00E+04(1.55E+05)
CEC2010F13	10	−5.56E+01(6.34E+00)	−5.58E+01(6.46E+00)	−5.61E+01(6.54E+00)	−5.61E+01(6.59E+00)
	30	−5.52E+01(4.71E+00)	−5.52E+01(4.71E+00)	−5.69E+01(4.12E+00)	−5.69E+01(4.12E+00)
CEC2010F14	10	5.89E+00(1.27E+00)	5.96E+00(1.20E+00)	5.81E+00(1.24E+00)	5.86E+00(1.31E+00)
	30	5.29E+01(7.65E+01)	5.49E+01(6.48E+01)	8.05E+01(9.09E+01)	9.19E+01(1.36E+02)
CEC2010F15	10	1.78E+01(2.23E+01)	1.78E+01(2.23E+01)	1.78E+01(2.23E+01)	1.78E+01(2.23E+01)
	30	5.06E+06(7.81E+06)	5.06E+06(7.81E+06)	6.53E+06(1.03E+07)	6.53E+06(1.03E+07)
CEC2017F1	10	6.73E−05(3.68E−04)	3.68E−05(2.01E−04)	2.51E−07(8.23E−07)	1.14E−05(6.19E−05)
	30	2.07E+03(4.81E+03)	1.18E+03(8.55E+02)	1.36E+03(9.37E+02)	1.69E+03(1.16E+03)
CEC2017F2	10	1.52E−07(2.46E−07)	2.25E−07(2.09E−07)	2.50E−07(1.92E−07)	2.43E−07(1.94E−07)
	30	1.03E+03(1.27E+03)	6.73E+02(4.00E+02)	8.20E+02(5.71E+02)	1.05E+03(4.05E+02)
CEC2017F4	10	4.46E+01(1.12E+01)	4.46E+01(1.12E+01)	4.41E+01(1.12E+01)	4.38E+01(1.10E+01)
	30	1.81E+02(2.01E+01)	1.81E+02(2.01E+01)	1.81E+02(2.01E+01)	1.80E+02(1.84E+01)
CEC2017F5	10	3.06E+00(5.81E−01)	3.96E+00(6.39E−01)	2.89E+00(4.59E−01)	3.12E+00(5.07E−01)
	30	2.92E+01(8.15E+00)	2.95E+01(9.61E+00)	2.88E+01(5.74E+00)	2.88E+01(5.76E+00)
CEC2017F13	10	1.13E+01(2.32E+01)	1.13E+01(2.32E+01)	1.12E+01(2.32E+01)	1.12E+01(2.32E+01)
	30	/	/	/	/
CEC2017F20	10	1.97E+00(4.98E−01)	1.98E+00(4.33E−01)	1.97E+00(5.44E−01)	2.01E+00(3.81E−01)
	30	9.77E+00(6.73E−01)	9.77E+00(6.59E−01)	9.77E+00(6.47E−01)	9.53E+00(6.50E−01)
AR		2.59	2.83	2.26	2.33

The value of Trigger is set to 0.6, 0.7, 0.8, and 0.9 respectively. Using elite search too early will increase the density of real evaluated individuals around the current optimal individual, negatively affecting the optimization. If the strategy is applied too late, the effect of accelerating convergence will be weakened. From Table 14, the AR value is the lowest when the Trigger value is 0.8, so the Trigger value of 0.8 is recommended.

The value of θ is set to 0.1, 0.2, 0.3, 0.4, respectively. The value of θ is positively correlated with the trigger frequency of the global surrogate model. A higher θ value can provide an earlier intervention when the algorithm falls into the local optimum, which is beneficial to enhance the global search ability of the SI-SADE. However, a high value of θ may reduce the convergence speed and affect the algorithm's efficiency. On the contrary, a lower value of θ will reduce the execution probability of URAM, which limits the adaptability of the algorithm to the diversification problem. From Table 15, the AR value increases as θ value increases, so the recommended value for θ is 0.1.

