Introduction

Evolutionary algorithms (EAs), including differential evolution (DE) [1,2,3], particle swarm optimization (PSO) [4, 5], and genetic algorithm (GA)[6, 7], have demonstrated remarkable search performance in numerous practical engineering problems [8], such as optimization design of internal combustion engine [9] and building designs [10]. However, real-world engineering problems often involve complex constraints, which are referred as constrained optimization problems (COPs). For example, in the topology optimization of 2D structures [11, 12], geometric size and load must be taken into account. The combination of EAs and constraint handling techniques (CHTs) shows a powerful ability to solve COPs, which can bias the population search to the feasible domain rather than the entire design space. Commonly used CHTs include penalty function methods [13,14,15], feasibility rules [16], stochastic ranking [17], and multi-objective optimization techniques [18]. In fact, the optimization objective and constraints of real-world engineering problems often involve expensive simulations (one simulation may take several minutes to several hours), which are referred as expensive constrained optimization problems (ECOPs), such as drug design [19], aerodynamic design [20], reliability analysis [21,22,23]. To address ECOPs, surrogate models, which are computationally cheap and able to provide accurate prediction on solutions, are used in EAs to replace many unnecessary time-consuming evaluations on poor solutions. This means that with the assistance of surrogate models, the number of evaluations for expensive simulations can be greatly reduced, thereby significantly reducing the computational cost required for the overall optimization process. Common surrogate models include radial basis functions (RBF) [24,25,26], Kriging models [27, 28], artificial neural networks (ANN) [29], and support vector machines (SVMs) [30, 31].

For complex applications such as topology optimization of sandwich structure design [12, 32], the design parameters that need to be considered often involve continuous variables (such as the size, thickness and displacement of materials) and integer variables (such as the number and the arrangement of materials). Such problems can be called expensive constrained problems with mixed-integer variables (ECOPs-MI). Figure 1 shows the projection diagrams of the feasible regions of an artificial ECOPs where the expressions of two inequality constraints are \({{x}_{3}}^{2}\le {\left(\left|{x}_{1}\right|-5\right)}^{2}+{\left(\left|{x}_{2}\right|-5\right)}^{2}\le {\left({x}_{3}+0.5\right)}^{2}\) under two different combinations of variable types. For the first combination, all the variables are continuous and the range of \({x}_{3}\) is \(\left[\mathrm{0,3}\right]\), thus the feasible region is composed of four disconnected sub-feasible regions shown in red line areas. By contrast to the first combination, only \({x}_{3}\) is changed into integer variable in the second combination and the range of \({x}_{3}\) is \(\{\mathrm{0,1},\mathrm{2,3}\}\), thus sixteen disconnected sub-feasible regions are generated. Therefore, it can be easily observed that the feasible region is divided into multiple disjoint sub-feasible regions by discrete variables, and the number of sub-feasible regions may increase as the number of discrete variables increases. This means that the adaptive ability to the real-time optimization state of the algorithm designed for solving such ECOPs-MI should be emphasized to avoid falling into these complex disjoint sub-feasible regions.

Fig. 1
figure 1

The projection diagram of the feasible region

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However, few of researches focus on solving ECOPs-MI, and these SAEAs mainly devote to designing efficient mutation and selection operations suitable for mixed-integer variables. Brownlee and Wright [10] used RBF as a surrogate model to assist the NSGA-II and applied it to typical building optimization problems, achieving better results. Pelamatti et al. [33] integrated the expected improvement criterion and the feasibility probability criterion to prescreen the offspring solutions. Liu et al. [34] proposed an adaptive prescreening by introducing a distance constraint, thus preventing the population from falling into the boundary regions. In a word, two findings can be summarized in the existing researches. First, the optimization process of these existing SAEAs cannot be adjusted based on real-time optimization state. Second, the existing studies are limited to developing single surrogate-assisted prescreening or local search strategy. To deal with the first finding, the real-time population state-driven surrogate-assisted optimization framework is designed in this paper to handling these complex landscapes caused by mixed-integer variables. Moreover, the collaborative framework between surrogate-assisted global and local searches has shown excellent abilities in dealing with other similar problems such as ECOPs with only continuous variables [12, 28, 32, 35,36,37,38]. In our previous work [12], a layer-by-layer feasible region-driven local search was designed to generate a high-quality initial population. Then three different search strategies were used. Zeng et al. [39] proposed a novel three-stage infilling framework to solve ECOPs where the three different search goals such as locating feasible areas, optimizing feasible objective values and improving accuracy were achieved in their respective stages. In Dong et al. [40], a Kriging-assisted two-phase TLBO optimization framework was constructed to alternately conduct local and global searches. Li et al. [38] proposed a three-level radial basis function (TLRBF)-assisted optimization algorithm for expensive optimization where three search procedures at each iteration such as global exploration, subregion search and local exploitation were included. Wang et al. [35] proposed a novel global and local surrogate-assisted differential evolution, in which generalized regression neural networks assisted prescreening and local RBF-based search strategies were designed. Chu et al. [32] utilized a feedback mechanism to drive an adaptive search for prescreening candidate solutions, followed by local exploitation of promising regions. Therefore, the collaborative framework of global and local surrogate-assisted searches is used in this paper to balance the global exploration and local exploitation.

