Dynamic multiobjective optimization is a significant challenge in accurately capturing changes in Pareto optimal sets (PS), encompassing both location and manifold changes. Existing approaches primarily focus on tracking changes in the location of the PS, often overlooking the potential impact of changes in the PS manifold, which can be decomposed into rotation and distortion changes. Such oversights can lead to a reduction in the overall performance of an algorithm. To address this issue, a prediction method based on joint subspace and correlation alignment (PSCA) is proposed. PSCA leverages a subspace alignment strategy to effectively capture rotation change in the PS manifold while employing a correlation alignment strategy to capture distortion change. By integrating these two strategies, a quasi-initial population is generated that embodies the captured rotation and distortion change patterns in a new environment. Then, the promising individuals are selected from this quasi-initial population based on their nondominated relations and crowding degree to form the initial population in the new environment. To evaluate the effectiveness of PSCA, we conduct experiments on fourteen benchmark problems. The experimental results demonstrate that PSCA achieves significant improvements over several state-of-the-art algorithms.
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Introduction
In the field of engineering and science, dynamic multiobjective optimization problems (DMOPs) are widely encountered. These problems exhibit various uncertainties and dynamics, such as multiple autonomous underwater vehicle (UAV) path planning in a dynamic environment [1], resilient scheduling problems in steelmaking plants [2], and multimedia data analysis [3]. Unlike static multiobjective optimization problems (MOPs), DMOPs have objectives and constraints that are not only dependent on decision variables but also change over time. Consequently, the Pareto sets (PSs) and Pareto fronts (PFs) in DMOPs may also undergo changes. Efficiently tracking the moving PSs and PFs becomes crucial in effectively solving DMOPs.
Over the past few decades, several methods have been proposed for dealing with complex DMOPs [4‐14]. Among these methods, prediction-based approaches have gained considerable attention and have become the focus of extensive research. The main idea of prediction-based approaches is to develop prediction models that utilize the information from Pareto solution sets in old environments to accurately forecast the behavior of Pareto solution sets in new and dynamically changing environments. By leveraging the knowledge gained from past experiences, prediction-based approaches offer a proactive strategy for dealing with uncertainties and dynamics in DMOPs.
Fig. 1
Decomposition of change for PS
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Anzeige
In general, prediction-based approaches can be broadly categorized into two types [15]: those based on linear models [13, 16‐20] and those based on nonlinear models [11, 21‐25]. The prediction approach based on linear models assumes a linear correlation between the previous optimal solutions and the next one. Notable examples of these approaches include the feed-forward prediction strategy (FPS) [16], the population-based prediction strategy (PPS) [13], the Kalman filter-based prediction method (KF) [18], and the multidirection prediction method (MDP) [19]. These methods leverage the linearity assumption to make accurate predictions in certain DMOP scenarios. The prediction approaches based on nonlinear models recognize the presence of a nonlinear correlation between historical and predicted optimal solutions. Machine learning-based methods have proven particularly useful within this category. The transfer learning-based prediction method (TL) and support vector regression-based prediction method (SVR) are two prominent examples of techniques that leverage machine learning to capture the nonlinear relationships within DMOPs. When faced with significant environmental changes, relying on a single prediction approach may not be sufficient to address the complexity of DMOPs. To overcome this limitation, multistrategy prediction methods have been introduced [9, 26‐31]. These methods involve combining multiple strategies to enhance the accuracy and robustness of prediction. By leveraging the strength of different strategies, multistrategy prediction methods offer a more comprehensive and flexible solution for addressing complex DMOPs.
Fig. 2
Approximate rotation change by subspace alignment
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While the aforementioned prediction-based approaches have demonstrated promising performance in addressing specific DMOPs, there is still substantial room for improvement. First, many existing methods primarily focus on the change in the location of the PS, neglecting the change in the PS manifold itself, thereby potentially compromising the accuracy of the prediction model. Second, in general, the manifold change can be further decomposed into two fundamental components: rotation change and distortion change (see Fig. 1). However, there is limited literature on considering the rotation change in PS manifolds. Thus far, only Rong et al. [32] and Wang et al. [22] have attempted to consider rotation changes in PS manifolds. In [32], Rong et al. divided the manifold into multiple submanifolds and then directly approximated the rotation change of each submanifold using the translation change of the corresponding submanifold, further approximating the change of the whole PS manifold using the location change of all sub-PSs. This may cause a large prediction error when the rotation change angle of the PS manifold is large. In [22], the rotation of only two principal dimensions in the PS manifold is considered. This may cause a large prediction error when dealing with high-dimensional manifolds. Third, when one manifold undergoes distortion change, the change in the correlation between individuals in the manifold is the key to the distortion change. However, most existing methods pay little attention to this issue.