Table 15. Experimental results of SI-SADE with different θ (0.1, 0.2, 0.3, 0.4)

Problem	D	θ = 0.1	θ = 0.2	θ = 0.3	θ = 0.4
		Mean(Std)	Mean(Std)	Mean(Std)	Mean(Std)
CEC2010F1	10	−4.65E−01(9.13E−02)	−5.13E−01(1.02E−01)	−5.10E−01(1.06E−01)	−5.30E−01(1.10E−01)
	30	−3.52E−01(2.92E−02)	−3.45E−01(3.86E−02)	−3.43E−01(3.43E−02)	−3.41E−01(3.48E−02)
CEC2010F7	10	9.60E+00(2.29E+01)	8.23E+00(1.47E+01)	1.48E+01(3.27E+01)	2.35E+01(9.71E+01)
	30	1.51E+03(1.83E+03)	1.10E+03(1.82E+03)	1.04E+03(1.89E+03)	1.10E+03(1.89E+03)
CEC2010F8	10	5.38E+01(7.45E+01)	1.22E+02(2.07E+02)	4.59E+02(1.83E+03)	2.28E+02(6.74E+02)
	30	8.12E+04(1.85E+05)	1.61E+05(2.28E+05)	4.14E+05(7.40E+05)	4.54E+05(8.26E+05)
CEC2010F13	10	−5.61E+01(6.54E+00)	−5.48E+01(6.68E+00)	−5.49E+01(6.53E+00)	−5.51E+01(6.71E+00)
	30	−5.69E+01(4.12E+00)	−5.60E+01(3.78E+00)	−5.62E+01(3.81E+00)	−5.57E+01(3.70E+00)
CEC2010F14	10	5.81E+00(1.24E+00)	6.17E+00(1.46E+00)	8.00E+00(1.08E+01)	6.24E+00(2.90E+00)
	30	8.05E+01(9.09E+01)	4.83E+01(3.07E+01)	7.88E+01(8.93E+01)	5.00E+01(3.14E+01)
CEC2010F15	10	1.78E+01(2.23E+01)	3.08E+02(1.46E+03)	3.09E+02(1.46E+03)	7.04E+01(1.77E+02)
	30	6.53E+06(1.03E+07)	9.49E+06(1.45E+07)	9.68E+06(1.70E+07)	1.00E+07(1.70E+07)
CEC2017F1	10	2.51E−07(8.23E−07)	1.33E−07(8.74E−08)	1.57E−07(1.12E−07)	1.57E−07(1.16E−07)
	30	1.36E+03(9.37E+02)	1.29E+03(7.87E+02)	1.33E+03(7.14E+02)	1.89E+03(2.28E+03)
CEC2017F2	10	2.50E−07(1.92E−07)	1.70E−04(9.30E−04)	2.66E−07(1.82E−07)	2.73E−07(1.78E−07)
	30	8.20E+02(5.71E+02)	8.33E+02(2.75E+02)	9.92E+02(6.86E+02)	9.90E+02(6.73E+02)
CEC2017F4	10	4.41E+01(1.12E+01)	4.57E+01(9.72E+00)	4.78E+01(1.04E+01)	4.76E+01(1.04E+01)
	30	1.81E+02(2.01E+01)	1.84E+02(1.66E+01)	1.77E+02(2.34E+01)	1.79E+02(2.25E+01)
CEC2017F5	10	2.89E+00(4.59E−01)	2.87E+00(4.41E−01)	2.92E+00(4.47E−01)	2.97E+00(4.61E−01)
	30	2.88E+01(5.74E+00)	2.88E+01(5.73E+00)	2.85E+01(5.10E+00)	2.86E+01(5.33E+00)
CEC2017F13	10	1.12E+01(2.32E+01)	7.52E+00(1.86E+01)	7.24E+00(1.87E+01)	5.20E+00(1.55E+01)
	30	/	/	/	/
CEC2017F20	10	1.97E+00(5.44E−01)	2.08E+00(4.25E−01)	2.04E+00(4.50E−01)	2.02E+00(3.81E−01)
	30	9.77E+00(6.47E−01)	9.79E+00(7.61E−01)	9.59E+00(6.62E−01)	9.59E+00(6.62E−01)
AR		2.11	2.43	2.70	2.76

Real-world application

The problem of polyline-based core sandwich structures (PCSSs) optimization is employed to validate the performance of SI-SADE in a real-world application. In PCSSs, a sandwich panel consists of two laminated faces and a core. In this optimization example, two panels are used to construct the design domain, but they are not included as parameter variables in the optimization process. The sandwich layer consists of a polyline-based core structure (PCS), which is composed of repeated unit cell arrays. Given the periodic structure of PCS, optimization can be performed by focusing on a single unit cell rather than the entire structure. In addition, the unit cell is composed of multiple 2D basic components, and the schematic diagram of the 2D basic components is plotted in Fig. 6. Adjusting the center point coordinates of the two ends and the thickness of the rectangular structure facilitates the movement and expansion of the 2D component. Finally, the upper and lower panels are assembled with PCS to form the final PCSS. The schematic diagram illustrating the construction of the PCSS is presented in Fig. 7.