More importantly, designing appropriate evolutionary operations for mixed-integer variables is also a key point to improve algorithm performance. But the evolutionary operations of the existing SAEAs are limited to two simple ways, i.e., continuous mutation + rounding [34, 41, 42], and hybrid mutation respectively for continuous and integer variables [43, 44]. A mixed-integer surrogate optimization algorithm named SO-MI [41] was proposed in which multiple sets of candidate solutions were generated by perturbing the continuous and integer variables of the current feasible solution, but only simple rounding was used to process integer variables. Liu et al. [34] utilized adaptive pre-screening operations and novel variation strategies to address ECOPs-MI where the integer constraints were satisfied by rounding the corresponding variable values to the closest integers. Xie et al. [42] introduced a dual-state-driven evolutionary optimization, where two different RBF-assisted ant colony optimization stages were integrated to enhance search performance in mixed and continuous variable spaces. Liu and Wang [43] combined the global and local modules to enhance the search capabilities on disconnected areas where the Gower distance based RBFN was employed in the global module to screen the solutions generated by hybrid evolutionary operations. Then the two different local search strategies were designed in the local module to refine the solutions in mixed and continuous variable space. In [44], a multisurrogate-assisted ant colony optimization algorithm was proposed where category and continuous variables were generated and selected in different ways. In a word, the above two types of evolutionary operations receive widespread attention, but there are still some challenges in searching mixed-integer space. First, although the first method is simple to implement, it is easy to fall into local optimization due to the lack of population diversity in the late stage of evolution. By contrast, the second method increases the diversity of the population, but the coupling relationship between continuous and integer variables is largely ignored, and the existing hybrid mutation mechanisms cannot achieve effective communication and inheritance between valuable parent individuals in integer space since the mutation is conducted in the encoded space instead of the original design space. However, such communication and inheritance can be easily achieved by the classical differential mutation on continuous variables. Therefore, to inherit the advantage of classical differential mutation while maintaining the diversity of solutions in integer space, the probability-driven mixed-integer mutation is designed where the individuals used in mutation and evaluation operations are handled in different manners, i.e., classical differential mutation maintains communication and the probability-driven evaluation increases diversity.

Based on the aforementioned discussions, this paper proposes a population state-driven surrogate-assisted differential evolution algorithm (PSSADE) where the population state information is identified and employed to adaptively adjust the optimization trajectory generated by the collaborative framework of global RBF pre-screening and RBF-based local search.

The main contributions of this paper can be summarized as follows:

  1. 1.

    By effectively combining the adaptive population update mechanism (APUM) and the collaborative framework of global and local surrogate-assisted search (CFGLS), PSSADE is able to escape local optima and find global optima, effectively balancing global exploration and local exploitation.

  2. 2.

    Probability-driven mixed-integer mutation (PMIU) is proposed, which mainly includes two parts, the adaptive individual construction strategy (AICS) for mutation stagnation and the probability-driven integer refinement operation (PIRO) for the serious loss of diversity.

  3. 3.

    In CFGLS, the PMIU is incorporated into the classical global DE/rand/2 and local DE/best/2 for improving the diversity and potentials of candidate solutions, and the collaborative framework further integrates both the superiority of global and local mutation to achieve a good balance between exploration and exploitation.

  4. 4.

    When the population distribution becomes too dense, PSSADE adaptively reselects the current population based on the efficient non-dominated sorting technique in the APUM, helping the algorithm escape local optima and find global optima.

  5. 5.

    Overall experiments have been conducted to study the performance of PSSADE and compare it with three state-of-the-art algorithms. Moreover, PSSADE is successfully applied to solve a practical ECOPs-MI, i.e., the design of infrared stealth material film.

The rest of the paper is organized as follows. The mathematical model and relative techniques are presented in “Problem statement and relative techniques”. A detailed description of the proposed method is presented in “The proposed algorithm”. The experimental results are discussed in “Experimental studies”. The effectiveness of PSADE in solving a real-world application problem is given in “Application in the design of infrared stealth material film”. Finally, the conclusions are given in “Conclusions”.

Problem statement and relative techniques

Problem statement

General mixed-integer constrained optimization problems with inequality constrains can be formulated as follows [45]:

$$ \begin{gathered} {\text{Minimize }}f\left( x \right) \hfill \\ {\text{subject to }}g_{j} \left( x \right) \le 0{ }j = 1,2, \ldots ,m \hfill \\ y_{{i_{1} }}^{l} \le y_{{i_{1} }} \le y_{{i_{1} }}^{u} { }i_{1} = 1,2, \ldots ,d_{1} \hfill \\ z_{{i_{2} }}^{l} \le z_{{i_{2} }} \le z_{{i_{2} }}^{u} { }i_{2} = 1,2, \ldots ,d_{2} \hfill \\ y \in R^{{d_{1} }} ,{ }z \in Z^{{d_{2} }} ,{ }x^{T} = \left( {y^{T} ,z^{T} } \right) \hfill \\ \end{gathered} $$
(1)

where \({y}_{{i}_{1}}^{l}\) and \({y}_{{i}_{1}}^{u}\) denote the lower and upper bounds, respectively, on the continuous variables \({y}_{{i}_{1}}\), \({i}_{1}=\mathrm{1,2},\dots ,{d}_{1}\), and where \({z}_{{i}_{2}}^{l}\) and \({z}_{{i}_{2}}^{u}\) denote the lower and upper bounds, respectively, on the integer variables \({z}_{{i}_{2}}\), \({i}_{2}=\mathrm{1,2},\dots ,{d}_{2}\). The \(j\) th computationally expensive constraint function is denoted by \({g}_{j}\left(x\right)\). The dimension of the mixed-integer optimization problem is denoted by \(d={d}_{1}+{d}_{2}\).