To address these concerns, a prediction method based on joint subspace and correlation alignment (PSCA) is proposed. As subspace alignment can approximate the rotation change between two high-dimensional data manifolds, we use it to approximate the rotation change between two PS manifolds in two adjacent environments. Specifically, we project a manifold and its rotated version onto their respective subspaces obtained by local principal component analysis (LPCA) [33]. By aligning all dimensions of their subspaces, we can approximate the alignment of these two manifolds, even manifolds with higher dimensions (see Fig. 2). This means that the mapping function aligning the subspace of a manifold with that of its rotated version can approximate the mapping function aligning the two manifolds. Consequently, the mapping function representing the rotation change between two manifolds, even manifolds with higher dimensions, can be approximated by the mapping function aligning all dimensions of their subspaces. In addition, the covariance is adept at describing a correlation between points in a manifold. Therefore, here, we align the correlation of individuals in one manifold with that of its distorted version by aligning the covariances (named correlation alignment [34]) between these two manifolds, further aligning these two manifolds. Following this approach, the rotation and distortion change patterns of the PS manifold in DMOPs are captured. This enables us to more accurately predict PS in a new environment. The main contributions are outlined as follows:
1.
Different from most existing methods that are only suitable for capturing a small rotation change of the PS manifold or rotation change of a low-dimensional PS manifold, the utilization of a subspace alignment approach enables the algorithm to capture a large rotation change of a high-dimensional PS manifold in a new environment.
2.
By employing statistical characteristic value covariance alignment, we capture the change in correlation between individuals in the manifold, enabling a better approximation of the distortion change in the PS manifold under the new environment.
3.
The joint application of subspace and correlation alignment techniques not only enhances the convergence but also improves the diversity, further enhancing the ability of the algorithm to accurately track changes in the PS manifold.
The subsequent sections of this paper are structured as follows: “Preliminaries and related work” provides the definition of DMOPs, accompanied by an overview of relevant prior research. In “Proposed method”, we introduce the principles of subspace alignment and statics alignment, followed by a detailed description of the proposed PCSA algorithm. “Experimental design” lists test problems, the compared algorithms, the experiment settings and performance metrics. “Discussion on experimental results” presents and discusses the results of the conducted experiment, providing a thorough analysis of the obtained data. “Conclusion” concludes the paper, summarizing our key contributions and presenting suggestions for future research.
Preliminaries and related work
Definition of DMOPs
Reference [35, 36] highlights the existence of numerous DMOPs, each characterized by its distinct attributes. In this paper, only the following specific type of DMOPs is focused:
where \(F(x,\alpha \left( t\right) ) \) represents the objective vector function, \(\alpha \left( t\right) \) denotes the environmental parameters associated with time t, m is the number of objective functions, \( R^{D} \) represents feasible region and D is dimension of this feasible region.
Defintion 1 (Pareto dominance in DMOPs) Let \( x_1,x_2 \) are two variable vectors in decision-space at time t, then \( x_1 \) is said to Pareto dominate \( x_2 \), which can be denoted by \(x _{1}\succ x_{2} \), if and only if
then the set consisting of all \(x_{*}^{t} \) meeting (2) in decision space \( \Omega \) is known as the Pareto optimal solution set, which is abbreviated to \( \textrm{PS}_{t} \) at time t.
Defintion 3 (Dynamic Pareto optimal front\( PF_{t} \)in DMOPs) The corresponding points for \( {\text {PS}}_{t} \) in the objective function space are called the Pareto optimal front set at time t.
Types of manifold changes
Defintion 1 (Rotation) A rotation change is said to exist between manifold B and C if B can be made more similar to C by a rotation.