[See PDF for image]

Fig. 6

Depiction for x–y-plane cross section of the unit cell. a 2D basic component, bx–y-plane cross section of the unit cell

[See PDF for image]

Fig. 7

Illustration for construction of PCSS: ax–y-plane cross section of the unit cell; bx–y-plane cross section of the polyline-based core; c polyline-based core; d assembling of the PCSS; e design of the PCSS

The design variables for the PCSSs are entirely derived from the unit cell. The optimization objective for the PCSSs is to minimize compliance, with constraints including maintaining a volume fraction below 0.3 and ensuring that the lower bound of the angle constraint is 0.2π. Notably, compliance is computed using the MATLAB simulation module, which is computationally expensive. The design domain of PCSSs is shown in Fig. 8.

[See PDF for image]

Fig. 8

Design domain in 3D PCSSs

In this application, the problem to be optimized is defined as follows:

21

$$\begin{array}{cc}\hfill cos\left({\alpha }_l\right)& =\frac{\left(x_{l-1}-x_l\right)\left(x_{l+1}-x_l\right)+\left(y_{l-1}-y_l\right)\left(y_{l+1}-y_l\right)}{\sqrt{{\left(x_{l-1}-x_l\right)}^2+{\left(y_{l-1}-y_l\right)}^2}\sqrt{{\left(x_{l+1}-x_l\right)}^2+{\left(y_{l+1}-y_l\right)}^2}},\hfill \\ \hfill & l=1,2,\dots ,n-1\hfill \end{array}$$

22

$$\begin{array}{cc}\hfill \text{Find}& \tilde{\text{D}}={\left(y_1^{1,1,1},y_1^{2,1,1},\dots ,y_1^{M,1,1},t_{2D}\right)}^T\hfill \\ \hfill \text{Min}& C={\int }_{D}H\left(\varphi (\text{x};\tilde{\text{D}})\right)\text{fu}dV+{\int }_{{\mathrm{\Gamma }}_t}\text{tu}dS\hfill \\ \hfill \text{S.t}.& {\int }_{D}H{\left(\varphi (\text{x};\tilde{\text{D}})\right)}^q\text{E}:\mathrm{\epsilon }(\text{u}):\mathrm{\epsilon }(\text{v})dV\hfill \\ \hfill & ={\int }_{D}H\left(\varphi (\text{x};\tilde{\text{D}})\right)\text{fv}dV+{\int }_{{\mathrm{\Gamma }}_t}\text{tv}dS,\forall \text{v}\in U_{\mathit{ad}}\hfill \\ \hfill & {\int }_{D}H\left(\varphi (\text{x};\tilde{\text{D}})\right)dV\le \overline{V}\hfill \\ \hfill & cos\left({\alpha }_l\right)\le cos\left(\underline{\alpha }\right)\hfill \\ \hfill & \tilde{\text{D}}\subset U_{\tilde{D}}\hfill \\ \hfill & \text{u}\subset \overline{\text{u}},\text{on}{\mathrm{\Gamma }}_u\hfill \\ \hfill & l=1,2,\dots ,M\hfill \\ \hfill \end{array}$$

The effectiveness of SI-SADE is fully verified by comparing it with five competitors, such as MPMLS [25], SParEA [54], FMSADE [18], and GLoSADE [22]. The maximum number of function evaluations is set to 3000, and the obtained structures for the proposed SI-SADE and compared algorithms are plotted in Fig. 9.

[See PDF for image]

Fig. 9

The structures optimized by the proposed SI-SADE and compared algorithms

The information in Table 16 proves that SI-SADE performs better than other comparison algorithms in dealing with a real-world problem. Moreover, in Fig. 9, although the three algorithms, SI-SADE, FMSADE, and GLoSADE, obtain similar structures, the compliance obtained by SI-SADE is significantly lower than all the compared algorithms.

