Abstract
In recent years, surrogate models derived from genuine data samples have proven to be efficient in addressing optimization challenges that are costly or time-intensive. However, the individuals in the population become indistinguishable as the curse of dimensionality increases in the objective space and the accumulation of surrogate approximated errors. Therefore, in this paper, each objective function is modeled using a radial basis function approach, and the optimal solution set of the surrogate model is located by the multi-objective evolutionary algorithm of strengthened dominance relation. The original objective function values of the true evaluations are converted to two indicator values, and then the surrogate models are set up for the two performance indicators. Finally, an adaptive infill sampling strategy that relies on approximate performance indicators is proposed to assist in selecting individuals for real evaluations from the potential optimal solution set. The algorithm is contrasted against several advanced surrogate-assisted evolutionary algorithms on two suites of test cases, and the experimental findings prove that the approach is competitive in solving expensive many-objective optimization problems.
0 Introduction
Multi-objective optimization problems (MOPs) , such as the design of a car's cab[1] and pressure vessel design[2], need to optimize two or more conflicting objectives concurrently. The mathematical formula of MOPs is as follows:
(1)
where fi (x) (i=1, 2, ···, M) represents the ith objective of the decision vector x.When the objective number exceeds 3, they are also described as many-objective optimization problems (MaOPs) . Researchers have developed a plethora of evolutionary algorithms aimed at identifying the best solutions of MOPs, which are made up of three types. The first category includes dominance-based multi-objective evolutionary algorithms (MOEAs) [3-4]. The second category features indicator-based MOEAs[5], which adopt an indicator, such as HV[6] or IGD [7], to evaluate the quality of the population. The third type of MOEAs is decomposition-based MOEAs, which classify the population into several subgroups[8] or translate the MOPs into multiple single-objective problems[9]. Multi-objective optimization algorithms aim to find a group of convergent and uniformly distributed optimal solutions referred to as the Pareto Set (PS) within the decision-making space and the Pareto Front (PF) within the space of objectives.
The methods above attained good results on common MOPs. However, they become vestigial on expensive optimization problems, in which a single evaluation of the function can be quite costly[10]. MOEAs involve massive function evaluations (FEs) to approach the optimal set, while there are limited computational resources in the expensive optimization problems during the optimization process. Surrogates are effective tools for solving expensive optimization problems[11]. Machine learning technologies, such as radial basis function network (RBFN) [12], polynomial regression (PR) [13], and the Gaussian process (GP) , often referred to as Kriging[14], are commonly used as surrogates. Different surrogates have different advantages[15-16]. For example, Kriging is a probabilistic model that predicts objective values based on the covariance function between the individuals and the true evaluated samples. Moreover, this approach offers an estimated outcome along with an estimate of the potential variability associated with the objective. Kriging performs well when the decision variable dimension is low, but the high complexity of hyper-parameters makes it difficult to build a Kriging surrogate when the decision variable dimension is greater than 10[17]. RBFN performs well for higher-order nonlinear problems, but it cannot provide approximate uncertainty[18]. In Ref.[19], Zhang et al. proposed using the heterogeneous database to train objective surrogates and constraint surrogates for solving expensive constrained multi-modal problems. In Ref.[20], Li et al. proposed combining surrogate-assisted evolution optimization and non-model evolutionary optimization to tackle complex expensive optimization problems. Among them, the global surrogate model established by all evaluated samples, the local model trained by a subset of the best solutions, and the local model trained by some of the newest samples are used for forecasting the fitness of individuals. In Ref.[21], Li et al. used the explanation model to assess the critical role of decision variables and integrated the information into the offspring reproduction. In Ref.[22], Pan et al. used the Kriging model as a surrogate model and proposed an angle penalty distance criterion to filter optimal solutions. More about the methods of surrogate-assisted evolutionary optimization algorithms (SAEAs) can be found in the overview article of Ref.[23].
