Model management is an essential component in data-driven surrogate-assisted evolutionary optimization. In model management, the solutions with a large degree of uncertainty in approximation play an important role. They can strengthen the exploration ability of algorithms and improve the accuracy of surrogates. However, there is no theoretical method to measure the uncertainty of prediction of Non-Gaussian process surrogates. To address this issue, this article proposes a method to measure the uncertainty. In this method, a stationary random field with a known zero mean is used to measure the uncertainty of prediction of Non-Gaussian process surrogates. Based on experimental analyses, this method is able to measure the uncertainty of prediction of Non-Gaussian process surrogates. The method's effectiveness is demonstrated on a set of benchmark problems in single surrogate and ensemble surrogates cases.

Data-driven optimization problems usually involve objective and constraint functions that are not available, and the evaluation of these functions is time-consuming and complex. There are only small data from physical experiments, numerical simulations, or daily life, and the evaluation of these functions involves a number of computationally expensive numerical simulations or costly physical experiments (Preen and Bull, 2016; Wang et al., 2016; Jin et al., 2018).

Evolutionary algorithms (EAs) are population-based search methods that mimic natural biological evolution and species' social behavior. They are promising in solving non-convex, constrained, multiobjective, or dynamic problems (Michalewicz and Schoenauer, 1996; Hart et al., 1998; Li et al., 2014; Zhang, Mei et al., 2021). However, most existing research on EAs usually assumes that the analytic objective and constraint functions are available, and evaluating these functions is cheap and simple. Therefore, EAs cannot be directly used to solve the data-driven optimization problems. Surrogate-assisted evolutionary algorithms (SAEAs) are considered to address the limitation of EAs in solving these problems (Jin et al., 2000; Tong et al., 2019; Zhang, Li et al., 2021; Wang et al., 2022). In SAEAs, many machine learning models can be used as surrogates to approximate the exact functions, including polynomial regression (PR), Gaussian process (GP), artificial neural network (ANN), radial basis function network (RBFN), support vector machine (SVM), and the ensemble of these surrogates. A limited number of exact function evaluations are carried out, and a small amount of data is used to train these surrogates (Braun et al., 2009; Jin et al., 2000; Chugh et al., 2019).

For all surrogates mentioned above, GP is usually used (Emmerich et al., 2006; Coelho and Bouillard, 2011; Chugh et al., 2016; Zhan and Xing, 2021). There is provided prediction and uncertainty information by GP, which is important in SAEAs. Then the existing infill sampling criteria can be used to guide the search of EAs and the update of surrogates, such as the lower confidence bound (LCB) (Torczon and Trosset, 1998), the expected improvement (EI) (Jones et al., 1998) and the probability of improvement (PoI) (Ulmer et al., 2003). On the contrary, although many Non-Gaussian process (Non-GP) surrogates can also provide a good prediction, they cannot provide the uncertainty of prediction of surrogates. In this case, these Non-GP surrogates have significant limitations: (1) Because there is no uncertainty information of prediction of surrogates, it is hard to improve the exploration of EAs and the accuracy of surrogates; (2) The existing infill sampling criteria cannot be used to guide the search of EAs.

It should be emphasized that the uncertainty information of prediction of surrogates plays an essential role in model management in SAEAs, because (1) solutions with a large degree of uncertainty indicate that the fitness landscape around them has not been well explored, and therefore the evaluation of these solutions is likely to find a better solution (Branke and Schmidt, 2005); (2) evaluating these solutions can most effectively improve the accuracy of surrogates (Jin, 2011).

Several methods are used to measure the uncertainty of prediction of Non-GP surrogates. For instance, Bayesian neural networks can measure the uncertainty of prediction of neural networks (Gal and Ghahramani, 2015). Cross-validation also can be used to measure the uncertainty of prediction of surrogates (Hutter et al., 2019). However, there is a significant limitation for the two methods: the accuracy of uncertainty highly depends on the size of training data. However, there is not much training data in data-driven optimization progress. Besides, the prior distribution also needs to be known for Bayesian Neural Networks. Based on the limitation, the two methods will not be investigated in the article.

In addition to the above methods, there are also three typical methods: (1) The distance from the solutions to the existing training data has been used as an uncertainty measure in Branke and Schmidt (2005). Since ensemble surrogates have been proven to provide uncertainty information, two methods have been proposed to measure the uncertainty of prediction of ensemble surrogates. (2) The literature (Wang et al., 2017) defined the uncertainty measurement to be the maximum difference between outputs of ensemble members. (3) The variance of predictions output by the base surrogates of ensemble is used to estimate the uncertainty of prediction of ensemble surrogates (Guo et al., 2018).

Among the three methods above, the first method is a qualitative uncertainty measurement method. In theory, it is not able to accurately measure the uncertainty of prediction of surrogates. Instead, it indicates only the crowded degree of the neighborhood of a solution. The second method is the disagreement among the outputs of ensemble members for the prediction of surrogates. This method was proposed based on Query-by-Committee (QBC) in active learning, which shows that the query with the maximum disagreement strategy can efficiently enhance the accuracy of surrogates (Wang et al., 2017). In essence, this method describes the difference of predictions among ensemble members in a solution. In the third method, the uncertainty of prediction of surrogates is defined by the variance of predictions output by the base surrogates of ensemble. It indicates the average squared deviation of the base surrogates about the output of ensemble. In the probability and statistic viewpoint, these methods for measuring the uncertainty of prediction of Non-GP surrogates are not a sound method. These methods mentioned above cannot address one important issue: to measure the uncertainty of prediction of Non-GP surrogates. Therefore, it can be confirmed that there is no a theoretical sound method to measure the uncertainty of prediction of Non-GP surrogates.

To address the issue mentioned above, this article proposes an uncertainty measure for the prediction (UMP) of Non-GP surrogates. This method can be written in the form of a random field model. In detail, it consists of two components: regression function (namely Non-GP surrogate) and residual variation (also known as uncertainty). In this method, two components are uncorrelated. In the first term, Non-GP surrogate as regression function only depends on decision variables, and the second term represents the uncertainty of prediction of Non-GP surrogate based on a stationary random field. Thus, based on the random field model, the uncertainty of prediction of Non-GP surrogate can be measured. Then, the existing infill sampling criteria can be used to guide the search of algorithms and the update of surrogates.

In this article, an uncertainty measure for the prediction of Non-GP Surrogates is proposed to overcome the drawbacks of existing uncertainty methods. The main contribution of this article can be summarized as follows:

(1) An uncertainty measure for the prediction of Non-GP surrogates is proposed, which overcomes the drawbacks of existing uncertainty methods;

(2) The effectiveness of the proposed method is investigated on a set of benchmark problems and analysed on Rastrigin function in both single surrogate and ensemble surrogates cases. The experimental results demonstrate that the proposed method is promising in solving data-driven optimization problems.

The rest of this article is structured as follows. Section 2 presents a brief review of the used surrogates, ensemble surrogates, and infill sampling criterion in this article. Section 3 presents the proposed uncertainty measure for the prediction of Non-GP surrogates. Section 4 demonstrates and discusses experimental results. Finally, Section 5 concludes the article with a summary and looks into the future work.

Surrogates and infill sampling criteria are essential components in online surrogate-assisted evolutionary algorithms. This section presents a brief review of surrogates and the infill sampling criterion involved in this article.

### 2.1 Polynomial Regression

Polynomial regression adopts the statistical tools of regression and analysis of variance to obtain the minimum variance of regression. It is widely used in approximating exact objective and constraint functions. The formulation of the polynomial regression at any untested $x$ is defined as follows
$f^(x)=β0+Σi=1dβixi+Σi=1,j=1,i≤jdβi,jxixj+Σi=1,j=1,k=1,i≤j≤kdβi,j,kxixjxk+⋯$
(1)
where $β0,βi,βi,j,βi,j,k$ are the coefficients to be estimated, $d$ is the dimension of problems; usually, the least square method (LSM) is often used to estimate these coefficients in the surrogate.

### 2.2 Radial Basis Function Network

Like other neural networks, RBFN has an input layer, hidden layer, and output layer. It uses radial basis functions as its activation functions. In RBFN, the input layer is directly connected to the hidden one, and the output of RBFN at an untested $x$ has the following expression
$f^(x)=∑i=1Mωiψ(∥x-ci∥p),$
(2)
where $ψ$ is activation function; $ci$ can be any point vector (e.g., origin or center); $M$ is the number of nodes of the hidden layer; and $ω$ is the unknown weights to be estimated, which can be determined by LSM or backpropagation based on gradient descent; $p$ is norm.