Relative techniques

Radial basis function (RBF)

The RBF [46] model is a machine learning model based on radial basis functions. It maps input data to a high-dimensional feature space and represents the similarity between data points using radial basis functions. The RBF model demonstrates good performance in handling nonlinear problems. It is widely applied in various fields such as pattern recognition, regression analysis, and time series forecasting. Given \(m\) distinct points \({x}_{1},...,{x}_{m}\in {R}^{n}\) and their function values \(f\left({x}_{1}\right),...,f\left({x}_{m}\right)\), the RBF approximation can be expressed as follows:

$$ \hat{f}\left( x \right) = w^{T} \varphi \mathop \sum \limits_{i = 1}^{m} \omega_{i} \phi \left( {\parallel x - x_{i} \parallel } \right) $$
(2)

where \(\parallel \cdot \parallel\) is the Euclidean norm, \({\omega }_{i}\in R\) for \(i=1,...,m\), \({x}_{i}\) denotes the \(i\) th of \(m\) basis function centers, and \(\varphi \) is an \(m\times 1\) vector containing the values of the basis function \(\phi \left( \cdot \right)\), and the \(i\) th value of \(\varphi \) is the Euclidean distance between the prediction site \(x\) and the center \({x}_{i}\) of the basis function. The cubic form \(\phi \left(r\right)={r}^{3}\) is used in this paper since it was widely employed in many SAEAs [47]. The unknown weight vector \(w={\left({\omega }_{1},...,{\omega }_{m}\right)}^{T}\) can be calculated as follows:

$$ w = \left( {\Phi^{T} \Phi } \right)^{ - 1} \Phi^{T} y $$
(3)

where \(y={\left(f\left({x}_{1}\right),...,f\left({x}_{m}\right)\right)}^{T}\), \(\Phi \) denotes the Gram matrix and it is defined as follow:

$$ \Phi = \left[ {\begin{array}{*{20}c} {\phi \left( {\parallel x_{1} - x_{1} \parallel } \right)} & \cdots & {\phi \left( {\parallel x_{1} - x_{m} \parallel } \right)} \\ \vdots & \ddots & \vdots \\ {\phi \left( {\parallel x_{m} - x_{1} \parallel } \right)} & \cdots & {\phi \left( {\parallel x_{m} - x_{m} \parallel } \right)} \\ \end{array} } \right] $$
(4)

Differential evolution (DE)

Differential evolution (DE) [48] is a population-based global optimization algorithm that finds wide applications in solving complex optimization problems. It emulates the evolutionary mechanism observed in nature, and the differential mutation is used to explore and search for the optimal solution.

DE/best/1:

$$ v_{i}^{g} = x_{best}^{g} + F\left( {x_{r1}^{g} - x_{r2}^{g} } \right) $$
(5)

DE/rand/1:

$$ v_{i}^{g} = x_{r1}^{g} + F\left( {x_{r2}^{g} - x_{r3}^{g} } \right) $$
(6)

DE/best/2:

$$ v_{i}^{g} = x_{best}^{g} + F\left( {x_{r1}^{g} - x_{r2}^{g} } \right) + F\left( {x_{r3}^{g} - x_{r4}^{g} } \right) $$
(7)

DE/rand/2:

$$ v_{i}^{g} = x_{r1}^{g} + F\left( {x_{r2}^{g} - x_{r3}^{g} } \right) + F\left( {x_{r4}^{g} - x_{r5}^{g} } \right) $$
(8)

DE/current-to-rand/1:

$$ v_{i}^{g} = x_{i}^{g} + F\left( {x_{r1}^{g} - x_{i}^{g} } \right) + F\left( {x_{r2}^{g} - x_{r3}^{g} } \right) $$
(9)

DE/current-to-best/1:

$$ v_{i}^{g} = x_{i}^{g} + F\left( {x_{best}^{g} - x_{i}^{g} } \right) + F\left( {x_{r1}^{g} - x_{r2}^{g} } \right) $$
(10)

where \({x}_{best}^{g}\) is the best individual of current population, and \(F\) is random parameter within [0,1]. \(r1, r2, r3, r4, r5,i\in \{\mathrm{1,2},\dots ,NP\}\) are distinct indices.

The formula of binomial crossover is expressed as Eq. (11).

$$ u_{i,j}^{g} = \left\{ {\begin{array}{*{20}c} {v_{i,j}^{g} {, } if{ }\ rand_{j} \le CR{ }\ or\ { }j = j_{rand} } \\ {x_{i,j}^{g} { , } else{ }} \\ \end{array} } \right.,{ }j = 1, \ldots ,n $$
(11)

The control parameter \(CR\in [\mathrm{0,1}]\) is the crossover rate, which determines the fraction \({u}_{i}^{g}\) from the mutant vector \({v}_{i}^{g}\). The integer \({j}_{rand}\in \{\mathrm{1,2},\dots ,d\}\) is chosen randomly and the condition \(j={j}_{rand}\) is designed to ensure that at least one dimension in \({u}_{i}^{g}\) originates from the vector \({v}_{i}^{g}\).

The selection operator is executed to select a better solution from \({u}_{i}^{g}\) and \({x}_{i}^{g}\) for surviving into the next generation, and the specific operation is shown as Eq. (12).

$$ x_{i}^{g + 1} = \left\{ {\begin{array}{ll} {u_{i}^{g} {, } if{ }f\left( {u_{i}^{g} { }} \right) \le f\left( {x_{i}^{g} { }} \right){ }} \\ {x_{i}^{g} {, } else{ }} \\ \end{array} } \right. $$
(12)

Feasibility rules (FR)

Classical feasibility rules (FR) [49] has attracted extensive attention because it does not use parameters and is easy to implement, and the detailed introduction is shown as follows:

  1. (i)

    When two feasible solutions are compared, the one with better objective function value is chosen.

  2. (ii)

    When one feasible and one infeasible solution are compared, the feasible solution is chosen.

  3. (iii)

    When two infeasible solutions are compared, the one with smaller constraint violation is chosen.

The proposed algorithm

Overall framework of the algorithm

The effective balance between global exploration and local exploitation plays a crucial role in improving the overall performance of the algorithm. Specifically, global exploration allows for the rapid identification of potentially promising regions; however, its solution accuracy is often insufficient since the global surrogate cannot provide accurate predictions for all the possible local regions. On the other hand, local exploitation can make it easier to locate the local optimal solution but is susceptible to getting trapped in local optima. Naturally, the idea arises to combine the strengths of both global exploration and local exploitation to ensure algorithmic convergence. Additionally, during the evolutionary process, the population distribution information has not been fully utilized. As a result, when the algorithm searches in a complex space and the population rapidly converges to a local optimum, the population individuals will be densely concentrated; then, it is difficult to escape from these local optima without suitable intervention.