In the SAEAs, surrogates are designated to replace true function evaluations in three ways. The first way is the regression model-based SAEAs[24-30], which train surrogate models to predict objective functions directly. In Ref.[31], Zhen et al. proposed utilizing a blend of extensive and targeted search tactics to obtain the optimal solution. Furthermore, this methodology leverages a Radial Basis Function (RBF) model to predicte the objective function of the candidates in the two stages. In Ref.[32], Wang et al. employed the RBF model and the Kriging model for the approximation of objectives in the global and local search processes, respectively. In Ref.[33], Song et al. built GP models tailored for individual objectives in the expensive MOPs by properly using influential points in training samples and adopted the two archive2 method[34] to optimize the population. In Ref.[35], Liu et al. used GP models to estimate each objective, and then used the true evaluated non-dominated solutions to adaptively reset the reference vector for uneven PF. In Ref.[36], Liu et al. selected Kriging models to estimate each objective and adopted two groups of reference vectors for convergence and diversity searching, respectively. In Ref.[37], Li et al. designed an RBFN model for each objective and used a local search strategy grounded in a PF model to exploit the sparse region. The approximate errors will accumulate with the increase of objectives, which increases the difficulties in selecting the authentic Pareto-optimal solutions throughout the course of evolution. The second way is to approximate the indicator function translated from the MOPs, using a surrogate [38]. In Ref.[39], Liu et al. trained the RBFN model to predict the hypervolume indicator (HVI) associated with each solution to assist the environmental selection. The third way is the classification model-based SAEAs, which use surrogates to classify[30, 40-45] or predict dominant relationships in the population[46]. In Ref.[44], Wei et al. trained the gradient boosting classifier (GBC) for determining the sorting layer of the individuals in the tiered learning framework of the swarm optimizer (LLSO) and enhanced the robustness of SAEAs. In Ref.[47], Pan et al. used RSEA[48] to construct the dominated boundary and selected a feedforward architecture within neural networks (FANNs) to evaluate the precedence among solutions. With the amplification of objectives, the absence of the selection pressure imposed by dominance relations will lead to poor classification performance. Then in Ref.[46], Yuan and Banzhat presented using deep learning techniques to estimate the Pareto and θ-dominance interactions between solutions. In Ref.[49], Zhang et al. picked two fuzzy classifiers to screen the individuals. One classifier was designed to ascertain the dominance relation, and the other was designed to measure the crowdedness of solutions. For the second and third surrogate-assisted optimization methods, the approximate errors will avoid cumulation with the increase of objectives, but the less output of information in the classification model or indicator model makes it difficult to discriminate the individuals in MOPs.
In the SAEAs, the majority of individuals are evaluated by the surrogates, while only a few individuals perform the expensive function evaluations in surrogate-assisted optimization methods. The criterion that picks individuals for real function evaluations is the infill sample strategy. Infill sample strategy is crucial for updating the surrogates and searching for the optimal solutions. There are two classes of individuals considered valid infill samples, one is the individuals with the lowest estimated fitness, while the other consists of those exhibiting the largest uncertainty of the approximated fitness[50]. The individuals with the lowest estimated fitness values will be selected for true function evaluations, as they can help exploit the promising region and accelerate the algorithm convergence[51]. Selecting individuals with the largest uncertainty for real evaluations can augment the prediction precision of the surrogates and facilitate the exploration of sparse regions[17]. It has been shown that the infill strategy, which only considers individuals with approximate optimum, may give rise to premature convergence or fall into a local optimum[50]. The infill strategy, which only considers the prediction uncertainty, will consume too many real function evaluations, leading to a slow convergence rate of the algorithm. Therefore, the above two criteria are unsuitable for use alone. The current infill strategies leveraging surrogate models in multi-objective optimization can be sorted into two main types. The first[52-53] of the infill sample strategy is extended by the expected improvement (EI) infill strategy[54] of the expensive single-objective optimization algorithms, which combine the approximate values with the uncertainty as an indicator. In Ref.[53], Zhang et al. used GP models to predict each objective function, and these approximate objective functions were used to derive the predictive distribution model for constructing the metric of the infill strategy. In Ref.[55], Wang et al. also selected the GP models to predict the objectives and then used the approximated objective functions to calculate the adaptive acquisition function to pick individuals for the true function evaluations. The second type of the infill strategy uses approximate values and uncertainty as two separate indicators. For the second category of the infill strategy, approximate value and uncertainty are adaptively considered for sampling according to the requirement on convergence or diversity of the population[25, 29, 56]. In Ref.[26], GP models were used to predict the objectives, and the convergence calculated by the forecasted objective values or the uncertainty of GP models is adaptively preferred for the infill strategy according to the crowding degree of the radial space. In Re.[33], each dimension of objectives is predicted by the pivotal point-insensitive surrogate model, and the convergence, diversity, or uncertainty is adaptively considered for the infill strategy according to the predominant demand of convergence, diversity, or uncertainty.