Table 16. Comparison results on solving PCSSs

Methods	SI-SADE	MPMLS	SParEA	FMSADE	GLoSADE
Comp	1488.02	1649.88	1926.65	1505.02	1504.64

Conclusions

This paper proposes a new surrogate-assisted evolutionary algorithm that fully utilizes the state information of the population. Specifically, SI-SADE categorizes the feasible states of the population into three states: infeasible, partially feasible, and fully feasible. Various evolutionary strategies and selection strategies are assigned based on individual information. In the infeasible state, the population is encouraged to enter and search the feasible region. In the partially feasible state, the exploitation of the region where the infeasible solution with excellent information is actively strengthened. In the fully feasible state, the feasible region is carefully exploited. Therefore, SI-SADE achieves a balance between diversity and convergence. In addition, the search ability of the algorithm is improved by expanding the parent and dynamically switching the global surrogate. Finally, the constraints and objectives are optimized sequentially in the infeasible state, and the algorithm can traverse the complex infeasible region.

The effectiveness of the proposed strategies in SI-SADE is validated by comparing different variants of the SI-SADE. The results show that: (1) the population can achieve a balance between diversity and convergence by assigning mutation operations that emphasize exploration or exploitation and environmental selection that relax constraints or not; (2) parent expansion and dynamic switching of the global surrogate significantly improve the search ability of the algorithm; (3) objective-based search has a significant effect in dealing with complex constraint problems.

Moreover, SI-SADE demonstrates strong competitiveness in benchmark problems and a real-world case. However, in the face of complex constraints, and the directions of constraints and the objective optimization overlap but point to the deeper infeasible region, both constraints and the objective will mislead SI-SADE, and solving such problems is still challenging.

Future work will study how to improve strategies in SI-SADE so that it can operate independently without relying on the population state.

Acknowledgements

This research is supported by the National Natural Science Foundation of China [grant numbers 52465029; 52305296]; Xiaolan Economic Development Zone of Nanchang County for the support of the 2024 Unveiling the List and Taking the Lead project [grant numbers ZJ-YF-001]; Young Talent Cultivation Innovation Fund Project of Nanchang University [grant numbers 9302-03740040]; Topology optimization design of multi-scale composite porous metamaterials [grant numbers BSKYCXZX 2023-07]; Natural Science Foundation of Hunan Province of China [grant number 2024JJ6501]; Guangdong Basic and Applied Basic Research Foundation [grant number 2024A1515011653].

Author contribution

ZZ: Conceptualization, Methodology, Validation, Formal analysis, Data Curation, Writing—Original Draft. ZY: Methodology, Validation, Formal analysis, Visualization, Supervision, Funding acquisition. ZL: Methodology, Writing—Review & Editing, Funding acquisition. LC: Formal analysis, Writing—Review & Editing, Funding acquisition. XC: Visualization, Writing—Review & Editing, Funding acquisition.

Data availability

Data will be made available on request.

Declaration

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Sobieszczanski-Sobieski, J; Haftka, RT. Multidisciplinary aerospace design optimization: survey of recent developments. Struct Optim; 1997; 14, pp. 1-23. [DOI: https://dx.doi.org/10.1007/BF01197554]

2. Papadrakakis, M; Lagaros, ND; Tsompanakis, Y. Structural optimization using evolution strategies and neural networks. Comput Methods Appl Mech Eng; 1998; 156, pp. 309-333. [DOI: https://dx.doi.org/10.1016/S0045-7825(97)00215-6]

3. Xu, P; Luo, W; Lin, X; Qiao, Y. Evolutionary continuous constrained optimization using random direction repair. Inf Sci (N Y); 2021; 566, pp. 80-102.4236418 [DOI: https://dx.doi.org/10.1016/j.ins.2021.02.055]

4. D’Angelo, G; Palmieri, F. GGA: A modified genetic algorithm with gradient-based local search for solving constrained optimization problems. Inf Sci (N Y); 2021; 547, pp. 136-162.4142180 [DOI: https://dx.doi.org/10.1016/j.ins.2020.08.040]

5. Jiao, R; Zeng, S; Li, C et al. A complete expected improvement criterion for Gaussian process assisted highly constrained expensive optimization. Inf Sci (N Y); 2019; 471, pp. 80-96. [DOI: https://dx.doi.org/10.1016/j.ins.2018.09.003]

6. Dong, H; Song, B; Dong, Z; Wang, P. SCGOSR: Surrogate-based constrained global optimization using space reduction. Appl Soft Comput; 2018; 65, pp. 462-477. [DOI: https://dx.doi.org/10.1016/j.asoc.2018.01.041]