Therefore, the PSSADE is designed to deal with the above challenges shown in ECOPs-MI. The main flow chart of PSSADE is shown in Fig. 2, and the pseudo-code is shown in Algorithm 1. The algorithm consists of two crucial components, i.e., APUM and GFGLS.

Fig. 2
figure 2

The main flow chart of PSSADE

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The algorithm begins with the initialization of a population using Latin Hypercube Sampling (LHS), followed by the actual evaluation for objective and constraints. Subsequently, in each generation, two essential operations are performed.

figure a

In APUM, when the population becomes too concentrated, the population reselect operation (PRS) is employed. Specifically, PRS adaptively reshapes the current population and increases population diversity by employing the efficient non-dominated sorting technique based on bi-objectives of fitness (individual quality information) and population distance (population diversity information).

In GFGLS, there are two steps: GSAS and LSAS. To address the issue of stagnation and diversity loss caused by conducting differential mutation for integer variables, a probability-driven mixed-integer mutation (PMIU) is designed. This mutation approach PMIU incorporates the adaptive individual construction strategy (AICS) and the probability-driven integer refinement operation (PIRO). This operation is integrated into GSAS and LSAS. In GSAS, a global RBF is employed to enhance global search capabilities. In LSAS, a local RBF is utilized to accelerate convergence speed.

Adaptive population update mechanism (APUM)

In the process of evolution, due to the complexity of the search space, the algorithm may fall into local optimization. If the population individuals are densely concentrated, the algorithm either falls into local optimization or has already found the global optimal solution. Therefore, the population distribution state information can be fully used to adjust algorithm search behavior in time. In APUM, there are three main steps: population distribution state evaluation, dynamic discriminant threshold setting, and population reselect operation. The population distribution state is evaluated by the mean distance between solutions. Then a discriminant threshold, which is dynamically decreased based on the current consumed number of function evaluations, is used to determine whether the population is over-concentrated or not. Finally, for the over-concentrated population, the population is reselected by employing the efficient non-dominated sorting technique.

Population distribution state evaluation

To assess the degree of aggregation in the population, the average ratio of the distance between each individual and the mean value across all dimensions, denoted as \(AP\), is computed in Eq. (13).

$$ AP = { }\mathop \sum \limits_{i = 1}^{NP} \mathop \sum \limits_{j = 1}^{D} \frac{{\left| {x_{i,j} - x_{mean,j} } \right|}}{{x_{max,j} - x_{min,j} }} $$
(13)

The parameter \(NP\) represents the population size, while \({x}_{max,j}\) and \({x}_{min,j}\) represent the upper and lower boundaries of the search space in the \(j\) th dimension, respectively. The ratio of distribution state between the current population and the initial population, denoted as \(DP\), is calculated using Eq. (14).

$$ DP = \frac{AP}{{AP_{init} }} $$
(14)

\(AP\) represents the distribution state value of the current population, while \({AP}_{{\text{init}}}\) represents the distribution state value of the initial population. Generally, the value of \(DP\) is restricted to the interval [0,1]. A low \(DP\) value indicates a highly clustered population, which suggests premature convergence.

Dynamic discriminant threshold

To accurate detect premature convergence, a dynamic discriminant threshold is utilized, denoted as \(AT\) and defined by Eq. (15).

$$ AT = AT_{init} \cdot GP $$
(15)

The parameter \(AT\) represents the current value of the discriminant threshold, \({AT}_{init}\) represents the initial value of \(AT\), and the parameter \(GP\) is defined in Eq. (16):

$$ GP = \frac{{FEs_{max} - FEs + 1.0}}{{FEs_{max} }} $$
(16)

The parameter \(GP\), which decreases linearly with the current consumed number of function evaluations, denoted as \(FEs\). \({FEs}_{max}\) represents the maximum number of function evaluations.

The value of \(AT\) decreases linearly with \(FEs\), following to the variation trend of the parameter \(GP\). The reason for utilizing a dynamic setting is that, as the optimization process proceeds, the algorithm gradually converges towards local regions, there by necessitating a higher tolerance for the aggregation degree of the population. In other words, this setting serves to fine-tune the delicate balance between global exploration and local exploitation throughout the optimization process of the algorithm.

Population reselect operation (PRS)

The population reselect algorithm, outlined in Algorithm 2, is triggered when \(DP\le AT\). This algorithm ensures that the population diversity is preserved by reselecting individuals with optimal fitness values and maximum total distance based on Pareto dominance and crowding distance.

The algorithm begins by calculating the total distance between each current individual and all individuals in the archived data. Individuals with both optimal fitness values and maximum total distance are selected. If the size of the selected Pareto front is smaller than the current population size, non-dominated solutions are selected from the remaining data. However, if the size of the selected Pareto front exceeds the current population size, individuals with the smallest crowding distance are removed, ensuring that the population size remains within the desired range.

figure b

Collaborative framework of global and local surrogate-assisted search (CFGLS)

Global surrogate pre-screening exhibits strong global search capabilities but suffers from low efficiency. Surrogate-based local search demonstrates fast convergence but may get trapped in local optima. Therefore, the combination of global surrogate pre-screening and surrogate-based local search may be more efficient in solving ECOPs-MI, as it can better balance global exploration and local exploitation. DE can effectively explore the search space by introducing differential vectors and mutation operations, which is usually used to solve continuous optimization problems. However, integer variables are often involved in practical problems, so it is necessary to design a mutation operation suitable for integer variables. In CFGLS, there are three main steps. Firstly, a probability-driven mixed-integer mutation in the differential mutation process is performed in the subsequent global and local searches. Secondly, global surrogate-assisted search is employed to enhance the global search performance. Lastly, local surrogate-assisted search is utilized to accelerate the convergence speed.