1 Motivation and Contributions
Although the above methods have revealed validity in tackling expensive multi-objective optimization problems (EMOPs) , many challenges remain. For the first category of SAEAs, the mistakes from approximations grow more significantly as the objective space expands in dimensionality. The mentioned methods of the infill strategies in SA-MOEAs use the performance indicators, which are calculated by the approximated objectives of surrogate models. When the objectives increase, the approximated errors will accumulate. And for the second and third categories of SAEAs, it is difficult to coordinate the convergence and diversity of high-dimensional space with a single performance indicator or classification method. Fig.1 provides an example to illustrate how the approximated errors induce the performance degradation of the SA-MOEAs. In Fig.1 (a) , the objective functions of A, B, and C are estimated by the surrogate models, and in Fig.1 (b) , the objective functions of A, B, and C are assessed through their actual objective functions. In Fig.1 (a) , suppose we need to choose one individual for the true function evaluations, and we utilize the performance indicator that quantifies the Euclidean distance from the candidates to the origin to assess the convergence quality of the individuals. The Euclidean distance of the individuals A, B, and C to the origin is evaluated by the approximated objective functions with 5.10, 2.23, and 4.21, respectively. Considering selecting the individual with the smallest convergence value for the true function evaluations, individual B with the least Euclidean distance of 2.23 to the origin will be picked. However, the true Euclidean distances of the individuals A, B, and C in Fig.1 (b) are4.71, 5.02, and 4.20, where the Euclidean distance of individual C to the origin is the minimum. In this way, the method of calculating the performance indicators with the approximated objective function values may select individuals with worse convergence performance because of the accumulation of approximate errors. To further verify the hypothesis in Fig.1, we conducted experimental on the DTLZ1 to DTLZ4 test problems with 10 objectives. The results are shown in Fig.2. Within Fig.2, we calculate the convergence of Fig.1 employing the fitness values derived from the surrogate model and the actual function evaluation, respectively, and then select individuals for the true evaluation based on these two results. The sum of individuals for each test problem is 20. In Fig.2, the blue bar chart represents the number of individuals whose selection results from the surrogate model and true function evaluations under the same selection criterion are identical, while the orange bar chart represents the number of individuals whose selection results are inconsistent. Fig.2 confirms that the assumption in Fig.1. Thus, to mitigate the negative effect of approximated error accumulation and coordinate convergence and diversity in high-dimensional objective space, this work designs an adaptive infill strategy combining the first and the second way of training surrogate models and proposes the two performance indicators assisted infill strategy for expensive many-objective optimization (TPAEMO) .A synthesis of the key contributions of this paper is provided in the subsequent section.
Fig.1An example demonstrating the negative effects that the approximated errors make on the algorithm
1) The surrogate models are established in two ways. The first way is to create a surrogate model for each individual objective, which aids in population optimization by coordinating the convergence and diversity within the complex, high-dimensional objective space. The second way is to train the surrogate model for two performance indicators, which is used in the infill criterion, and reduces the negative impact of error accumulation.
2) An adaptive infill strategy is designed in which the candidates with good convergence or diversity are picked out for expensive function evaluations based on the level of convergence observed within the group. In this way, it is effective to enhance trade-offs between diversity and convergence.
The subsequent sections of this paper are projected as follows. Section 2 gives concise descriptions of the NSGA-II/SDR algorithm and RBF networks. Section 3 specifically introduces the proposed TPAEMO algorithm. Section 4 displays the outcomes of experiments and parameter analysis. The conclusions and prospect of subsequent research endeavors are given in Section 5.
Fig.2Selection results based on real function evaluation and predicting of surrogate model under the same convergence criterion
2 Preliminaries
2.1 NSGA-II/SDR
NSGA-II[3] is a representative of dominance-based MOEAs that uses the dominance relation to evaluate and compare solutions in multi-objective optimization tasks. When dealing with a higher number of objectives, NSGA-II becomes invalid due to insufficient selection pressure. In Ref.[4], Tian et al. introduced an enhanced version of the dominance relationship known as SDR to bolster the selection pressure of the dominance-based MOEAs for many-objective problems. The standard definition of this enhanced dominance relationship is provided further in the text.