### 2.3 Support Vector Machine

SVM is one of the popular surrogates based on statistical learning theory and is often used as a surrogate by constructing a hyperplane in high-dimensional space. The SVM at an untested $x$ is expressed as
$f^(x)=ωTϕ(x)+b,$
(3)
where $ϕ(x)$ is feature vector; coefficient vector $ω$ and coefficient $b$ need to be estimated.
The unknown parameter $ω$ and $b$ can be obtained by optimizing a constrained optimization problem (Cristianini and Shawe-Taylor, 2000) based on observed values $yi$ at $xi$ for $i=1,···,N$, which is shown as
$min12∥ω∥2+L∑i=1N(ξi+ξi')styi-ωTϕ(xi)-b≤ɛ+ξiωTϕ(xi)+b-yi≤ɛ+ξi'ξi',ξi≥0,$
(4)
where $L=1.0$ and $ɛ=0.1$ are prespecified values in this article, and $ξi$ and $ξi'$ are slack variables representing upper and lower constraints.

### 2.4 Ensemble Surrogates

The ensemble surrogates have been proven to outperform most of the single surrogates. They are able to generate more reliable predictions of fitness landscape of problems than single surrogates (Liu et al., 2000; Queipo et al., 2005), when little is known about the problem to be optimized at hand. The prediction of ensemble surrogates $f^ens(x)$ is formulated as
$f^ens(x)=∑i=1Kwif^i(x),∑i=1Kwi=1,$
(5)
where $f^i(x)$ represents the output of the $ith$ member in the ensemble; $K$ is the number of members in the ensemble, $K=3$ in this article; in this article, $wi$ is the weight of the $ith$ member defined by
$wi=0.5-ei2(∑j=1Kej),$
(6)
where $ei$ and $ej$ are the root mean square error (RMSE) of the $ith$ and $jth$ member in the ensemble, respectively.
The ensemble surrogates also have been proven to provide uncertainty information of prediction of ensemble surrogates, and two methods have been proposed to measure the uncertainty information. The literature (Wang et al., 2017) defined the uncertainty measurement to be the maximum difference between outputs of ensemble members, as shown in Eq. (7).
$U(x)=max(f^i(x)-f^j(x)),$
(7)
where the uncertainty $U(x)$ at $x$ is the maximum difference between the outputs of two ensemble members $f^i(x)$ and $f^j(x)$.
The literature (Guo et al., 2018) used the variance of predictions output by the base members of ensemble to estimate the uncertainty of prediction of ensemble surrogates, as shown in Eq. (8).
$U(x)=1K-1∑i=1K(f^i(x)-f^ens(x))2.$
(8)

### 2.5 Lower Confidence Bound

The LCB was suggested (Lewis et al., 2000; Emmerich et al., 2002) to select potential candidate solutions, especially in solving multimodal optimization problems. LCB can prevent premature convergence and enhance the search toward less explored regions in search space. The expression of LCB is
$fLCB(x)=f^(x)-ωs^(x),$
(9)
where $f^(x)$ and $s^(x)$ are prediction mean and variance (uncertainty degree) from surrogates, respectively; the parameter $ω$ scales the impact of the variance; a reasonable choice is $ω=2$, which leads to a high confidence probability (around $97%$) (Emmerich et al., 2002).

We aim to address the issue that there is no theoretical method to measure the uncertainty of prediction of Non-GP surrogates. Hence, an uncertainty measure for prediction of Non-GP surrogates is proposed in this article. This method can be written in the form of random field model. In detail, it consists of two components: regression function (namely Non-GP surrogate) and residual variation (the uncertainty of prediction of Non-GP surrogate). In this method, the two components are uncorrelated. In the first term, Non-GP surrogate as regression function only depends on decision variables, and the second term represents the uncertainty of prediction of Non-GP surrogate based on a stationary random field.

### 3.1 Formulation for Uncertainty Measure for Prediction

The UMP is formulated as
$F(x)=m(x)+ε(x),$
(10)
where $m(x)$ as a regression function can be any Non-GP surrogate or ensemble Non-GP surrogates, $ε(x)$ is a mean 0 random field with distribution $N(0,σ2)$.
In this article, the UMP makes the assumptions in building a cheap surrogate for an expensive function $y=f(x),x∈Rd$, $F(x)∼N(m(x),σ2)$ is a random variable, and $ε(x)∼N(0,σ2)$. For any $x$, $x'∈Rd$, the correlation between $ε(x)$ and $ε(x'),$ depends on the distance between $x$ and $x'$. The correlation function $c(x,x')$ in this article is shown in Eq. (11).
$c(x,x'θ)=exp-∑i=1dθi|xi-xi'|2,$
(11)
where $d$ is dimension of problems; $θ=[θ1,···,θd]T$ measures the importance or activity of the variable $x$.

#### 3.1.1 Hyperparameter Estimation

In UMP, the hyperparameters $σ2$, and $θ$ can be determined by maximizing the log likelihood function based on observe values $yi$ at $xi(i=1,···,N)$, which is shown as
$-12[Nlog(2πσ2)+log(det(C))+(y-m)TC-1(y-m)/σ2],$
(12)
where $m=(m(xi)),i=1,···,N$, is a known N-dimensional column vector of Non-GP surrogate among training data; $C$ is a known $N×N$ correlation matrix among training data; $y$ is a N-dimensional column observed vector among training data.
The estimation of $σ2$ can be obtained by taking the partial derivative of Eq. (12) with respect to $σ2$
$σ^2=(y-m)TC-1(y-m)N.$
(13)
Substituting Eq. (13) into Eq. (12), the maximum of log likelihood over $σ^2$ is
$-Nlog2πσ^2+log(det(C))+N2,$
(14)
since Eq. (14) depends only on parameters within $C$, thus above the maximum of log likelihood can be
$-Nlog2πσ^2-log(det(C)),$
(15)

#### 3.1.2 Prediction Distribution

When all unknown hyperparameters are determined, then the prediction distribution at any untested point $x$ can be obtained by using conditional distribution. The uncertainty (conditional variance) of prediction of Non-GP surrogates is
$s^(x)=σ^2[1-rTC-1r],$
(16)
where $r$ is a known $N×1$ correlation matrix of the untested point $x$ with training data.

### 3.2 Instantiation of UMP Framework

In UMP, any Non-GP surrogate can be considered to be the first term of the UMP. In this article, the first term will be instantiated with RBFN, QP, and SVM as a surrogate, respectively.

#### 3.2.1 UMP with RBFN

The form of RBFN is described in Eq. (2). Here, the cubic kernel function is used as its activation function
$ψ(∥x-c∥p)=∥x-c∥p3,$
(17)
where $c$ is a center point vector, $p=2$ in this article.

In this article, $2d+1$ cubic kernel functions are considered, based on the suggestion in Sprecher (1993). Based on the self-organizing method, $2d+1$ center point vectors are obtained by k-means algorithm (MacQueen, 1967).

#### 3.2.2 UMP with Quadratic Polynomial

The quadratic polynomial (QP, second-order polynomial) is one of the most widely used polynomial regression models. Due to its simplicity and flexibility, QP is usually used as a surrogate and has a wide range of applications in various fields of science and engineering. It can be expressed as follows
$f^(x)=β0+Σi=1dβixi+Σi=1,j=1,i≤jdβi,jxixj,$
(18)
where $β0$, $βi$ and $βi,j$ are the coefficients to be estimated; $x=[x1,…,xd]T$; QP is implemented using the Python tool-box (Pedregosa et al., 2011) in this article.

#### 3.2.3 UMP with SVM

SVM is one of the regression techniques that have been introduced in Section II. In this article, the radial basis kernel function is adopted in SVM
$κ(x,x')=<ϕ(x),ϕ(x')>=exp(-γ∥x-x'∥2),$
(19)
where the SVM is carried out using the Python tool-box (Pedregosa et al., 2011), the parameters $γ$ is set as $'scale'$ in this article.

### 3.3 Workflow of UMP

The pseudocode for the workflow of UMP is presented in the Algorithm 1. Initially, $11d-1$ samples in the search space are generated using Latin hypercube sampling (LHS) (Stein, 1987) and evaluated by exact functions. Then these samples are archived in an initial database. $τ$ latest samples in the database are selected as training data to train the Non-GP surrogate. The Non-GP surrogate replace the exact functions in evolving a population of $NP$ individuals for $T$ generations with a DE. Then a potential candidate solution is selected in the population by using LCB and evaluated by exact functions. After that, the solution is added to the database. Finally, when the computational budget is exhausted, the best solution in the database is chosen as the output.

To investigate the performance of the proposed UMP, a set of experiments is carried out in both single and ensemble surrogates by Algorithm 1, respectively. The Non-GP surrogates involving RBFN, QP, and SVM are considered in this article.