Probability-driven mixed-integer mutation (PMIU)

To inherit the search greedy brought by high-value parent individuals and the diversity of differential vectors in differential mutation, classical DE-based algorithms usually round the integer dimensions of the mutation individuals to make the overall search biased towards the mixed-integer variable design space. This method can avoid unnecessary redundant resources allocated by the algorithm to explore the non-integer design space to a certain extent. However, there are still two limitations on these algorithms, i.e., the optimization stagnation brought by conducting differential mutation on integer dimensions, and the reduction of diversity caused by rounding the decimal into integers directly. Therefore, a probability-driven mixed-integer mutation (PMIU) is designed where the optimization stagnation and diversity loss are well considered by the adaptive individual construction strategy (AICS) for mutation stagnation and the probability-driven integer refinement operation (PIRO) for diverse evaluations respectively.

In AICS, each individual constructed in the population is composed of three parts, namely, the continuous value of the integer variable, the continuous variable value, and the rounded value of the integer variable. Parts of the individual are selected to mutate and evaluate. The continuous value of the integer variable and the continuous variable value are used for mutation, making full use of the excellent characteristics of continuous variable mutation. The rounded value of integer variable and the continuous variable value are used for function evaluation, model construction and prediction, and thus the accuracy of the surrogate on the original mixed-integer optimization prediction can be maintained. To illustrate the mutation and evaluation process in detail, a simple example is shown in Fig. 3. This example takes Eq. (8) as an example, where the scaling factor \(F\) is taken to be 0.5. As shown in Fig. 3, five individuals \({x}_{r1},{x}_{r2},{x}_{r3},{x}_{r4},{x}_{r5}\) are selected from the current population, and the differential mutation is conducted to obtain \({v}_{i}\). It can be seen that PMIU still provides diversity in the later stages of evolution, rather than stagnating, while the conventional rounding method may not be able to generate new solutions in integer dimensions.

Fig. 3
figure 3

Comparison of integer variable mutation

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In PIRO, to avoid rounding the decimal to the same value, the random values based on the uniform distribution are viewed as the probability to determine the rounding ways. For example, for the conventional rounding method, the round down is fixed when the decimal part is smaller than 0.5; otherwise, the round up is employed. This means that the diversity will be greatly reduced when the decimal parts of different values are smaller or larger than 0.5. In contrast to PIRO, the concrete expression is shown in Eq. (17) where the comparison result between the probability and the decimal part determines whether the value is rounded down or up. Therefore, the randomness of probability provides higher diversity for mutation on integer dimensions.

$$ \overline{{y_{i} }} = \left\{ {\begin{array}{ll} {y_{i} , {\text{if}}\ y_{i}\ {\text{is}}\ {\text{interger}} } \\ {y_{i} + 1, {\text{if}}\ r\;{\text{and}}\ \left( {0,1} \right) y_{i} - y_{i} } \\ {y_{i} , {\text{if}}\ r{\kern 1pt} \;{\text{and}}\ \left( {0,1} \right) \ge y_{i} - y_{i} } \\ \end{array} } \right. $$
(17)

where \(\lceil{y}_{i}\rceil\) is the integer part of \({y}_{i}\).

Global surrogate-assisted search (GSAS)

The GSAS uses all archived data to construct the RBF model. By learning and fitting the sample data, the RBF model approximates and predicts the objective function and constraints, so that some local minima can be ignored and potential feasible areas can be quickly obtained in the search space. During the optimization process, \(\lambda \) trial vectors are generated for each individual in the population using the DE/rand/2 strategy. These trial vectors are then predicted using the RBF surrogate model, and the FR [49] is employed to select the best candidate vector for actual function evaluation. The feasibility of the trial vectors is considered in FR to ensure that the population is guided to the appropriate region. The pseudo-code for the global surrogate-assisted search algorithm is presented as Algorithm 3.

figure c

Local surrogate-assisted search (LSAS)

To further accelerate the convergence speed, the LSAS is proposed in this study. The region near the current optimal solution is likely to contain the global optimal solution. In LSAS, the \(15*D\) individuals closest to the current optimal solution are identified, and these local regions are exploited by constructing a more accurate local RBF model through these individuals. For each individual, \(\lambda \) trial vectors are generated using the DE/best/2 strategy. The local RBF surrogate model is employed to predict the objective function values for these trial vectors. Based on feasibility rules, the best candidate vector is selected for actual function evaluation. Simultaneously, the current best individual is updated. The pseudo-code for the algorithm is presented as Algorithm 4.

figure d

Experimental studies

Experimental settings

In this subsection, 10 benchmark problems are used to validate the effectiveness of the algorithm. Table 1 presents the main characteristics of these ten benchmark problems. For more detailed information, please refer to Appendix A.

Table 1 The main feature of the ten benchmark problems
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The main parameters of the algorithm are set as follows: \(NP = max(d, 50)\), \(MaxFEs = 1000\), \(\lambda = (50*d, 500)\). The value of \({AT}_{init}\) is set between 0.1 and 1, with experiments conducted at 0.1 intervals. The study determines that a value between 0.2 and 0.6 is appropriate, and this article uses a value of 0.4. Other parameters related to the differential evolution algorithm are set as follows: the scaling factor \(F = 0.5 + 0.5*rand\), and the mutation probability \(CR = 0.5 + 0.5*rand\), rand is a random number between 0 and 1. All experimental results are obtained over 20 independent runs in Matlab R2021b. To facilitate the comparison of PSSADE with other algorithms, the effective rate (ER) is used, which represents the ratio of the number of effective runs to the total runs. A run is considered effective if it can find at least one feasible solution within the predetermined maximum evaluation time (\(MaxFEs\)). Additionally, in the following tables, the terms "Best," "Mean," "Worst," and "Std" represent the best, average, worst, and standard deviation of the optimal objective values for all effective runs, respectively. Also, these tables include the results of the Wilcoxon rank sum test calculated at a significant level of 0.05, where the symbols, i.e., "-", "≈", and " + ", respectively indicate that the performance of the corresponding algorithm is worse than, similar to, and better than that of PSSADE.