Definition 1 (Strengthened dominance relation) . An individual x is said to have an enhanced dominance over another individual y (denoted as ) if the following conditions are met:
(2)
where represents the angular separation between the individual x and y in the objective space, θ symbolizes a threshold parameter set to the -th minimum element of
(3)
NSGA-II/SDR embeds the SDR in NSGA-II, in which the conventional Pareto dominance relationship is substituted by the SDR. And the NSGA-II/SDR maintains the same overall structure as the standard NSGA-II.
2.2 Radial Basis Function Network
Radial basis function network (RBFN) [15] is a kind of artificial neural network, and it has three fully connected levels, which contain an input level, a hidden level, and an output level. Given N samples of D decision variables xj (j=1, 2，···, N) and their objective values yj (j=1，2，···，N) , the objectives for a novel instance x can be predicted using the following formula:
(4)
where φ ( x, ci) represents the kernel function used in the hidden level, and ci is the centroid of the i-th neuron, which is determined through the K-means clustering method. In this paper, we choose the Gaussian kernel as the activation function, which is defined below:
(5)
The parameters βi and η in Eq. (4) need to satisfy the following equation:
(6)
which can also be expressed in a matrix form below:
(7)
Eq. (7) can alternatively be written as φ·B = Y, and BT=[β1, ···, βn, η] can be derived using the method of least squares:
(8)
After obtaining B , RBFN makes a prediction as Eq. (4) .
3 Proposed Algorithm
3.1 Overall Framework
RBFN with lower complexity in hyper-parameter optimization can better fit the global depiction of the fitness landscape and exhibit better performance for complex problems with high-dimensional and high-order nonlinear aspects[15]. Thus, in this paper, we adopt RBFN models as surrogate models for each objective function and the two performance indicators. Algorithm 1 indicates the pseudocode of the proposed TPAEMO algorithm, and the flowchart is displayed in Fig.3. The NSGA-II/SDR severs as the basic optimization framework in TPAEMO. In the main loop, the Latin hypercube sampling (LHS) method is used to create the initial population containing11D-1 individuals. All the individuals in the initial population are calculated by the true objective functions and kept in the archive Arc. Then N individuals with the smallest convergence and biggest diversity are selected from Arc by the environmental selection in NSGA-II/SDR as the initial population. After that, the N individuals are optimized for several generations. At last, the optimized population is used for the proposed infill strategy, several individuals are recalculated by the true expensive objective functions, and then they are added to the database Arc. This process will be executed repeatedly if the true function evaluations are not up to the maximum number, and the converged non-dominated optimal solution set will be the output.
Fig.3Flowchart of TPAEMO
3.2 Infill Strategy
In SAEAs, general individuals are predicted by the surrogate models, and the approximated errors will accumulate with the increase of objectives. To coordinate the convergence and diversity of expensive MaOPs and make the best use of the limited computing resources, an adaptive infill sample strategy is put forward in this article. In the phase of the infill sample strategy, τ individuals will be selected from the optimized population P′ for the expensive function evaluations in the TPAEMO. Algorithm 2 displays the pseudocode of the infill sample strategy put forward in TPAEMO. First, all individuals already evaluated by the expensive functions in the population P′ will be identified and deleted. Second, the remaining individuals in P′ will be divided into τ clusters using k-means clustering, according to their approximated objective functions by RBFobj. Then, the objective function values of the true samples in Arc will be translated to the performance indicators of the convergence and diversity, which are defined as Eq. (9) and Eq. (10) :
(9)
(10)
where expresses the Euclidean distance between x and p in the objective space, and dj ( x) denotes the Euclidean distance between x and the j-th closest data point to x in Arc in the objective space. The true samples and their performance indicators are added to archive Arc1 and the data in Arc1 will be used to train two RBFindicator models. After that, the RBFindicator models will be adopted to assess the convergence and diversity performance indicators of the population P′. Finally, one individual with either the minimal convergence performance or the maximum diversity performance, as approximated by the RBFindicator models, will be selected from each cluster according to the convergence degree of the population P′. Here, a straightforward method is used to estimate the convergence degree of the population. The non-dominated ranking is used to sort the population P′ into several non-dominated levels. If there is more than one non-dominated level, which implies that the population shows poor convergence, the individual with the minimal convergence value Con will be selected from each cluster. Otherwise, the individual with the maximum diversity value Div will be chosen from each cluster.