7. Liu, Y; Liu, J; Jin, Y et al. A surrogate-assisted two-stage differential evolution for expensive constrained optimization. IEEE Trans Emerg Top Comput Intell; 2023; 7, pp. 715-730. [DOI: https://dx.doi.org/10.1109/TETCI.2023.3240221]

8. Wang, B-C; Li, H-X; Li, J-P; Wang, Y. Composite Differential Evolution for Constrained Evolutionary Optimization. IEEE Trans Syst Man Cybern Syst; 2019; 49, pp. 1482-1495. [DOI: https://dx.doi.org/10.1109/TSMC.2018.2807785]

9. Yang, Z; Qiu, H; Gao, L et al. Surrogate-assisted classification-collaboration differential evolution for expensive constrained optimization problems. Inf Sci (N Y); 2020; 508, pp. 50-63.3997614 [DOI: https://dx.doi.org/10.1016/j.ins.2019.08.054]

10. Sun, G; Yang, G; Zhang, G. Two-level parameter cooperation-based population regeneration framework for differential evolution. Swarm Evol Comput; 2022; 75, [DOI: https://dx.doi.org/10.1016/j.swevo.2022.101122] 101122.

11. Deng, L-B; Li, C-L; Sun, G-J. An adaptive dimension level adjustment framework for differential evolution. Knowl Based Syst; 2020; 206, [DOI: https://dx.doi.org/10.1016/j.knosys.2020.106388] 106388.

12. Zhao, K; Wang, P; Tong, X. An adaptive two-population evolutionary algorithm for constrained multi-objective optimization problems. IEEE Access; 2023; 11, pp. 82118-82131. [DOI: https://dx.doi.org/10.1109/ACCESS.2023.3300590]

13. Wang, Y; Cai, Z; Zhou, Y; Zeng, W. An adaptive tradeoff model for constrained evolutionary optimization. IEEE Trans Evol Comput; 2008; 12, pp. 80-92. [DOI: https://dx.doi.org/10.1109/TEVC.2007.902851]

14. Deb K, Agrawal S, Pratap A, Meyarivan T (2000) A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II, pp 849–858. https://doi.org/10.1007/3-540-45356-3_83

15. Wang, Y; Cai, Z. Constrained evolutionary optimization by means of (μ + λ)-differential evolution and improved adaptive trade-off model. Evol Comput; 2011; 19, pp. 249-285. [DOI: https://dx.doi.org/10.1162/EVCO_a_00024]

16. Gong, W; Cai, Z; Liang, D. Adaptive ranking mutation operator based differential evolution for constrained optimization. IEEE Trans Cybern; 2015; 45, pp. 716-727. [DOI: https://dx.doi.org/10.1109/TCYB.2014.2334692]

17. Jia, G; Wang, Y; Cai, Z; Jin, Y. An improved (μ+λ)-constrained differential evolution for constrained optimization. Inf Sci (N Y); 2013; 222, pp. 302-322.2998515 [DOI: https://dx.doi.org/10.1016/j.ins.2012.01.017]

18. Chu, S; Yang, Z; Xiao, M et al. Explicit topology optimization of novel polyline-based core sandwich structures using surrogate-assisted evolutionary algorithm. Comput Methods Appl Mech Eng; 2020; 369, 4117353 [DOI: https://dx.doi.org/10.1016/j.cma.2020.113215] 113215.

19. Li, C; Sun, G; Deng, L et al. A population state evaluation-based improvement framework for differential evolution. Inf Sci (N Y); 2023; 629, pp. 15-38. [DOI: https://dx.doi.org/10.1016/j.ins.2023.01.120]

20. Zhao, X; Zhang, K; Chen, G et al. Surrogate-assisted differential evolution for production optimization with nonlinear state constraints. J Pet Sci Eng; 2020; 194, [DOI: https://dx.doi.org/10.1016/j.petrol.2020.107441] 107441.