For the single surrogate, two experiments are carried out. First, the experiment compares Non-GP surrogates with and without UMP, and they are named UMP/RBFN, UMP/QP, UMP/SVM, RBFN, QP, and SVM, respectively. Second, the UMP compares with the existing uncertainty method in Branke and Schmidt (2005), which is the distance from the solutions to the existing training data (DUM). The three algorithms with the UMP are named UMP/RBFN, UMP/QP, and UMP/SVM, and the compared algorithms are named DUM/RBFN, DUM/QP, and DUM/SVM.

Regarding ensemble surrogates, the proposed UMP compares with method $Uens$ which is the maximum difference between the outputs of the ensemble members (Wang et al., 2017) and VUM which is variance of predictions output by the base surrogates of the ensemble (Guo et al., 2018), respectively. In this article, the ensemble surrogates consist of three surrogates: RBFN, QP, and SVM. These algorithms with the proposed method and two compared methods are named UMP/ensemble, $Uens$/ensemble, and VUM/ensemble, respectively.

### 4.1 Parameter Settings

There are several parameters in experiments. The setting of these parameters is given below.

(1) The computational budget with exact function evaluations $FEs=100$ was performed in this article, based on the assumption that the optimization algorithm is only allowed to evaluate a small number of candidate solutions during optimization. The number of the run was 25.

(2) DE parameters: DE/rand/1/bin was employed in this article. The evolution generations $T=100$, population size $NP=20$, scaling factor $F=0.5$, and the crossover rate $CR=0.9$.

(3) Initial samples $11d-1$ were randomly generated by LHS.

(4) The range of values for parameters $θ$ was $[1.0e-6,20]$.

(5) Training data $τ=50$ for dimension $d=2$, $τ=11d-1$ for $d=5,10$ was considered. $τ$ training data in the database was selected under considering both the quality and the computational cost of Non-GP surrogate.

### 4.2 Test Problems

The effectiveness of the proposed method is verified on benchmark problems CEC 2014 (Liu et al., 2014) with 2, 5, and 10 dimensions. The benchmark problems are listed in Table 1.

Table 1:

Test problems.

ProblemObjective function name$f*$Property
$F1$ Shifted Sphere Unimodal
$F2$ Shifted Ellipsoid Unimodal
$F3$ Shifted and Rotated Ellipsoid Unimodal
$F4$ Shifted Step Unimodal, Discontinuous
$F5$ Shifted Ackley Multi-modal
$F6$ Shifted Griewank Multi-modal
$F7$ Shifted and Rotated Rosenbrock Multi-modal with very narrow valley
$F8$ Shifted and Rotated Rastrigin Very complicated multi-modal
ProblemObjective function name$f*$Property
$F1$ Shifted Sphere Unimodal
$F2$ Shifted Ellipsoid Unimodal
$F3$ Shifted and Rotated Ellipsoid Unimodal
$F4$ Shifted Step Unimodal, Discontinuous
$F5$ Shifted Ackley Multi-modal
$F6$ Shifted Griewank Multi-modal
$F7$ Shifted and Rotated Rosenbrock Multi-modal with very narrow valley
$F8$ Shifted and Rotated Rastrigin Very complicated multi-modal

### 4.3 Comparison on Single Non-GP Surrogate

#### 4.3.1 Effect of the UMP

To investigate the effectiveness of the proposed UMP, a set of comparative experiments are carried out for algorithms with and without UMP. Table 2 presents all algorithms' average best fitness values on test problems with 2, 5, and 10 dimensions. Figures 1, 2, and 3 present the comparison of convergence curves of different algorithms on $F1,F5$, and $F8$ test problems with different dimensions, respectively.
Figure 1:

Comparison of with and without UMP on convergence curves for $F1$, $F5$, and $F8$ on 2d, respectively.

Figure 1:

Comparison of with and without UMP on convergence curves for $F1$, $F5$, and $F8$ on 2d, respectively.

Close modal
Table 2:

Comparing the averages fitness values (shown as Avg $±$ Std) of UMP/RBFN, UMP/QP, UMP/SVM, RBFN, QP, and SVM. Statistically significant results evaluated using a Friedman test.