4 Experimental Studies
In this section, the performance metric of experiment comparison and the parameter settings of TPAEMO and other comparison algorithms are presented first. Then, experiments with different values of τ in TPAEMO are conducted to examine the effects of different τ values. After that, the effects of approximate performance indicators are investigated. At last, we contrast TPAEMO with the advanced SA-MOEAs algorithms, including K-RVEA, CSEA, and KTA2. We also compare TPAEMO with NSGA-II/SDR without using the surrogates. All these algorithms are operated on the PlatEMO[57] to obtain the results. The Wilcoxon rank-sum test at a significance of 0.05 is adopted to compare the results of TPAEMO and other algorithms. Symbols “+”, “-” and “≈” represent that TPAEMO has achieved better, worse, and similar results with the compared algorithms, respectively. We execute experiments on the test suites of DTLZ[58] and MaF[59] to examine the optimization capability of TPAEMO.
4.1 Performance Metrics
In this paper, we use the performance metric of Inverted Generational Distance (IGD) [60] to compare the optimization capability of different algorithms. IGD uses a group of uniformly distributed reference points sampled along the true Pareto front (PF) to assess the convergence and diversity of the algorithm. An algorithm performs better on convergence and diversity when it has a lower Inverted Generational Distance (IGD) value. The formula of IGD is shown below:
(11)
where P* represents the set of reference points sampled from the true Pareto front, Ω represents the approximate Pareto optimal set achieved by running the algorithm, and d (r, Ω) represents the smallest Euclidean distance from the point r on the true PF to the individual in Ω .
4.2 Comparison Algorithms and Parameter Configuration
4.2.1 Comparison Algorithms
To test the viability of the proposed algorithm, we contrast TPAEMO against CSEA, K-RVEA, KTA2, and NSGA-II/SDR. A concise description of each comparison algorithm is given below.
(1) CSEA[47]: The method leverages feed-forward neural networks (FNNs) to develop the classification criterion, which divides true samples into non-dominated and dominated categories to determine the dominant relationship between newly generated solutions and reference solutions.
(2) K-RVEA[25]: It adopts the Kriging model to estimate each objective function. Then, in managing the Kriging models, uncertainty information brought by the Kriging models, the distribution information of the reference vectors, and the position of the individuals are considered for selecting individuals to reevaluate by an expensive function.
(3) KTA2[33]: This method constructs one influential point-insensitive model to predict each objective function and considers convergence, diversity, and model accuracy in selecton new individuals for reevaluation according to the foremost demand for convergence, diversity, or uncertainty of the surrogate model.
(4) NSGA-II/SDR[4]: The method is advanced for tackling many-objective optimization problems without surrogate models, and an adaptive niching technique of strengthened dominance relation (SDR) is advanced in view of the angle niches between the individuals, where only the best converged individual is installed at the first non-dominated level in each niche.
4.2.2 Parameter Configuration
Parameter configuration of TPAEMO and the compared algorithms is given below:
(1) In our experiments, reproduction operators containing simulated binary crossover and polynomial mutation are employed in all the comparison algorithms to create the offspring solutions. The parameters of reproduction operators are configured as pc=1.0, pm=1/D, ηc=20, ηm=20.
(2) The maximum expensive function evaluations of 300 times serve as the ending condition for all comparison algorithms in this paper.
(3) The population scale of TPAEMO is configured as 50. The initial population scale of the compared algorithms is configured as 50.
(4) The related parameters of the compared algorithms are configured as suggested in Refs.[4, 25, 33, 47].
4.3 Sensitivity Analysis of τ
The parameter τ of the number to pick individuals for actual function evaluations may affect the optimization capability of TPAEMO, so we set different values of 1, 3, 5, 8, and 10 for parameter τ to conduct experiments. The mean IGD results of different parameter τ over 20 independent executions on DTLZ1 and DTLZ6 with 10 objectives are shown in Fig4. From Fig.4, it can be observed that when τ=3, the mean IGD values of DTLZ1 and DTLZ6 with 10 objectives are minimal. Thus, we set τ=3.
4.4 Effect of Approximate Performance Indicators
To study the effect of approximate performance indicators predicted by the surrogate models directly, we compare TPAEMO with another variant, TPAEMO-I, where candidate performance indicator values are calculated form approximate objectives rather than being directly predicted by the surrogate model. The average IGD results of TPAEMO and TPAEMO-I according to 20 independent executions on DTLZ1-DTLZ7 problems with 3, 4, 6, 8, and 10 objectives are shown in Table1. The best median result in each row is marked with a gray-colored backdrop.TPAEMO obtained 11 better results and 7 worse results than TPAEMO-I, and TPAEMO showed better performance on all objectives of DTLZ5, and high-dimensional objectives of DTLZ2. The results reflect that our TPAEMO shows better performance than TPAEMO-I.