Comparison with classical algorithms designed for ECOPs-MI

In order to fully validate the performance of PSSADE, three algorithms where two of them, i.e., MI-EDDE [50] and MI-BEXPM [51] are both recently developed evolutionary algorithms for mixed-integer constrained optimization problems, and the other algorithm named SADE-MI [34] is a well-known surrogate-assisted evolutionary algorithm for ECOPs-MI. Table 2 lists the comparison results among PSSADE, SADE-MI, MI-EDDE and MI-BEXPM on test problems.

Table 2 Experimental values of PSSADE, SADE-MI, MI-EDDE and MI-BEXPM on ten test problems
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From Table 2, it can be observed that both PSSADE and SADE-MI outperform MI-EDDE and MI-BEXPM on all 10 problems. PSSADE and SADE-MI are optimization algorithms based on surrogate models, while MI-EDDE and MI-BEXPM are not. This implies that by utilizing surrogate models in evolutionary algorithms, the search performance of the algorithms can be greatly improved within a certain number of effective evaluations. Furthermore, PSSADE is able to find the known optimal solution for all problems, whereas SADE-MI fails to find the known optimal solution for problems P1 and P9. This indicates that PSSADE performs better than SADE-MI in terms of global search performance. In terms of the mean value, PSSADE outperforms SADE-MI in problems P1, P2, P5, P9, and P10, while slightly worse in problems P3, P4, and P6. Due to the simplicity of problems P7 and P8, both PSSADE and SADE-MI are able to consistently find the approximate optimal values. This suggests that PSSADE exhibits better local search capability than SADE-MI in most problems.

Notably, PSSADE is able to find solutions that are better than the known best solution for problems 9 and 10, as detailed in Table 3. Based on the above discussion, it can be concluded that PSSADE not only efficiently locates feasible solutions, but also achieves more accurate solutions.

Table 3 Solutions better than the known best solutions found by PSSADE
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Comparison with classical algorithms designed for ECOPs-C

To clearly demonstrate the performance difference between the proposed PSSADE and other classical algorithms designed for expensive constrained optimization problems with only continuous variables (ECOPs-C), two different types of surrogate-assisted evolutionary algorithms such as GLoSADE [35] and SACCDE [46] are selected as the competitors in solving ECOPs-MI where the rounding technique is utilized to handle integer variables. In order to eliminate the influence of parameters as much as possible, all the parameter settings are set the same as their original papers, and the source codes of both GLoSADE and SACCDE come from their respective developers. Table 4 gives the comparison results among PSSADE, GLoSADE and SACCDE.

Table 4 Experimental values of PSSADE, GLoSADE and SACCDE on ten test problems
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As shown in Table 4, PSSADE can find feasible solutions on all test functions, while SACCDE fails to achieve stable performance in findind feasible solutions on P1 and P3. GLoSADE performs even worse and fails to find any feasible solution on P1, P3, and P9. This indicates that the two algorithms that are not designed specifically for the characteristics of mixed-integer variables cannot stably locate a feasible region. More importantly, in terms of the Wilcoxon rank sum test, PSSADE significantly outperforms the other two methods on most test functions, but none of them perform better than PSSADE on any problem. Therefore, the combination of the classical SAEAs designed for ECOPs-C and the rounding technique cannot provide satisfactory performance in solving ECOPs-MI.

Effectiveness of the three key strategies in PSSADE

In this subsection, the parameter settings remain the same as those mentioned earlier. Table 5 presents the experimental results of PSSADE and its three variants. In Table 5, PSSADE_noAPUM indicates PSSADE without the APUM, PSSADE_noGSAS represents PSSADE without GSAS, and PSSADE_noLSAS denotes PSSADE without LSAS. From Table 5, it can be observed that PSSADE with the three strategies outperforms the three corresponding variant algorithms in terms of the mean value. Additionally, from the mean value, and ER of problems P1, P3, P4, P6, P8, and P10, it can be observed that PSSADE_noAPUM may not be able to find the global optimal value and there is even a possibility of not finding feasible solutions. This suggests that the APUM mechanism can help the algorithm escape local optima and find the global optimal solution. By recording the population distribution state value DP and the threshold value AT of P3 and plotting Fig. 4, it is evident that with the involvement of APUM, if the population distribution becomes too concentrated, the population is reselected, ensuring population diversity and facilitating the search for the global optimal solution.

Table 5 Experimental function values of PSSADE, PSSADE_noAPUM, PSSADE_noGLAS and PSSADE_noLSAS on ten test problems
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Fig. 4
figure 4

The population diversity curve

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In Table 5, except for problem P2, the performance of PSSADE_noGSAS is significantly weakened compared to PSSADE, considering the mean value, ER, and Wilcoxon rank sum test. This indicates the crucial role of GSAS in aiding the search for the global optimal solution. Furthermore, it is observed that PSSADE_noLSAS is able to find feasible solutions in all problems, but its performance is worse than the PSSADE algorithm in problems P1, P3, P4, and P10. This suggests that the GSAS strategy plays a certain role in enhancing local search, helping the algorithm find local optima and improving population search performance. To further illustrate the impact of GSAS and LSAS, the optimal values of problem P3 after applying these two strategies are recorded and plotted in Fig. 5. The initialization did not yield a feasible solution, but through GSAS, the algorithm quickly locates the feasible region, and each significant improvement is achieved under the GSAS strategy. Additionally, LSAS contributes to improving local optima. By combining these two strategies, the algorithm is able to rapidly localize the feasible region and converge to the global optimal solution.