ProblemdUMP/RBFNRBFNUMP/QPQPUMP/SVMSVM
F1 9.8806E-3$±$ 0.0156 0.0642 $±$ 0.1018 1.4462E-4$±$ 2.4305E-4 2.8772 $±$ 2.8031 6.0986E-4$±$ 7.6748E-4 0.8707 $±$ 1.5511
0.0331$±$ 0.0328 0.2288 $±$ 0.1966 0.7640$±$ 0.7112 54.0371 $±$ 23.7337 0.0264$±$ 0.0169 25.1719 $±$ 16.9316
10 1.9803$±$ 1.0066 9.6927 $±$ 5.0127 67.9611$±$ 21.5325 287.0700 $±$ 76.2774 21.5962$±$ 11.4967 87.2006 $±$ 37.7374
F2 0.0182$±$ 0.0342 0.2342 $±$ 0.5029 3.0383E-4$±$ 2.5810E-4 2.8759 $±$ 2.6865 1.0445E-3$±$ 1.3433E-3 5.2747 $±$ 8.8904
0.4057$±$ 0.3093 2.3933 $±$ 1.7200 14.0614$±$ 19.1254 155.4160 $±$ 68.8183 0.0799$±$ 0.0468 173.4310 $±$ 82.1052
10 32.4434$±$ 16.5484 129.7015 $±$ 51.3120 307.0399$±$ 97.6482 1425.5204 $±$ 393.2456 109.8823$±$ 53.2732 555.1530 $±$ 268.1075
F3 0.0437$±$ 0.0915 1.2991 $±$ 1.8210 9.9812E-3$±$ 0.0320 1.3025 $±$ 1.2417 0.0011$±$ 0.0010 5.4055 $±$ 7.0946
0.4611$±$ 0.4911 3.2056 $±$ 4.6624 5.6759$±$ 5.0457 39.0094 $±$ 21.5417 0.0307$±$ 0.0185 38.9898 $±$ 25.2550
10 243.0518$±$ 84.7628 823.7360 $±$ 378.8218 477.9377$±$ 182.8839 1921.1620 $±$ 503.9823 559.6709$±$ 316.1527 2451.7201 $±$ 795.8515
F4 0.0$±$ 0.0 0.0$±$ 0.0 0.0$±$ 0.0 2.2400 $±$ 2.1029 0.0$±$ 0.0 0.6800 $±$ 0.6144
0.1200$±$ 0.3249 0.2000 $±$ 0.4000 4.4000$±$ 4.9799 49.0000 $±$ 24.5715 0.7600$±$ 1.0688 26.2000 $±$ 24.6398
10 2.5200$±$ 1.0998 9.5200 $±$ 4.3185 65.2000$±$ 22.7385 230.8000 $±$ 77.1305 23.5200$±$ 10.9986 76.1600 $±$ 34.7754
F5 2.7644$±$ 1.0851 10.4969 $±$ 4.7438 2.7476$±$ 0.8620 8.0343 $±$ 2.2641 1.8426$±$ 0.6829 2.2079 $±$ 1.1024
3.6208$±$ 0.7392 12.9774 $±$ 1.5847 4.0928$±$ 0.912 16.1825 $±$ 1.9481 3.5112$±$ 0.5640 6.3787 $±$ 2.4737
10 4.2216$±$ 0.3694 13.2189 $±$ 1.1633 10.6836$±$ 6.2570 18.4682 $±$ 1.0169 3.7534$±$ 0.3806 9.8341 $±$ 1.4496
F6 0.1410$±$ 0.0035 0.1600 $±$ 0.3545 0.4851$±$ 0.2165 1.1618 $±$ 0.7925 0.0602$±$ 0.0401 0.1971 $±$ 0.1836
0.6092$±$ 0.2302 0.7121 $±$ 0.8439 1.1118$±$ 0.2986 16.7889 $±$ 7.6244 0.5397$±$ 0.1432 3.9278 $±$ 3.3254
10 1.1185$±$ 0.0837 1.2103 $±$ 0.1364 14.2539$±$ 4.4617 75.7570 $±$ 21.1887 3.7237$±$ 1.0159 18.4265 $±$ 6.9328
F7 0.1219$±$ 0.1055 0.2861 $±$ 0.2670 0.0732$±$ 0.0671 0.0773 $±$ 0.0712 0.0131$±$ 0.0142 0.2308 $±$ 0.2212
43.3089$±$ 30.4430 70.2126 $±$ 47.2680 66.0930$±$ 34.6149 72.0384 $±$ 30.1393 67.1543$±$ 27.5345 126.1766 $±$ 79.2998
10 204.2603$±$ 65.2703 316.0039 $±$ 125.3541 353.6906$±$ 159.2818 728.8562 $±$ 328.2364 708.1847$±$ 356.1181 719.1093 $±$ 365.0644
F8 3.1274$±$ 1.4961 3.3412 $±$ 3.0295 3.7549$±$ 2.0459 3.8671 $±$ 1.3504 2.3542$±$ 1.9835 3.1703 $±$ 2.2549
18.5786$±$ 3.9201 19.3769 $±$ 4.4019 22.5314$±$ 7.7902 29.2176 $±$ 5.6230 21.0797$±$ 4.3804 22.6443 $±$ 7.0308
10 59.4989$±$ 11.4535 59.9236 $±$ 6.4440 76.0060$±$ 16.9585 96.6026 $±$ 16.0349 72.4144$±$ 13.7050 90.2328 $±$ 12.0011
Average rank 1.98 3.35 3.35 5.46 2.02 4.83
ProblemdUMP/RBFNRBFNUMP/QPQPUMP/SVMSVM
F1 9.8806E-3$±$ 0.0156 0.0642 $±$ 0.1018 1.4462E-4$±$ 2.4305E-4 2.8772 $±$ 2.8031 6.0986E-4$±$ 7.6748E-4 0.8707 $±$ 1.5511
0.0331$±$ 0.0328 0.2288 $±$ 0.1966 0.7640$±$ 0.7112 54.0371 $±$ 23.7337 0.0264$±$ 0.0169 25.1719 $±$ 16.9316
10 1.9803$±$ 1.0066 9.6927 $±$ 5.0127 67.9611$±$ 21.5325 287.0700 $±$ 76.2774 21.5962$±$ 11.4967 87.2006 $±$ 37.7374
F2 0.0182$±$ 0.0342 0.2342 $±$ 0.5029 3.0383E-4$±$ 2.5810E-4 2.8759 $±$ 2.6865 1.0445E-3$±$ 1.3433E-3 5.2747 $±$ 8.8904
0.4057$±$ 0.3093 2.3933 $±$ 1.7200 14.0614$±$ 19.1254 155.4160 $±$ 68.8183 0.0799$±$ 0.0468 173.4310 $±$ 82.1052
10 32.4434$±$ 16.5484 129.7015 $±$ 51.3120 307.0399$±$ 97.6482 1425.5204 $±$ 393.2456 109.8823$±$ 53.2732 555.1530 $±$ 268.1075
F3 0.0437$±$ 0.0915 1.2991 $±$ 1.8210 9.9812E-3$±$ 0.0320 1.3025 $±$ 1.2417 0.0011$±$ 0.0010 5.4055 $±$ 7.0946
0.4611$±$ 0.4911 3.2056 $±$ 4.6624 5.6759$±$ 5.0457 39.0094 $±$ 21.5417 0.0307$±$ 0.0185 38.9898 $±$ 25.2550
10 243.0518$±$ 84.7628 823.7360 $±$ 378.8218 477.9377$±$ 182.8839 1921.1620 $±$ 503.9823 559.6709$±$ 316.1527 2451.7201 $±$ 795.8515
F4 0.0$±$ 0.0 0.0$±$ 0.0 0.0$±$ 0.0 2.2400 $±$ 2.1029 0.0$±$ 0.0 0.6800 $±$ 0.6144
0.1200$±$ 0.3249 0.2000 $±$ 0.4000 4.4000$±$ 4.9799 49.0000 $±$ 24.5715 0.7600$±$ 1.0688 26.2000 $±$ 24.6398
10 2.5200$±$ 1.0998 9.5200 $±$ 4.3185 65.2000$±$ 22.7385 230.8000 $±$ 77.1305 23.5200$±$ 10.9986 76.1600 $±$ 34.7754
F5 2.7644$±$ 1.0851 10.4969 $±$ 4.7438 2.7476$±$ 0.8620 8.0343 $±$ 2.2641 1.8426$±$ 0.6829 2.2079 $±$ 1.1024
3.6208$±$ 0.7392 12.9774 $±$ 1.5847 4.0928$±$ 0.912 16.1825 $±$ 1.9481 3.5112$±$ 0.5640 6.3787 $±$ 2.4737
10 4.2216$±$ 0.3694 13.2189 $±$ 1.1633 10.6836$±$ 6.2570 18.4682 $±$ 1.0169 3.7534$±$ 0.3806 9.8341 $±$ 1.4496
F6 0.1410$±$ 0.0035 0.1600 $±$ 0.3545 0.4851$±$ 0.2165 1.1618 $±$ 0.7925 0.0602$±$ 0.0401 0.1971 $±$ 0.1836
0.6092$±$ 0.2302 0.7121 $±$ 0.8439 1.1118$±$ 0.2986 16.7889 $±$ 7.6244 0.5397$±$ 0.1432 3.9278 $±$ 3.3254
10 1.1185$±$ 0.0837 1.2103 $±$ 0.1364 14.2539$±$ 4.4617 75.7570 $±$ 21.1887 3.7237$±$ 1.0159 18.4265 $±$ 6.9328
F7 0.1219$±$ 0.1055 0.2861 $±$ 0.2670 0.0732$±$ 0.0671 0.0773 $±$ 0.0712 0.0131$±$ 0.0142 0.2308 $±$ 0.2212
43.3089$±$ 30.4430 70.2126 $±$ 47.2680 66.0930$±$ 34.6149 72.0384 $±$ 30.1393 67.1543$±$ 27.5345 126.1766 $±$ 79.2998
10 204.2603$±$ 65.2703 316.0039 $±$ 125.3541 353.6906$±$ 159.2818 728.8562 $±$ 328.2364 708.1847$±$ 356.1181 719.1093 $±$ 365.0644
F8 3.1274$±$ 1.4961 3.3412 $±$ 3.0295 3.7549$±$ 2.0459 3.8671 $±$ 1.3504 2.3542$±$ 1.9835 3.1703 $±$ 2.2549
18.5786$±$ 3.9201 19.3769 $±$ 4.4019 22.5314$±$ 7.7902 29.2176 $±$ 5.6230 21.0797$±$ 4.3804 22.6443 $±$ 7.0308
10 59.4989$±$ 11.4535 59.9236 $±$ 6.4440 76.0060$±$ 16.9585 96.6026 $±$ 16.0349 72.4144$±$ 13.7050 90.2328 $±$ 12.0011
Average rank 1.98 3.35 3.35 5.46 2.02 4.83
Figure 2:

Comparison of with and without UMP on convergence curves for $F1$, $F5$, and $F8$ on 5d, respectively.

Figure 2:

Comparison of with and without UMP on convergence curves for $F1$, $F5$, and $F8$ on 5d, respectively.

Close modal
Figure 3:

Comparison of with and without UMP on convergence curves for $F1$, $F5$, and $F8$ on 10d, respectively.

Figure 3:

Comparison of with and without UMP on convergence curves for $F1$, $F5$, and $F8$ on 10d, respectively.

Close modal

From Table 2 and Figures 13, the results of UMP/RBFN, UMP/QP, and UMP/SVM are significantly better than those of the other algorithms in most test problems. The performance of the three algorithms (UMP/RBFN, UMP/QP, and UMP/SVM) is always better than the other on both unimodal or multimodal problems, especially in 5- and 10-dimensional test problems. The results are mainly attributed to the fact that there is uncertainty information in the three algorithms that can be used to guide the search of algorithms and the update of surrogates. The test problems are more complex as the number of dimensions increases, and there are many local optima for multimodal problems. The uncertainty information is able to strengthen the exploration ability of algorithms and improve the accuracy of the surrogates.

#### 4.3.2 Comparison with Peer Algorithms

To further investigate the effectiveness of the proposed UMP, the proposed UMP is compared with the uncertainty method DUM, which is shown in Eq. (20). Table 3 presents the average best fitness values obtained by proposed algorithms UMP/RBFN, UMP/QP, and UMP/SVM, and the compared algorithms DUM/RBFN, DUM/QP, and DUM/SVM. Figures 4, 5, and 6 present the comparison of convergence curves of different algorithms on $F1,F5$, and $F8$ with different dimensions, respectively.
$U(x)=1∑i=1L1dxxi',$
(20)
where $U(x)$ represents the uncertainty of prediction of a solution $x$, $dxxi'$ is the Euclidean distance from solution $x$ to solution $xi'$ in the training data $τ$, and $L$ is the number of solutions in the neighborhood used for estimation; $L$ is equal to the number of training data $τ$ in this article.
Figure 4:

Comparison of UMP and DUM on convergence curves for $F1$, $F5$, and $F8$ on 2d, respectively.

Figure 4:

Comparison of UMP and DUM on convergence curves for $F1$, $F5$, and $F8$ on 2d, respectively.