Fig.4Mean IGD values attained by different values of τ
Table1IGD statistical results attained by TPAEMO-I and TPAEMO for DTLZ1-DTLZ7 with 3, 4, 6, 8, and 10 objectives
4.5 Effect of the Infill Strategy
To discuss the effect of the infill strategy, we compare TPAEMO with another variant of the algorithm, named TPAEMO-II, in which only the performance indicator of convergence predicted by the surrogate model is employed to choose individuals for expensive function evaluations. The average IGD results of TPAEMO-II and TPAEMO based on 20 independent executions of DTLZ1-DTLZ7 test problems with 3, 4, 6, 8, and 10 objectives are shown in Table2 TPAEMO obtained 9/35 better results than TPAEMO-II and 0/35 worse results than TPAEMO-II, which indicates that the infill strategy used in TPAEMO is more effective than that in TPAEMO-II. TPAEMO performed better than TPAEMO mainly on DTLZ2 which showed that the performance indicator of diversity used in TPAEMO is effective.
Table2IGD statistical results attained by TPAEMO-II and TPAEMO for DTLZ1-DTLZ7 with 3, 4, 6, 8, and 10 objectives
4.6 Comparison to Other Algorithms
We test the optimization capability of TPAEMO on DTLZ1-DTLZ7 problems with 3, 4, 6, 8, and 10 objectives and MaF1-MaF7 problems with 3, 5, and 10 objectives. Regarding all of the test cases, the dimension of the decision variable is 10.
1) Results on DTLZ benchmarks: the IGD scores attained by the five compared algorithms across 20 separate executions on DTLZ1 through DTLZ7 are collected in Table3, where the best median result in each row is marked with a gray-colored backdrop. From Table3, it can be seen that TPAEMO obtained 25 better results and 2 worse results than NSGA-II/SDR within a group of 35 test examples, which indicates that TPAEMO is more efficient than NSGA-II/SDR, which has no surrogate model to participate in tackling the computationally expensive many-objective test problems. Table3 reveals the following results that TPAEMO obtained 15 best median results within the35 test instances, KTA2 ranked second with the10 most successful results, CSEA ranked third with 5 superior results, and K-RVEA ranked fourth with 5 finest results.
Table3IGD statistical results attained by CSEA, K-RVEA, KTA2, NSGA-II/SDR, and TPAEMO for DTLZ1-DTLZ7 with 3, 4, 6, 8, and 10 objectives
TPAEMO performed the best among other comparison algorithms in the context of DTLZ2, DTLZ4, DTLZ5, and DTLZ6, and performed worse than other comparison algorithms on DTLZ1, DTLZ3, and DTLZ7. DTLZ1 and DTLZ3 contain a lot of locally non-dominated solutions, making them difficult to converge. KTA2 performed best on DTLZ1 and DTLZ3, and CSEA ranked second. The call for convergence frequently emerges in the optimization of local Pareto optimal problems, so KTA2 frequently selects well-converged solutions in the sampling strategy. CSEA takes convergence into consideration, so CSEA selected potentially better-converged solutions for expensive evaluations. K-RVEA used a group of well-distributed reference vectors to navigate the search, so K-RVEA obtained a set of uniformly distributed solutions. TPAEMO tries to coordinate convergence and diversity, and when facing multi-local optimal solutions, the diversity of infill strategy is frequently used.