21. Regis, RG. Evolutionary programming for high-dimensional constrained expensive black-box optimization using radial basis functions. IEEE Trans Evol Comput; 2014; 18, pp. 326-347. [DOI: https://dx.doi.org/10.1109/TEVC.2013.2262111]

22. Wang, Y; Yin, D-Q; Yang, S; Sun, G. Global and local surrogate-assisted differential evolution for expensive constrained optimization problems with inequality constraints. IEEE Trans Cybern; 2019; 49, pp. 1642-1656. [DOI: https://dx.doi.org/10.1109/TCYB.2018.2809430]

23. Zeng, Y; Cheng, Y; Liu, J. A surrogate-assisted constrained optimization evolutionary algorithm by searching multiple kinds of global and local regions. IEEE Trans Evol Comput; 2023; 29, pp. 61-75. [DOI: https://dx.doi.org/10.1109/TEVC.2023.3346435]

24. Ong, YS; Nair, PB; Keane, AJ. Evolutionary optimization of computationally expensive problems via surrogate modeling. AIAA J; 2003; 41, pp. 687-696. [DOI: https://dx.doi.org/10.2514/2.1999]

25. Li, G; Zhang, Q. Multiple penalties and multiple local surrogates for expensive constrained optimization. IEEE Trans Evol Comput; 2021; 25, pp. 769-778. [DOI: https://dx.doi.org/10.1109/TEVC.2021.3066606]

26. Yang, Z; Qiu, H; Gao, L et al. A general framework of surrogate-assisted evolutionary algorithms for solving computationally expensive constrained optimization problems. Inf Sci (N Y); 2023; 619, pp. 491-508. [DOI: https://dx.doi.org/10.1016/j.ins.2022.11.021]

27. Su, Y; Xu, L; Goodman, ED. Hybrid surrogate-based constrained optimization with a new constraint-handling method. IEEE Trans Cybern; 2022; 52, pp. 5394-5407. [DOI: https://dx.doi.org/10.1109/TCYB.2020.3031620]

28. Wang, Y; Cai, Z. A dynamic hybrid framework for constrained evolutionary optimization. IEEE Trans Syst Man Cybern Part B (Cybern); 2012; 42, pp. 203-217. [DOI: https://dx.doi.org/10.1109/TSMCB.2011.2161467]

29. Pan, J-S; Liang, Q; Chu, S-C et al. A multi-strategy surrogate-assisted competitive swarm optimizer for expensive optimization problems. Appl Soft Comput; 2023; 147, 110733. [DOI: https://dx.doi.org/10.1016/j.asoc.2023.110733]

30. Mahdavi, S; Rahnamayan, S; Deb, K. Opposition based learning: a literature review. Swarm Evol Comput; 2018; 39, pp. 1-23. [DOI: https://dx.doi.org/10.1016/j.swevo.2017.09.010]

31. Wu, M; Xu, J; Wang, L et al. Adaptive multi-surrogate and module-based optimization algorithm for high-dimensional and computationally expensive problems. Inf Sci (N Y); 2023; 645, 119308. [DOI: https://dx.doi.org/10.1016/j.ins.2023.119308]

32. Ray T, Singh HK, Isaacs A, Smith W (2009) Infeasibility driven evolutionary algorithm for constrained optimization. pp 145–165. https://doi.org/10.1007/978-3-642-00619-7_7

33. Yang, Z; Kumar, A; Mallipeddi, R; Lee, D-G. Spherical search with epsilon constraint and gradient-based repair framework for constrained optimization. Swarm Evol Comput; 2023; 82, [DOI: https://dx.doi.org/10.1016/j.swevo.2023.101370] 101370.

34. Jiao, R; Zeng, S; Li, C. A feasible-ratio control technique for constrained optimization. Inf Sci (N Y); 2019; 502, pp. 201-217.3963685 [DOI: https://dx.doi.org/10.1016/j.ins.2019.06.030]

35. Deb, K. An efficient constraint handling method for genetic algorithms. Comput Methods Appl Mech Eng; 2000; 186, pp. 311-338. [DOI: https://dx.doi.org/10.1016/S0045-7825(99)00389-8]

36. Zhang, Z; Wang, Y; Liu, J et al. A two-phase kriging-assisted evolutionary algorithm for expensive constrained multiobjective optimization problems. IEEE Trans Syst Man Cybern Syst; 2024; 54, pp. 4579-4591. [DOI: https://dx.doi.org/10.1109/TSMC.2024.3383219]

37. Zhang, K; Xu, Z; Yen, GG; Zhang, L. Two-stage multiobjective evolution strategy for constrained multiobjective optimization. IEEE Trans Evol Comput; 2024; 28, pp. 17-31. [DOI: https://dx.doi.org/10.1109/TEVC.2022.3202723]

38. Ma, H; Wei, H; Tian, Y et al. A multi-stage evolutionary algorithm for multi-objective optimization with complex constraints. Inf Sci (N Y); 2021; 560, pp. 68-91.4217881 [DOI: https://dx.doi.org/10.1016/j.ins.2021.01.029]