Close modal
Table 3:

Comparing the averages fitness values (shown as Avg $±$ Std) of UMP/RBFN, UMP/QP, UMP/SVM, DUM/RBFN, DUM/QP, and DUM/SVM. Statistically significant results evaluated using a Friedman test.

ProblemdUMP/RBFNDUM/RBFNUMP/QPDUM/QPUMP/SVMDUM/SVM
F1 9.8806E-3$±$ 0.0156 0.0502 $±$ 0.0802 1.4462E-4$±$ 2.4305E-4 4.1442 $±$ 3.3661 6.0986E-4$±$ 7.6748E-4 0.0180 $±$ 0.0232
0.0331$±$ 0.0328 0.1686 $±$ 0.1147 0.7640$±$ 0.7112 140.5686 $±$ 65.2623 0.0264$±$ 0.0169 26.6025 $±$ 25.5855
10 1.9803$±$ 1.0066 9.4756 $±$ 4.5445 67.9611$±$ 21.5325 477.5787 $±$ 130.4803 21.5962$±$ 11.4967 84.9443 $±$ 29.7331
F2 0.0182$±$ 0.0342 0.2449 $±$ 0.3725 3.0383E-4$±$ 2.5810E-4 6.7903 $±$ 6.7277 1.1499E-3$±$ 1.0445E-3 0.3620 $±$ 0.7392
0.4057$±$ 0.3093 4.1801 $±$ 3.7846 14.0614$±$ 19.1254 365.6379 $±$ 146.3157 0.0799$±$ 0.0468 148.4953 $±$ 124.1047
10 32.4434$±$ 16.5484 119.4572 $±$ 49.2795 307.0399$±$ 97.6482 2484.1205 $±$ 510.5520 109.8823$±$ 53.2732 638.3514 $±$ 180.0408
F3 0.0437$±$ 0.0915 0.5505 $±$ 0.6879 9.9812E-3$±$ 0.0320 4.7287 $±$ 2.9526 0.0053$±$ 0.0069 0.2930 $±$ 0.5269
0.4611$±$ 0.4911 2.0510 $±$ 1.1517 5.6759$±$ 5.0457 110.6858 $±$ 48.6745 0.0307$±$ 0.0185 35.7253 $±$ 33.3522
10 243.0518$±$ 84.7628 848.8346 $±$ 268.9680 477.9377$±$ 182.8839 3141.0693 $±$ 956.7205 559.6709$±$ 316.1527 2204.2445 $±$ 1091.0657
F4 0.0$±$ 0.0 0.0$±$ 0.0 0.0$±$ 0.0 4.0800 $±$ 3.6212 0.0$±$ 0.0 0.0$±$ 0.0
0.1200$±$ 0.3249 0.1600 $±$ 0.3666 4.4000$±$ 4.9799 141.1600 $±$ 55.9783 0.7600$±$ 1.0688 31.6800 $±$ 26.8264
10 2.5200$±$ 1.0998 8.6000 $±$ 3.1874 65.2000$±$ 22.7385 439.3200 $±$ 112.3308 23.5200$±$ 10.9986 76.6000 $±$ 30.6111
F5 2.7644$±$ 1.0851 9.7490 $±$ 3.8671 2.7476$±$ 0.8620 8.1565 $±$ 2.7408 1.8426 $±$ 0.6829 0.9289$±$ 0.5018
3.6208$±$ 0.7392 11.8476 $±$ 2.1805 4.0928$±$ 0.9124 17.0522 $±$ 1.5073 3.5112$±$ 0.5640 6.0516 $±$ 1.4182
10 4.2216$±$ 0.3694 12.7883 $±$ 1.5767 10.6836$±$ 6.2570 19.2076 $±$ 0.7207 3.7534$±$ 0.3806 9.1504 $±$ 1.2157
F6 0.1410$±$ 0.0035 0.1494 $±$ 0.0676 0.4851$±$ 0.216 2.0738 $±$ 1.0517 0.0602$±$ 0.0401 0.1313 $±$ 0.1128
0.6092$±$ 0.2302 0.6143 $±$ 0.1708 1.1118$±$ 0.2986 33.2851 $±$ 11.7161 0.5397$±$ 0.1432 5.2599 $±$ 4.2937
10 1.1185$±$ 0.0837 1.1323 $±$ 0.0466 14.2539$±$ 4.4617 117.7120 $±$ 20.8241 3.7237$±$ 1.0159 17.1081 $±$ 6.6650
F7 0.1219$±$ 0.1055 0.2439 $±$ 0.2661 0.0732$±$ 0.0671 0.1221 $±$ 0.1160 0.0131$±$ 0.0142 0.0974 $±$ 0.0516
43.3089$±$ 30.4430 88.9265 $±$ 47.3513 66.0930$±$ 34.6149 113.1713 $±$ 55.7217 67.1543$±$ 27.5345 104.1294 $±$ 65.0487
10 204.2603$±$ 65.2703 311.2954 $±$ 159.9181 353.6906$±$ 159.2818 1876.2670 $±$ 884.2830 708.1847$±$ 356.1181 719.7931 $±$ 303.3686
F8 3.1274$±$ 1.4961 3.3755 $±$ 2.4286 3.7549$±$ 2.0459 4.1802 $±$ 2.0134 2.3542$±$ 1.9835 2.4814 $±$ 2.0600
18.5786$±$ 3.9201 18.6270 $±$ 5.6959 22.5314$±$ 7.7902 31.8924 $±$ 8.3335 21.0797$±$ 4.3804 21.1876 $±$ 6.2968
10 59.4989$±$ 11.4535 60.8467 $±$ 6.7949 76.0060$±$ 16.9585 123.9444 $±$ 13.2013 72.4144$±$ 13.7050 83.7737 $±$ 13.1376
Average rank 2.08 3.42 3.38 5.92 2.04 4.17
ProblemdUMP/RBFNDUM/RBFNUMP/QPDUM/QPUMP/SVMDUM/SVM
F1 9.8806E-3$±$ 0.0156 0.0502 $±$ 0.0802 1.4462E-4$±$ 2.4305E-4 4.1442 $±$ 3.3661 6.0986E-4$±$ 7.6748E-4 0.0180 $±$ 0.0232
0.0331$±$ 0.0328 0.1686 $±$ 0.1147 0.7640$±$ 0.7112 140.5686 $±$ 65.2623 0.0264$±$ 0.0169 26.6025 $±$ 25.5855
10 1.9803$±$ 1.0066 9.4756 $±$ 4.5445 67.9611$±$ 21.5325 477.5787 $±$ 130.4803 21.5962$±$ 11.4967 84.9443 $±$ 29.7331
F2 0.0182$±$ 0.0342 0.2449 $±$ 0.3725 3.0383E-4$±$ 2.5810E-4 6.7903 $±$ 6.7277 1.1499E-3$±$ 1.0445E-3 0.3620 $±$ 0.7392
0.4057$±$ 0.3093 4.1801 $±$ 3.7846 14.0614$±$ 19.1254 365.6379 $±$ 146.3157 0.0799$±$ 0.0468 148.4953 $±$ 124.1047
10 32.4434$±$ 16.5484 119.4572 $±$ 49.2795 307.0399$±$ 97.6482 2484.1205 $±$ 510.5520 109.8823$±$ 53.2732 638.3514 $±$ 180.0408
F3 0.0437$±$ 0.0915 0.5505 $±$ 0.6879 9.9812E-3$±$ 0.0320 4.7287 $±$ 2.9526 0.0053$±$ 0.0069 0.2930 $±$ 0.5269
0.4611$±$ 0.4911 2.0510 $±$ 1.1517 5.6759$±$ 5.0457 110.6858 $±$ 48.6745 0.0307$±$ 0.0185 35.7253 $±$ 33.3522
10 243.0518$±$ 84.7628 848.8346 $±$ 268.9680 477.9377$±$ 182.8839 3141.0693 $±$ 956.7205 559.6709$±$ 316.1527 2204.2445 $±$ 1091.0657
F4 0.0$±$ 0.0 0.0$±$ 0.0 0.0$±$ 0.0 4.0800 $±$ 3.6212 0.0$±$ 0.0 0.0$±$ 0.0
0.1200$±$ 0.3249 0.1600 $±$ 0.3666 4.4000$±$ 4.9799 141.1600 $±$ 55.9783 0.7600$±$ 1.0688 31.6800 $±$ 26.8264
10 2.5200$±$ 1.0998 8.6000 $±$ 3.1874 65.2000$±$ 22.7385 439.3200 $±$ 112.3308 23.5200$±$ 10.9986 76.6000 $±$ 30.6111
F5 2.7644$±$ 1.0851 9.7490 $±$ 3.8671 2.7476$±$ 0.8620 8.1565 $±$ 2.7408 1.8426 $±$ 0.6829 0.9289$±$ 0.5018
3.6208$±$ 0.7392 11.8476 $±$ 2.1805 4.0928$±$ 0.9124 17.0522 $±$ 1.5073 3.5112$±$ 0.5640 6.0516 $±$ 1.4182
10 4.2216$±$ 0.3694 12.7883 $±$ 1.5767 10.6836$±$ 6.2570 19.2076 $±$ 0.7207 3.7534$±$ 0.3806 9.1504 $±$ 1.2157
F6 0.1410$±$ 0.0035 0.1494 $±$ 0.0676 0.4851$±$ 0.216 2.0738 $±$ 1.0517 0.0602$±$ 0.0401 0.1313 $±$ 0.1128
0.6092$±$ 0.2302 0.6143 $±$ 0.1708 1.1118$±$ 0.2986 33.2851 $±$ 11.7161 0.5397$±$ 0.1432 5.2599 $±$ 4.2937
10 1.1185$±$ 0.0837 1.1323 $±$ 0.0466 14.2539$±$ 4.4617 117.7120 $±$ 20.8241 3.7237$±$ 1.0159 17.1081 $±$ 6.6650
F7 0.1219$±$ 0.1055 0.2439 $±$ 0.2661 0.0732$±$ 0.0671 0.1221 $±$ 0.1160 0.0131$±$ 0.0142 0.0974 $±$ 0.0516
43.3089$±$ 30.4430 88.9265 $±$ 47.3513 66.0930$±$ 34.6149 113.1713 $±$ 55.7217 67.1543$±$ 27.5345 104.1294 $±$ 65.0487
10 204.2603$±$ 65.2703 311.2954 $±$ 159.9181 353.6906$±$ 159.2818 1876.2670 $±$ 884.2830 708.1847$±$ 356.1181 719.7931 $±$ 303.3686
F8 3.1274$±$ 1.4961 3.3755 $±$ 2.4286 3.7549$±$ 2.0459 4.1802 $±$ 2.0134 2.3542$±$ 1.9835 2.4814 $±$ 2.0600
18.5786$±$ 3.9201 18.6270 $±$ 5.6959 22.5314$±$ 7.7902 31.8924 $±$ 8.3335 21.0797$±$ 4.3804 21.1876 $±$ 6.2968
10 59.4989$±$ 11.4535 60.8467 $±$ 6.7949 76.0060$±$ 16.9585 123.9444 $±$ 13.2013 72.4144$±$ 13.7050 83.7737 $±$ 13.1376
Average rank 2.08 3.42 3.38 5.92 2.04 4.17
Figure 5:

Comparison of UMP and DUM on convergence curves for $F1$, $F5$, and $F8$ on 5d, respectively.

Figure 5:

Comparison of UMP and DUM on convergence curves for $F1$, $F5$, and $F8$ on 5d, respectively.

Close modal
Figure 6:

Comparison of UMP and DUM on convergence curves for $F1$, $F5$, and $F8$ on 10d, respectively.

Figure 6:

Comparison of UMP and DUM on convergence curves for $F1$, $F5$, and $F8$ on 10d, respectively.

Close modal

The Table 3 and Figures 46 show that UMP/RBFN, UMP/QP, and UMP/SVM achieve significantly better performance than the other algorithms on most test problems. There is a significant difference in 10 dimensional test problems, especially for $F3$, $F5$, and $F7$, which maybe due to the accuracy of DUM that is unreliable. Thus, the DUM cannot efficiently guide the search of algorithms when there are many local optima on $F5$ and $F7$ in 10-dimensional test problems, and it cannot accurately fit the fitness landscape of $F3$, $F5$, and $F7$ in 10-dimensional test problems. Instead, the proposed UMP still has good performance in these problems with 10 dimension.

A comparison experiment is carried out to analyze further the proposed method on $F8$, with the results shown in Figure 7. In the figure, the RBFN surrogate is used as a regression function to approximate the exact function, and the UMP and DUM methods are used to estimate the uncertainty of prediction of RBFN. From Figure 7a, there is a significant error between the prediction values of RBFN and the exact function. The proposed UMP is able to measure the error more accurately than DUM. Besides, the values adjusting to the prediction of RBFN from the proposed UMP are better at approximating the exact function values than DUM, according to the assumption of random field model as Eq. (10). Therefore, the proposed UMP has better performance in the measurement of uncertainty for the prediction of RBFN than DUM. The validity of the proposed UMP also is investigated in Figure 7 b, and it shows that $fLCB$ of UMP is a better approximation to the lower bound for the exact function values than that of DUM. The DUM method indicates the crowded degree of the neighborhood of a solution. Thus it cannot accurately measure the uncertainty of prediction of RBFN.
Figure 7:

(a) Illustration of UMP and DUM methods on a 1-d toy example with $F8$ function; the number of training points 100 is considered; red curve represents the exact function; green dashed curve represents the prediction from RBFN; shaded regions represent the confidence interval of the prediction of RBFN; (b) $y-fLCB$ plot on $F8$ in single surrogate RBFN case.

Figure 7:

(a) Illustration of UMP and DUM methods on a 1-d toy example with $F8$ function; the number of training points 100 is considered; red curve represents the exact function; green dashed curve represents the prediction from RBFN; shaded regions represent the confidence interval of the prediction of RBFN; (b) $y-fLCB$ plot on $F8$ in single surrogate RBFN case.

Close modal

### 4.4 Comparison on Ensemble Surrogates

An experiment is carried out to show the effectiveness of the proposed UMP on ensemble surrogates case. The proposed method is compared with the uncertainty method $Uens$ in Eq. (7) and method VUM in Eq. (8), respectively. Table 4 presents the average best fitness values obtained by UMP/ensemble, $Uens$/ensemble, and VUM/ensemble algorithms on a set of test problems. Figures 8, 9, and 10 present the comparison of convergence curves of different algorithms on $F1,F5$, and $F8$ with different dimensions, respectively.
Figure 8:

Comparison of UMP, $Uens$ and VUM on convergence curves for $F1$, $F5$, and $F8$ on 2d, respectively.

Figure 8:

Comparison of UMP, $Uens$ and VUM on convergence curves for $F1$, $F5$, and $F8$ on 2d, respectively.

Close modal
Table 4:

Comparing the averages fitness values (shown as Avg $±$ Std) of UMP/ensemble, $Uens$/ensemble, and VUM/ensemble. Statistically significant results evaluated using a Friedman test.