For DTLZ2, TPAEMO performed the best, followed by KTA2. This might be due to the infill sampling strategies used in TPAEMO and KTA2, which try to support the co-development of convergence and diversity. The parallel axis plot depicting non-dominated solutions in the run with the best IGD values for K-RVEA, CSEA, KTA2 NSGA-II/SDR, and TPAEMO for 8-objective DTLZ2 is displayed in Fig.5. We can discover from Fig.5 that the upper bound of objectives for the non-dominated solution set acquired by TPAEMO exhibits lower values than the compared algorithms, and the non-dominated solutions of TPAEMO possess a more intense distribution than NSGA-II/SDR. This means that TPAEMO exhibits superior results regarding convergence and diversity compared to the algorithms K-RVEA, CSEA, KTA2 and NSGA-II/SDR, and TPAEMO can coordinate convergence and diversity commendably. Since DTLZ4 is a biased multi-objective optimization problem, it is difficult for the algorithms to ensure diversity. As indicated in Table3, for DTLZ4, TPAEMO performed best. To conduct an in-depth analysis of different algorithms' performance on DTLZ4, the non-dominated solutions of the run with the best IGD values for CSEA, NSGA-II/SDR, K-RVEA, TPAEMO, and KTA2 for DTLZ4 are displayed in Fig.6. From Fig.6, we can observe that the Pareto-front solutions generated by TPAEMO are the closest to the true PF of DTLZ4, followed by KTA2. And the diversity of TPAEMO is better than that of KTA2 on DTLZ4. Although the diversity of K-RVEA on DTLZ4 is the best, owing to the uniformly distributed reference vectors used in K-RVEA, the Pareto-front solutions generated by K-RVEA have a farther distance to the true PF than TPAEMO and KTA2. For NSGA-II/SDR, it maintained the worst diversity.
DTLZ5 and DTLZ6 are degenerate test problems, and TPAEMO performed best on these two kinds of problems. The majority of reference vectors used in K-RVEA are empty on the degenerate problems, which makes it not easy to approach the true Pareto front fast. For DTLZ7 with a disconnected Pareto front, K-RVEA performed best, followed by TPAEMO. Although TPAEMO does not perform the best on DTLZ1, DTLZ3, and DTLZ7, it still exhibits a comparative advantage in contrast with the other four algorithms.
Fig.5The parallel coordinates graph for the Pareto-optimal solutions attained by K-RVEA, CSEA, KTA2, NSGA-II/SDR, and TPAEMO in the executions with the best IGD value on 8-objective DTLZ2
2) Results on MaF benchmarks: additionally, we compare the performance of TPAEMO with other algorithms on MaF test problems, and the results of IGD values attained by the four algorithms under comparison over 20 independent executions on MaF1 to MaF7 are collected in Table4, where the best median result in each row is marked with a gray background. Among these problems[59], MaF1 with an inverted PF is translated from DTLZ1, and MaF2 is the transformed version of DTLZ2. MaF3 and MaF4 are translated from DTLZ3. MaF5, MaF6, and MaF7 are translated from DTLZ4, DTLZ5 and DTLZ7 respectively.
Fig.6Pareto-optimal solutions attained with CSEA, NSGA-II/SDR, K-RVEA, TPAEMO, and KTA2 of the executions with the best IGD value for three-objective DTLZ4
From Table4, we can discover that TPAEMO attained 13 superior outcomes than CSEA and 10 superior outcomes than K-RVEA among21 test problems. Compared with NSGA-II/SDR, TPAEMO obtained 16 better results and 0 worse results, which indicates our method is more valid than the compared algorithms.
Table4IGD statistical results attained by CSEA, K-RVEA, NSGA-II/SDR, and TPAEMO for MaF1-MaF7 with 3, 5, and 10 objectives
5 Conclusions
In this work, a new surrogate-assisted many-objective evolutionary optimization algorithm is proposed to tacke expensive many-objective optimization problems.The surrogate models are constructed to estimate the objective functions and performance indicators of convergence and diversity. In the surrogate-assisted optimization phase, the objective functions approximated by the surrogate models are used for searching globally optimal areas. Then, to prevent the accumulation of approximated errors, the approximated objectives are aligned with approximated performance indicators of convergence and diversity during the selection phase for true objective function evaluations. In addition, the approximated performance indicators of convergence and diversity are adaptively considered in the infill strategy according to the distribution of the population.
To examine the effectiveness of TPAEMO, two sets of test problems are adopted to contrast the proposed method with four algorithms, including CSEA, K-RVEA, KTA2, and NSGA-II/SDR. The results of the experiments confirm that TPAEMO performs relatively better and converges more rapidly than the three most advanced surrogate-assisted multi-objective optimization algorithms and one MOEA algorithm without surrogate models.
Although TPAEMO has shown good performance in dealing with EMOPs, its performance in tackling disconnected expensive many-objective optimization problems must still be improved. In future work, we may explore growing a neural gas network (iGNG) [35] or distributing information of the PFs[61] in the decision and objective space for model management to further improve the performance of handling EMOP problems.