Defintion 2 (Distortion) If a manifold cannot change into another by translation and rotation, the type of the change between two manifolds is called distortion.
Related work
The process of solving DMOPs involves the rapid and accurate tracking of the entire changing PS or PF. In order to achieve accurate PS or PF, convergence is crucial, while capturing comprehensive PS or PF requires maintaining diversity. To address these requirements, the literature has proposed three types of methods: Diversity-based methods, memory-based methods, and prediction-based methods.
1.
Diversity-based methods: Researchers have proposed various approaches to address the issue of population diversity loss after convergence in DMOPs. One such approach is the Dynamic NSGA-II (DNSGA-II), introduced by Deb et al. [36]. This approach maintains diversity by replacing a fixed proportion of the population with randomly generated or mutated solutions derived from previous solutions. Yang et al. [37] proposed a reference point-based prediction approach, where a random number following a uniform distribution was added to the predicted centre to preserve diversity in the predicted PF. In the steady-state and generational evolutionary algorithm (SGEA) [12], promising diversity is maintained by incorporating information from both previous and new environments. In [17], A gradual search technique is utilized to generate solutions with a good distribution based on user-defined minimum and maximum PS points. Randomly produced individuals are then added to the population in the next environment to enhance diversity. In [38], a dynamic multiobjective evolutionary algorithm based on a dynamic evolutionary environment model (DEE-DMOEA) was proposed. This algorithm leverages reported knowledge and information from the population in dynamic environments to guide the evolution of diverse subpopulations in new environments, thus improving the approximate PS or PF diversity. In [14], Zhang et al. proposed a classification strategy that categorizes decision variables into principal and non-principal groups to generate well-diversified points effectively. In [39], Ahrari et al. developed an adaptive variation operator that enhances diversity in the reinitialized population while maintaining the predictor’s accuracy following a change.
2.
Memory-based methods: Memory-based methods have been proposed when dealing with environmental changes that exhibit periodicity. These methods utilize historical information from previous environments to locate optimal individuals in subsequent environments. For instance, Wang et al. [17] proposed a multi-strategy ensemble evolutionary algorithm (MS-MOEA), which combines differential operators and adaptive genetics to generate new offspring and expedite convergence. The Gaussian mutation operator is employed to address premature convergence, while the memory approach helps obtain a better initial population during environmental transitions. In [40], an adaptive hybrid method incorporating memory, random strategies, and local search was proposed to handle environmental changes effectively. The memory strategy enhances convergence when the change in the environment is minimal. Liang et al. [41] proposed a hybrid approach that combines memory and prediction methods. When consecutive environments exhibit similar changes, a differential prediction approach is utilized to predict optimal points in the new environment. Conversely, when the changes are distinct, a memory-driven approach is employed to search for the optimal points after a change.
3.
Prediction-based method: When the change of PS exhibits regularity, we can learn knowledge from the historical PS information to predict the PS under new environment. Hence the prediction-based method is proposed.Zhou et al. proposed an individual-based prediction method [42] that constructs time series for each individual in the population. However, when dealing with a large population, the computational costs become extensive. To address this limitation, Zhou et al. introduced a population-based prediction strategy (PPS) [13], which focuses on reducing computation complexity by building a time series solely for the center point of the PS. While effective for cases of PS translation, this approach lacks accuracy when faced with PS rotation in a dynamic environment. To overcome the limitation of PS rotation prediction accuracy, Rong et al. proposed a multidirection prediction method (MDP) [19]. MDP adaptively clusters the population into multiple subpopulations, taking into account the severity of each environmental change. For each subpopulation, a time series is constructed exclusively for the center point corresponding to the cluster. This approach achieves a fine balance between prediction accuracy and computational cost.
In MDP, the PS prediction is often guided by the center point of the PS manifold. However, it is important to note that the centre point may only partially capture some of the characteristics of the PS. To overcome this limitation, researchers have proposed prediction methods based on other special points, such as a hybrid method based on pivot points [43], a quantile point-based method [44] and knee point-based methods [45‐47]. These special points offer unique perspectives for guiding the search for the PS or PF and enrich the overall understanding of the PS or PF characteristics.