ProblemdUMP/ensembleVUM/ensemble$Uens$/ensemble
F1 7.7161E-5$±$ 1.5035E-4 1.3604 $±$ 1.2453 9.3382E-3 $±$ 0.0197
0.0011$±$ 0.0012 12.1513 $±$ 8.6543 0.1555 $±$ 0.0832
10 1.0012$±$ 0.7830 23.5616 $±$ 9.8029 9.3609 $±$ 3.5025
F2 2.0820E-4$±$ 2.2511E-4 3.7257 $±$ 2.9982 0.0810 $±$ 0.1031
0.0526$±$ 0.0614 77.1360 $±$ 38.0107 2.6106 $±$ 2.0893
10 33.0773$±$ 25.3703 284.9453 $±$ 95.0346 141.5993 $±$ 40.6508
F3 4.1281E-4$±$ 5.1014E-4 3.4762 $±$ 2.0919 0.6851 $±$ 1.3881
0.1005$±$ 0.1455 28.3199 $±$ 11.8856 2.5193 $±$ 1.5458
10 537.8770$±$ 150.8156 1204.2153 $±$ 381.1870 889.4860 $±$ 271.9350
F4 0.0$±$ 0.0 1.3200 $±$ 1.3181 0.0$±$ 0.0
0.1200$±$ 0.3249 12.600 $±$ 11.0381 0.1200$±$ 0.3249
10 3.2400$±$ 1.9448 21.0000 $±$ 8.4332 10.2000 $±$ 4.8249
F5 2.7002$±$ 1.1667 10.3965 $±$ 4.9964 3.8192 $±$ 1.8471
3.5092$±$ 0.7333 13.8540 $±$ 2.8998 8.4735 $±$ 1.5095
10 5.5079$±$ 0.5751 13.1730 $±$ 0.9270 10.8165 $±$ 1.1127
F6 0.0872$±$ 0.0544 0.6791 $±$ 0.3800 0.1351 $±$ 0.0789
0.6518 $±$ 0.2003 1.3107 $±$ 0.3293 0.6159$±$ 0.1419
10 1.0563$±$ 0.1378 1.3724 $±$ 0.3217 1.4022 $±$ 0.3057
F7 0.0076$±$ 0.0090 0.1469 $±$ 0.1503 0.2916 $±$ 0.2351
63.8693 $±$ 30.8029 81.6944 $±$ 35.7565 59.3588$±$ 38.5356
10 235.0830$±$ 119.8799 585.8805 $±$ 231.1875 334.1423 $±$ 157.4397
F8 1.7132$±$ 1.1547 3.3034 $±$ 1.8626 2.7242 $±$ 1.3798
19.9698 $±$ 5.5287 24.4794 $±$ 5.4224 17.1605$±$ 4.8970
10 69.4359 $±$ 7.8951 74.3970 $±$ 11.7897 58.7404$±$ 8.5256
Average rank 1.27 2.83 1.90
ProblemdUMP/ensembleVUM/ensemble$Uens$/ensemble
F1 7.7161E-5$±$ 1.5035E-4 1.3604 $±$ 1.2453 9.3382E-3 $±$ 0.0197
0.0011$±$ 0.0012 12.1513 $±$ 8.6543 0.1555 $±$ 0.0832
10 1.0012$±$ 0.7830 23.5616 $±$ 9.8029 9.3609 $±$ 3.5025
F2 2.0820E-4$±$ 2.2511E-4 3.7257 $±$ 2.9982 0.0810 $±$ 0.1031
0.0526$±$ 0.0614 77.1360 $±$ 38.0107 2.6106 $±$ 2.0893
10 33.0773$±$ 25.3703 284.9453 $±$ 95.0346 141.5993 $±$ 40.6508
F3 4.1281E-4$±$ 5.1014E-4 3.4762 $±$ 2.0919 0.6851 $±$ 1.3881
0.1005$±$ 0.1455 28.3199 $±$ 11.8856 2.5193 $±$ 1.5458
10 537.8770$±$ 150.8156 1204.2153 $±$ 381.1870 889.4860 $±$ 271.9350
F4 0.0$±$ 0.0 1.3200 $±$ 1.3181 0.0$±$ 0.0
0.1200$±$ 0.3249 12.600 $±$ 11.0381 0.1200$±$ 0.3249
10 3.2400$±$ 1.9448 21.0000 $±$ 8.4332 10.2000 $±$ 4.8249
F5 2.7002$±$ 1.1667 10.3965 $±$ 4.9964 3.8192 $±$ 1.8471
3.5092$±$ 0.7333 13.8540 $±$ 2.8998 8.4735 $±$ 1.5095
10 5.5079$±$ 0.5751 13.1730 $±$ 0.9270 10.8165 $±$ 1.1127
F6 0.0872$±$ 0.0544 0.6791 $±$ 0.3800 0.1351 $±$ 0.0789
0.6518 $±$ 0.2003 1.3107 $±$ 0.3293 0.6159$±$ 0.1419
10 1.0563$±$ 0.1378 1.3724 $±$ 0.3217 1.4022 $±$ 0.3057
F7 0.0076$±$ 0.0090 0.1469 $±$ 0.1503 0.2916 $±$ 0.2351
63.8693 $±$ 30.8029 81.6944 $±$ 35.7565 59.3588$±$ 38.5356
10 235.0830$±$ 119.8799 585.8805 $±$ 231.1875 334.1423 $±$ 157.4397
F8 1.7132$±$ 1.1547 3.3034 $±$ 1.8626 2.7242 $±$ 1.3798
19.9698 $±$ 5.5287 24.4794 $±$ 5.4224 17.1605$±$ 4.8970
10 69.4359 $±$ 7.8951 74.3970 $±$ 11.7897 58.7404$±$ 8.5256
Average rank 1.27 2.83 1.90
Figure 9:

Comparison of UMP, $Uens$ and VUM on convergence curves for $F1$, $F5$, and $F8$ on 5d, respectively.

Figure 9:

Comparison of UMP, $Uens$ and VUM on convergence curves for $F1$, $F5$, and $F8$ on 5d, respectively.

Close modal
Figure 10:

Comparison of UMP, $Uens$ and VUM on convergence curves for $F1$, $F5$, and $F8$ on 10d, respectively.

Figure 10:

Comparison of UMP, $Uens$ and VUM on convergence curves for $F1$, $F5$, and $F8$ on 10d, respectively.

Close modal

From Table 4 and Figures 810, the UMP/ensemble significantly outperforms $Uens$/ensemble and VUM/ensemble on most test problems. In the results, the performance of VUM/ensemble is worst. The performance of UMP/ensemble and $Uens$/ensemble is similar on $F4$ in 2 and 5 dimensions, and $Uens$/ensemble is slightly better than UMP/ensemble on $F6$, $F7$, and $F8$ in 5 dimension. However, UMP/ensemble is worse than $Uens$/ensemble on $F8$ in 10 dimension, probably because the number of computational budget $FEs=100$ is too small. Most of the reevaluation solutions are consumed due to a large degree of uncertainty, resulting in less opportunity for UMP/ensemble to exploit the search space sufficiently.

The effectiveness is further analyzed on ensemble surrogates case in Figure 11. In the figure, the ensemble of RBFN, QP, and SVM surrogates is used as regression function to approximate the exact $F8$ function, and the UMP, VUM, and $Uens$ methods are used to estimate the uncertainty of prediction of the ensemble, respectively. From Figure 11a, there is a significant error between the prediction values of the ensemble and the exact function. The proposed UMP is best to approximate the error, and the uncertainty of the ensemble from VUM and $Uens$ methods is similar in general. Besides, the values obtained by UMP are the best approximation of the exact function values than VUM and $Uens$ in general, according to the assumption of random field model as Eq. (10). Figure 11b also verifies $fLCB$ of UMP is a best approximation to the lower bound for the exact function values.
Figure 11:

(a) Illustration of UMP, $Uens$ and VUM methods on a 1-d toy example with $F8$ function; the number of training points 100 is considered; red curve represents the exact function; green dashed curve represents the ensemble prediction from the weight sum of prediction of RBFN, QP and SVM; shaded regions represent confidence interval of the ensemble prediction; (b) $y-fLCB$ plot on $F8$ in ensemble surrogates case.

Figure 11:

(a) Illustration of UMP, $Uens$ and VUM methods on a 1-d toy example with $F8$ function; the number of training points 100 is considered; red curve represents the exact function; green dashed curve represents the ensemble prediction from the weight sum of prediction of RBFN, QP and SVM; shaded regions represent confidence interval of the ensemble prediction; (b) $y-fLCB$ plot on $F8$ in ensemble surrogates case.

Close modal

Regarding the uncertainty method $Uens$, it is maximum difference among the outputs of the ensemble members for the prediction of a solution. In essence, this method describes the prediction difference among ensemble members in a solution. Likewise, for the method VUM, the uncertainty of prediction of a solution is measured by the variance of the prediction output by the base surrogates of the ensemble. This method indicates the average squared deviation of the base surrogates about the output of the ensemble surrogates. From the probability and statistic viewpoint, these methods measuring the uncertainty of prediction can be insufficient. Therefore, these compared methods are potentially unreliable in measuring the uncertainty of the prediction of surrogates.

This article mainly addresses the issue that there is no theoretical method to measure the uncertainty of prediction of Non-GP surrogates. This article proposes a theoretical method to measure the uncertainty. In this method, a stationary random field with a known zero mean is used to measure the uncertainty of prediction of Non-KGP surrogates. The method's effectiveness has been verified based on some experiments in single and ensemble surrogates cases. The experimental results demonstrate that the proposed method is more promising than other methods on a set of test problems.

Although the performance of the proposed method is competitive, the computational cost of this method is higher than others. The computational cost of UMP, DUM, $Uens$ and VUM is $O(N3)$, $O(N)$, $O(1)$, and $O(1)$, respectively. $N$ is the number of training samples. Thus, the main limitation of the method is that it needs more computational cost in measuring the uncertainty for the prediction of Non-GP surrogates. Therefore, our future work is to solve this drawback using transfer learning.

This work was supported in part by the National Natural Science Foundation of China under Grants 62076226, in part by the Fundamental Research Funds for the Central Universities China University of Geosciences (Wuhan) under Grant $CUGGC02$, and in part by the 111 project under Grant $B17040$.