39. Song, Z; Wang, H; Xue, B et al. Balancing objective optimization and constraint satisfaction in expensive constrained evolutionary multiobjective optimization. IEEE Trans Evol Comput; 2024; 28, pp. 1286-1300. [DOI: https://dx.doi.org/10.1109/TEVC.2023.3300181]

40. Wang, Y; Li, J-P; Xue, X; Wang, B. Utilizing the correlation between constraints and objective function for constrained evolutionary optimization. IEEE Trans Evol Comput; 2020; 24, pp. 29-43. [DOI: https://dx.doi.org/10.1109/TEVC.2019.2904900]

41. Gutmann, H-M. A radial basis function method for global optimization. J Global Optim; 2001; 19, pp. 201-227.1833217 [DOI: https://dx.doi.org/10.1023/A:1011255519438]

42. Takahama T, Sakai S (2010) Efficient constrained optimization by the ε constrained adaptive differential evolution. In: Evolutionary computation (CEC), 2010 IEEE Congress on. IEEE, pp 1–9. https://doi.org/10.1109/CEC.2010.55 86484

43. Runarsson, TP; Yao, X. Stochastic ranking for constrained evolutionary optimization. IEEE Trans Evol Comput; 2000; 4, pp. 284-294. [DOI: https://dx.doi.org/10.1109/4235.873238]

44. Wang, B-C; Li, H-X; Feng, Y; Shen, W-J. An adaptive fuzzy penalty method for constrained evolutionary optimization. Inf Sci (N Y); 2021; 571, pp. 358-374.4259251 [DOI: https://dx.doi.org/10.1016/j.ins.2021.03.055]

45. Rahimi, I; Gandomi, AH; Chen, F; Mezura-Montes, E. A review on constraint handling techniques for population-based algorithms: from single-objective to multi-objective optimization. Arch Comput Methods Eng; 2023; 30, pp. 2181-2209. [DOI: https://dx.doi.org/10.1007/s11831-022-09859-9]

46. Zhong, J-H; Shen, M; Zhang, J et al. A differential evolution algorithm with dual populations for solving periodic railway timetable scheduling problem. IEEE Trans Evol Comput; 2013; 17, pp. 512-527. [DOI: https://dx.doi.org/10.1109/TEVC.2012.2206394]

47. Mckay, MD; Beckman, RJ; Conover, WJ. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics; 2000; 42, pp. 55-61. [DOI: https://dx.doi.org/10.1080/00401706.2000.10485979]

48. Mallipeddi R, Suganthan PN, Mallipeddi R, Suganthan PN (2010) Problem definitions and evaluation criteria for the CEC 2010 competition on constrained real-parameter optimization

49. Zhang, J; Sanderson, AC. JADE: adaptive differential evolution with optional external archive. IEEE Trans Evol Comput; 2009; 13, pp. 945-958. [DOI: https://dx.doi.org/10.1109/TEVC.2009.2014613]

50. Tian, Y; Zhang, Y; Su, Y et al. Balancing objective optimization and constraint satisfaction in constrained evolutionary multiobjective optimization. IEEE Trans Cybern; 2022; 52, pp. 9559-9572. [DOI: https://dx.doi.org/10.1109/TCYB.2020.3021138]

51. Huang, H; Xu, Y; Xiang, Y; Hao, Z. Correlation-based dynamic allocation scheme of fitness evaluations for constrained evolutionary optimization. IEEE Trans Evol Comput; 2024; 28, pp. 1250-1264. [DOI: https://dx.doi.org/10.1109/TEVC.2023.3302897]

52. Song, Z; Wang, H; Jin, Y. A surrogate-assisted evolutionary framework with regions of interests-based data selection for expensive constrained optimization. IEEE Trans Syst Man Cybern Syst; 2023; 53, pp. 6268-6280. [DOI: https://dx.doi.org/10.1109/TSMC.2023.3281822]

53. Wu G, Mallipeddi R, Suganthan PN (2017) Problem definitions and evaluation criteria for the CEC 2017 competition and special session on constrained single objective real-parameter optimization

54. Rahi, KH; Singh, HK; Ray, T. Partial evaluation strategies for expensive evolutionary constrained optimization. IEEE Trans Evol Comput; 2021; 25, pp. 1103-1117. [DOI: https://dx.doi.org/10.1109/TEVC.2021.3078486]