Most of the aforementioned prediction methods are based on linear correlation between solutions in different environments. However, in DMOPs, there is a possibility of a nonlinear correlation between historical points and predicted points. To address this, researchers have proposed nonlinear model-based prediction approaches. For instance, Wang et al. developed a prediction strategy based on Gaussian mixture models, which captures the nonlinear change in the PS when transitioning to a new environment [22]. Similarly, Sun et al. proposed a self-evolving fuzzy system online prediction algorithm that employs a self-evolving fuzzy system to predict the PS with nonlinear change in the next environment [25]. In addition, due to the challenges of accurately capturing the nonlinear correlations between historical and predicted points, classification-based prediction methods have been introduced [48, 49]. These methods identify optimal solutions from rough optimal solutions using a classifier, in order to act as the initial population in the new environment. The rough optimal solutions are obtained by a simple prediction model approximating nonlinear relationships, and the classifier is trained using historical information.
Among these nonlinear prediction methods, there is considerable focus on machine learning-based approaches [4, 9‐11, 21, 50‐52], particularly transfer learning-based methods [9, 11, 48‐50, 52‐54]. Transfer learning aims to transfer knowledge learned from previous environments into the next one to exploit optimal solutions. However, the presence of low-quality individuals in the initial population can lead to negative transfer and result in high computational costs. To mitigate this, several methods have been proposed to reduce negative transfer. For example, [53] introduces a clustering-based method that transfers useful knowledge between similar clusters from the previous environment to the new one. [54] presents a multiple source transfer learning method for DMOEA, which performs two transfer learning procedures to fully utilize historical information from all previous environments. This effectively suppresses negative transfer.
The aforementioned prediction methods are primarily suitable for DMOPs with deterministic changes in the environment. However, when the environment in DMOPs undergoes stochastic changes, additional prediction methods have been developed. [9] proposes a knowledge-guided transfer strategy for DMOEA with stochastic change. This strategy involves extracting and preserving knowledge from historical environments, eliminating redundant knowledge, evaluating knowledge representatives in new environments, and employing a hybrid transfer strategy to select appropriate knowledge for generating a new initial population. In addition, [55] introduces a Mahalanobis Distance-based approach (MDA) that utilizes stored information and assesses search environments to understand the relationship between new and historical environments. This enables a change response strategy to enhance convergence and population diversity, while considering the change degree of decision variables to mitigate the impact of stochastic changes on the population.
Most existing methods primarily focus on changes in the PS and overlook the changes in the PS manifold. This oversight can limit the prediction model’s ability to track changes in the PS. In general, the changes in the PS manifold can be decomposed into two fundamental components: rotation change and distortion change. Therefore, we believe that there are two mapping functions corresponding to rotation change and distortion change, respectively, which can be approximated by the proposed method.
Proposed method
In this study, we focus exclusively on DMOPs where the change patterns of the PS in adjacent environments exhibits similarity but not complete identity. In general, the change of the PS in a new environment can be deconstructed into translation, rotation, and distortion changes. Here we assume that the translation and rotation changes of the PS in adjacent environments are identical, while the distortion change differs and the degree of the distortion is slight. To address this assumption, we derive the translation and rotation transformation matrices of the PS in the current environment by learning from historical PS information in the previous two environments. However, obtaining the distortion transformation function presents a significant challenge. This is because learning the distortion transformation function necessitates access to the PS in the new environment, which is unavailable. To overcome this obstacle, the most promising individuals from the population obtained by combining the translation and rotation change patterns are used to approximate the PS in the new environment. Subsequently, the distortion transformation function is learned by the PS in the previous environment and the approximated PS in the new environment.
The framework of PSCA
First, we employ the subspace alignment strategy to capture the rotation transformation matrix of the PS manifold in the new environment, generating half of the quasi-initial population. Second, the other half of the quasi-initial population is generated using the correlation alignment strategy, which captures the distortion transformation function of the PS manifold in the new environment. Finally, the most promising solutions in the quasi-initial population are extracted to serve as the initial population in the new environment, and these solutions evolve into the true PS by applying a static multiobjective evolutionary algorithm (MOEA). The whole framework of PSCA is presented in Algorithm 1.