Fig. 5
figure 5

The optimal value curve under two strategies

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Effectiveness of the probability-driven mixed-integer mutation

To further illustrate the effectiveness of the probability-driven mixed-integer mutation (PMIU), three classical mutation operations, namely rounding technique [52], separate method shown in [53] where different mutation operations are designed for continuous and integer variables respectively, and random-key method [54] where three different rounding ways are used based on the decimal of integer variable. To provide a fair comparison, three variants, i.e., Rounding, Separation, and Random-key are constructed based on the framework of PSSADE. The only difference between PSSADE and three variants is the inconsistency of the mutation operations on mixed-integer variables. For instance, the variant such as Rounding means that the treatment on integer variables shown in PMIU of PSSADE is replaced by the rounding technique shown in [52]. Table 6 presents the experimental results of PSSADE with four different mutation operations.

Table 6 Experimental values of PMIU, Rounding, Separation and Random-key based on the framework of PSSADE on ten test problems
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As shown in Table 6, all the algorithms achieve 100% ER in locating feasible regions, which means that the ability of PSSADE in locating feasible regions may not be affected by different mutation operations. Moreover, in terms of the mean value and Wilcoxon rank sum test, the convergence ability of PMIU in finding better feasible solution is superior to other methods on most test problems. Therefore, the PMIU designed in this paper performs best among these compared methods.

Effectiveness of the population reselect operation

To demonstrate the effectiveness of the population reselect operation (PRS), we compared PRS with two adaptive population strategies, namely restart mechanism (RM) in [55] and population selection strategy (PSS) in [12].

In RM, when the population converges to an infeasible region, the population will be regenerated in the design space. In PSS, the ranked values of each individual can be obtained based on specific sorting rules and affinity propagation clustering, and then the top-ranked individuals in each cluster are selected to reform the population. For fairness, we replace the PRS with RM and PSS in PSSADE to construct two competitor algorithms namely RM and PSS, respectively. The comparison results are shown in Table 7. In terms of the mean value and Wilcoxon rank sum test in Table 7, it can be seen that PRS designed in this article is superior to RM and PSS.

Table 7 Experimental values of PRS, RM and PSS based on the framework of PSSADE on ten test problems
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Comparison of numerical engineering cases

In this subsection, we apply our method to two specific engineering cases: the reliability-redundancy allocation of a full bridge strain gauge and the structural design of a truss structure, as shown in Figs. 6 and 7. The optimization of the reliability-redundancy allocation aims to maximize reliability while satisfying constraints such as weight, cost, and volume. Strain gauges R1, R4 and R2, R3 are attached to the upper and lower surfaces of the cantilever beam, respectively. The gravity of the object to be measured will cause the cantilever beam to deform, resulting in changes in the resistance of the strain gauges. These changes will then be converted into electrical signals through the full-bridge strain gauges shown in Fig. 5 for output. Here, a reliability redundancy allocation method is used to maximize the reliability of the full-bridge strain gauge. The structural design of a truss structure focuses on minimizing mass while meeting displacement constraints. The structure has 11 elements. There are three load cases, F1 = 280 kN acting on nodes 1 and 5, F2 = 210 kN acting on node 3, and F3 = 310 kN acting on node 7. The coordinates of nodes 1, 3, 5, and 7 are fixed, but the height of the structure is variable. The integer design variables are the side length of 11 angle steels, and the continuous design variables are the height of nodes 2, 4, and 6. The constraint condition is that the maximum displacement of all nodes does not exceed 8 mm. For each set of design variables, both the mass and the displacement constraints of the structure can be obtained within 1 s. Therefore, both problems are constrained optimization problems with mixed-integer variables. For a detailed case study, please refer to reference [34].

Fig. 6
figure 6

The full bridge strain gauge design

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Fig. 7
figure 7

The plane truss structure design

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The optimization results of the reliability allocation problem are listed in Table 8. The optimization results of the truss structure design problem are listed in Table 9. PSSADE, SADE-MI, MI-EDDE, and MI-BEXPM were employed to conduct 20 independent optimizations for each of the two problems, and the maximum number of function evaluations (\({FEs}_{Max}\)) is set to 1000. The results presented in the two tables indicate that PSSADE outperforms the other three algorithms in terms of best value, mean value, worst value, and standard deviation. The efficiency rate (ER) of PSSADE and SADE-MI both achieved 100%. This indicates that feasible solutions can be consistently obtained. These findings demonstrate that PSSADE excels among the four algorithms in terms of global search capability and algorithm stability.

Table 8 Optimization results of reliability allocation problem
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Table 9 Optimization results of the truss structure design problem
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The optimal solution is

$$\begin{aligned}&\left\{{x}_{1},{x}_{2},{x}_{2},{x}_{4},{x}_{5},{y}_{1},{y}_{2},{y}_{3},{y}_{4},{y}_{5}\right\}\\ & =\left\{\mathrm{0.7731,0.8674,0.8890,0.7112,0.7637,3},\mathrm{3,3},\mathrm{3,1}\right\}\end{aligned}$$

The optimal solution is

$$\begin{aligned}&\left\{{x}_{1},{x}_{2},{x}_{2},{y}_{1},{y}_{2},{y}_{3},{y}_{4},{y}_{5},{y}_{6},{y}_{7},{y}_{8},{y}_{9},{y}_{10},{y}_{11}\right\}\\ & \quad =\Bigg\{\mathrm{3.00,3.20,2.10,43,34,41,42,10,41,10,}\\ & \qquad\quad \mathrm{41,31,33,40}\Bigg\}\end{aligned}$$

Application in the design of infrared stealth material film

To improve the concealment and survival ability of the equipment, infrared stealth generally reduces the possibility of the protected target being detected by reducing, eliminating, simulating or changing the difference in the infrared radiation characteristics of the two atmospheric windows (3–5 µm and 8–14 µm) between the target and the background [56]. The design of spectral selective emission materials mainly achieves infrared stealth by reducing the emissivity in the atmospheric window, while increasing the emissivity in the non-atmospheric window (5–8 µm) for radiation cooling [57, 58]. The theoretical basis for the design of infrared stealth materials is the ultrathin metal tunneling effect and the impedance matching principle. The infrared stealth can be truly realized by finding the optimal combination between high emissivity metal films and radiation tunable dielectric materials. The structure design of a four-layer film spectral selective infrared stealth material is shown in Fig. 8. The simulation model is shown in Fig. 9. Commonly utilized high emissivity metal film materials include Ag, Au, Mo, Cu and Al, while frequently used dielectric materials consist of Ge, Si, and ZnS. The key to optimizing the design lies in selecting the appropriate material combination to meet dimensional constraints and achieve low emissivity for the atmospheric window, while maximizing the emissivity of the non-atmospheric window. The mathematical representation of this problem is provided as follows.