Branke
,
J.
, and
Schmidt
,
C
. (
2005
).
Faster convergence by means of fitness estimation
.
Soft Computing
,
9
(
1
):
13
20
.
Braun
,
J.
,
Krettek
,
J.
,
Hoffmann
,
F.
, and
Bertram
,
T
. (
2009
).
Multi-objective optimization with controlled model assisted evolution strategies
.
Evolutionary Computation
,
17
(
4
):
577
593
.
Chugh
,
T.
,
Jin
,
Y.
,
Miettinen
,
K.
,
Hakanen
,
J.
, and
Sindhya
,
K
. (
2016
).
A surrogate-assisted reference vector guided evolutionary algorithm for computationally expensive many-objective optimization
.
IEEE Transactions on Evolutionary Computation
,
22
(
1
):
129
142
.
Chugh
,
T.
,
Sindhya
,
K.
,
Hakanen
,
J.
, and
Miettinen
,
K
. (
2019
).
A survey on handling computationally expensive multiobjective optimization problems with evolutionary algorithms
.
Soft Computing
,
23
(
9
):
3137
3166
.
Coelho
,
R. F.
, and
Bouillard
,
P
. (
2011
).
Multi-objective reliability-based optimization with stochastic metamodels
.
Evolutionary Computation
,
19
(
4
):
525
560
.
Cristianini
,
N.
, and
Shawe-Taylor
,
J
. (
2000
).
An introduction to support vector machines and other kernel-based learning methods
.
Cambridge
:
Cambridge University Press
.
Emmerich
,
M.
,
Giannakoglou
,
K.
, and
Naujoks
,
B
. (
2006
).
Single-and multiobjective evolutionary optimization assisted by Gaussian random field metamodels
.
IEEE Transactions on Evolutionary Computation
,
10
(
4
):
421
439
.
Emmerich
,
M.
,
Giotis
,
A.
,
Özdemir
,
M.
,
Bäck
,
T.
, and
Giannakoglou
,
K
. (
2002
). Metamodel-assisted evolution strategies. In
2002 International Conference on Parallel Problem Solving from Nature
, pp.
361
370
.
Gal
,
Y.
, and
Ghahramani
,
Z
. (
2015
). Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In
2015 International Conference on Machine Learning
, pp.
1050
1059
.
Guo
,
D.
,
Jin
,
Y.
,
Ding
,
J.
, and
Chai
,
T
. (
2018
).
Heterogeneous ensemble-based infill criterion for evolutionary multiobjective optimization of expensive problems
.
IEEE Transactions on Cybernetics
,
49
(
3
):
1012
1025
.
Hart
,
E.
,
Ross
,
P.
, and
Nelson
,
J
. (
1998
).
Solving a real-world problem using an evolving heuristically driven schedule builder
.
Evolutionary Computation
,
6
(
1
):
61
80
.
Hutter
,
F.
,
Kotthoff
,
L.
, and
Vanschoren
,
J
. (
2019
).
Automated machine learning: Methods, systems, challenges
.
Berlin
:
Springer Nature
.
Jin
,
Y
. (
2011
).
Surrogate-assisted evolutionary computation: Recent advances and future challenges
.
Swarm and Evolutionary Computation
,
1
(
2
):
61
70
.
Jin
,
Y.
,
Olhofer
,
M.
, and
Sendhoff
,
B
. (
2000
). On evolutionary optimization with approximate fitness functions. In
2000 IEEE Genetic and Evolutionary Computation Conference (GECCO)
, pp.
786
793
.
Jin
,
Y.
,
Wang
,
H.
,
Chugh
,
T.
,
Guo
,
D.
, and
Miettinen
,
K
. (
2018
).
Data-driven evolutionary optimization: An overview and case studies
.
IEEE Transactions on Evolutionary Computation
,
23
(
3
):
442
458
.
Jones
,
D. R.
,
Schonlau
,
M.
, and
Welch
,
W. J
. (
1998
).
Efficient global optimization of expensive black-box functions
.
Journal of Global Optimization
,
13
(
4
):
455
492
.
Lewis
,
R. M.
,
Torczon
,
V.
, and
Trosset
,
M. W
. (
2000
).
Direct search methods: Then and now
.
Journal of Computational and Applied Mathematics
,
124
(
1-2
):
191
207
.
Li
,
C.
,
Yang
,
S.
, and
Yang
,
M
. (
2014
).
An adaptive multi-swarm optimizer for dynamic optimization problems
.
Evolutionary Computation
,
22
(
4
):
559
594
.
Liu
,
B.
,
Chen
,
Q.
,
Zhang
,
Q.
,
Liang
,
J.
,
Suganthan
,
P. N.
, and
Qu
,
B
. (
2014
). Problem definitions and evaluation criteria for computational expensive optimization. In
2014 IEEE Congress on Evolutionary Computation
, pp.
2081
2088
.
Liu
,
Y.
,
Yao
,
X.
, and
Higuchi
,
T
. (
2000
).
Evolutionary ensembles with negative correlation learning
.
IEEE Transactions on Evolutionary Computation
,
4
(
4
):
380
387
.
MacQueen
,
J
. (
1967
). Some methods for classification and analysis of multivariate observations. In
1967 5th Berkeley Symposium on Mathematical Statistics and Probability
, pp.
281
297
.
Michalewicz
,
Z.
, and
Schoenauer
,
M
. (
1996
).
Evolutionary algorithms for constrained parameter optimization problems
.
Evolutionary Computation
,
4
(
1
):
1
32
.
Pedregosa
,
F.
,
Varoquaux
,
G.
,
Gramfort
,
A.
,
Michel
,
V.
,
Thirion
,
B.
,
Grisel
,
O.
,
Blondel
,
M.
,
Prettenhofer
,
P.
,
Weiss
,
R.
, and
Dubourg
,
V.
(
2011
).
Scikit-learn: Machine learning in Python
.
Journal of Machine Learning Research
,
12
:
2825
2830
.
Preen
,
R. J.
, and
Bull
,
L
. (
2016
).
Design mining interacting wind turbines
.
Evolutionary Computation
,
24
(
1
):
89
111
.
Queipo
,
N. V.
,
Haftka
,
R. T.
,
Shyy
,
W.
,
Goel
,
T.
,
Vaidyanathan
,
R.
, and
Tucker
,
P. K
. (
2005
).
Surrogate-based analysis and optimization
.
Progress in Aerospace Sciences
,
41
(
1
):
1
28
.
Sprecher
,
D. A
. (
1993
).
A universal mapping for Kolmogorov's superposition theorem
.
Neural Networks
,
6
(
8
):
1089
1094
.
Stein
,
M
. (
1987
).
Large sample properties of simulations using Latin hypercube sampling
.
Technometrics
,
29
(
2
):
143
151
.
Tong
,
H.
,
Huang
,
C.
,
Liu
,
J.
, and
Yao
,
X
. (
2019
). Voronoi-based efficient surrogate-assisted evolutionary algorithm for very expensive problems. In
2019 IEEE Congress on Evolutionary Computation
, pp.
1996
2003
.
Torczon
,
V.
, and
Trosset
,
M. W
. (
1998
). Using approximations to accelerate engineering design optimization. In
1998 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization
, p. 4800.
Ulmer
,
H.
,
Streichert
,
F.
, and
Zell
,
A
. (
2003
). Evolution strategies assisted by Gaussian processes with improved preselection criterion. In
2003 IEEE Congress on Evolutionary Computation
, pp.
692
699
.
Wang
,
H.
,
Jin
,
Y.
, and
Doherty
,
J
. (
2017
).
Committee-based active learning for surrogate-assisted particle swarm optimization of expensive problems
.
IEEE Transactions on Cybernetics
,
47
(
9
):
2664
2677
.
Wang
,
H.
,
Jin
,
Y.
, and
Jansen
,
J. O
. (
2016
).
Data-driven surrogate-assisted multiobjective evolutionary optimization of a trauma system
.
IEEE Transactions on Evolutionary Computation
,
20
(
6
):
939
952
.
Wang
,
W.
,
Liu
,
H.-L.
, and
Tan
,
K. C.
(
2022
).
A surrogate-assisted differential evolution algorithm for high-dimensional expensive optimization problems
.
IEEE Transactions on Cybernetics
,
1
13
.
Zhan
,
D.
, and
Xing
,
H
. (
2021
).
A fast Kriging-assisted evolutionary algorithm based on incremental learning
.
IEEE Transactions on Evolutionary Computation
,
25
(
5
):
941
955
.
Zhang
,
F.
,
Mei
,
Y.
,
Nguyen
,
S.
,
Zhang
,
M.
, and
Tan
,
K. C
. (
2021
).
Surrogate-assisted evolutionary multitask genetic programming for dynamic flexible job shop scheduling
.
IEEE Transactions on Evolutionary Computation
,
25
(
4
):
651
665
.
Zhang
,
M.
,
Li
,
H.
,
Pan
,
S.
,
Lyu
,
J.
,
Ling
,
S.
, and
Su
,
S
. (
2021
).
Convolutional neural networks based lung nodule classification: A surrogate-assisted evolutionary algorithm for hyperparameter optimization
.
IEEE Transactions on Evolutionary Computation
,
25
(
5
):
869
882
.