Fig. 8
figure 8

Design of stealth material film

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Fig. 9
figure 9

FDTD simulation model

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Consider

$$ x = \left[ {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} ,x_{8} } \right] $$
(18)

Minimize

$$ {\text{RMSD}} = \sqrt {\frac{{\mathop \sum \nolimits_{{3{\upmu }m}}^{{14{\upmu }m}} \left( {\varepsilon_{exp,\lambda } - \varepsilon_{ideal,\lambda } } \right)^{2} }}{N}} $$
(19)

Subject to:

$$ x_{1} + x_{2} + x_{3} + x_{4} \le 1500 $$
(20)
$$ \varepsilon_{{3 - 5{\mu m}}} = \frac{{\mathop \sum \nolimits_{{3{\upmu }m}}^{{5{\upmu }m}} \varepsilon_{exp,\lambda } }}{{N_{{3 - 5{\mu m}}} }} \le 0.3 $$
(21)
$$ \varepsilon_{{8 - 14{\mu m}}} = \frac{{\mathop \sum \nolimits_{{8{\upmu }m}}^{{14{\upmu }m}} \varepsilon_{exp,\lambda } }}{{N_{{8 - 14{\mu m}}} }} \le 0.3 $$
(22)

The former four parameters, i.e., \({x}_{1}\), \({x}_{2}\), \({x}_{3}\), and \({x}_{4}\), are continuous variables, and they represent the thicknesses of the four-layer film spectral selective infrared stealth material respectively. The latter four parameters, i.e., \({x}_{5}\), \({x}_{6}\), \({x}_{7}\), and \({x}_{8}\), are integer variables, and they represent the material types shown in Fig. 8. The symbol \(\varepsilon \) represents the spectral emissivity of the designed material, which is obtained from the simulation software FDTD solutions. Each evaluation takes approximately 3–5 min to complete. RMSD (root mean square deviation) is used to measure the degree of spectral selectivity between the designed selective emission multilayer film and the ideal selective emission infrared stealth material spectral curve. \({\varepsilon }_{{\text{exp}},\uplambda }\) and \({\varepsilon }_{{\text{ideal}},\uplambda }\) respectively represent the spectral emissivity of the designed multilayer film and the ideal selective infrared stealth material at different wavelengths of 3–14 µm, and \(N\) represents the number of wavelength samples taken. The emissivity of the ideal selective infrared stealth material is 0 at 3–5 µm and 8–14 µm, and 1 at 5–8 µm [59].

Considering that the SADE-MI performs best in all the compared algorithms, SADE-MI is selected as the competitor algorithm for validating the performance of PSSADE in the optimization design of infrared stealth material film. The comparison results of PSSADE and SADE-MI are obtained over 20 independent runs where the maximum number of function evaluations is set to 1000, and all of the results are shown in Table 10 and Table 11, respectively. The optimal infrared stealth material film designed by the two algorithms is shown in Fig. 10, and the corresponding emissivity curve is shown in Fig. 11.

Table 10 Optimization results of the optimization design of infrared stealth material film
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Table 11 The comparison of optimal results between PSSADE and SADE-MI
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Fig. 10
figure 10

The obtained infrared stealth material film

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Fig. 11
figure 11

The emissivity of the designed infrared stealth material film by PSSADE and SADE-MI

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In Table 10, PSSADE performance is superior to SADE-MI in terms of the best, mean, worst values and Wilcoxon rank sum test. From Table 11, the optimized material obtained by PSSADE has a lower emissivity in the atmospheric window (\({\varepsilon }_{3-5\mu m}=0.1132\)) and a higher emissivity in non-atmospheric windows (\({\varepsilon }_{5-8\mathrm{\mu m}}=0.8152\)). The results indicate that PSSADE can effectively solve optimization problems in practical engineering applications.

Conclusions

In this paper, a population state-driven surrogate-assisted differential evolution algorithm is proposed to solve the ECOPs-MI. The main contribution is that this paper effectively combines APUM and CFGLS to deal with mixed-integer variables, which improves the efficiency and accuracy of the algorithm.

There are three key theoretical contributions in this paper. Firstly, in probability-driven mixed-integer mutation, an adaptive individual construction strategy for mutation and evaluation is designed to inherit the advantage of the original differential mutation, and a probability-driven integer refinement operation is employed to alleviate the loss of population diversity brought by conventional rounding. Secondly, the probability-driven mixed-integer mutation is applied to both GSAS and LSAS, which effectively balances global exploration and local exploitation. Finally, the population state information is used to adjust the population in real time, which effectively avoids falling into the local optimal problem.

Moreover, PSSADE is highly competitive in 10 benchmark problems and 2 numerical engineering cases. It is verified that PSSADE has superior search performance. Furthermore, the effectiveness of PSSADE is verified that each theoretical part contributes to the improvement of the algorithm. Finally, PSSADE shows excellent performance in the design of infrared stealth material film.

However, there are still areas for improvement. For instance, exploring individual-based updates to reduce the number of actual function evaluations and extending the algorithm to handle high-dimensional or multi-objective problems.