Algorithm 1
PCSA-MOEA/DE
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Translation change of PS
Definition (Translation) A translation change is said to exist between \( {\text {PS}}_{t-1} \) and \( {\text {PS}}_{t} \) if \( {\text {PS}}_{t-1} \) can be made closer to \( {\text {PS}}_{t} \) by a translation.
The translation change of PS is the location change of PS. According to the above-given assumption that the translation changes of PS in adjacent environments are identical, the transformation matrix of the translation change of PS in t time step (new environment) can be expressed as
where \( c_{t-1} \) and \( c_{t-2} \) are the center of PS in \( t-1 \) and \( t-2 \) time step, respectively. If \( B_{t} \) is the obtained approximation set of the PS in t time step,then \( c_{t} \) can be expressed as
Generating quasi-initial population by subspace alignment strategy
Subspace alignment
When there is a large difference in distribution between source data and target one in the transfer learning community, domain shifting arises, reducing transfer learning performance.To address the issue, the common idea is to align the two distributions using a map function. However, obtaining this map function is a great challenge. To address the issue, subspace alignment is proposed [56]. In this approach, the two manifolds corresponding to source data and target one are projected on their respective subspaces \( S_{s} \) and \( S_{t} \) obtained by Principal component analysis (PCA). Then the map function aligning these two subspaces is used to approximate the map function aligning these two manifolds. The detail is as follows: first, \( Y_{s}=\left( \varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{D}\right) \) and \( Y_{t}=\left( \varepsilon '_{1},\varepsilon '_{2},\ldots ,\varepsilon '_{D}\right) \) are obtained by PCA, where \( Y_{s} \) and \( Y_{t} \) are orthogonal matrices constructed by the orthogonal basis of \( S_{s} \) and \( S_{t} \),respectively, \( \varepsilon _{i} \) is the ith basis vector of \( S_{s} \) and equal to the eigenvector corresponding to the ith largest eigenvalue of the covariance matrix for source data, and \( \varepsilon '_{i} \) is that of \( S_{t} \) and equal to that for target one. Second, the optimization model for obtaining the transformation matrix \( M^{*} \) corresponding to this map function is built as
where \(\parallel \cdot {\parallel }_{F}^{2}\) is the Frobenius norm. Finally, the optimal solution for the optimization model, namely, the transformation matrix \( M^{*} \) is obtained by
Remark:In PCA, both source data and target one are translated into where their centers coincide with the origin before being projected into their respective subspaces.
Generating quasi-initial population
As \( Y_{s} \) and \( Y_{t} \) in (5) are orthogonal matrices,So
According to (5) and (7), \( M^{*} \) is also an orthogonal matrix. Since a linear transformation corresponding to a orthogonal matrix is a rotational one around the origin, \( M^{*} \) can describe the rotation change of a manifold around the origin.In DMOPs, if \( {\text {PS}}_{t-2} \) translates and rotates into \( {\text {PS}}_{t-1} \), we take \( {\text {PS}}_{t-2} \) and \( {\text {PS}}_{t-1} \) as the source data manifold and target one in the transfer learning community, respectively. First, both \( {\text {PS}}_{t-2} \) and \( {\text {PS}}_{t-1} \) are translated into the place where their centers coincide with origin. Then the map matrix \( M_{t-1} \) that the translated \( {\text {PS}}_{t-2} \) (denoted \(( {\text {PS}}_{t-2}-C_{t-2}) \) where \(C_{t-2} \) consists of the center of \( {\text {PS}}_{t-2}\))rotate into the translated \( {\text {PS}}_{t-1} \) (denoted \( ({\text {PS}}_{t-1}-C_{t-1}) \) ) around the origin (is also the center of the two translated manifolds \(( {\text {PS}}_{t-2}-C_{t-2}) \) and \(( {\text {PS}}_{t-1}-C_{t-1}) \))can be approximately obtained by (5) in subspace alignment, and expressed as
where \( Y_{t-2} \) and \( Y_{t-1} \) are the matrices constructed by the orthogonal basis of subspaces \( S_{t-2}\), \( S_{t-1}\) of \(( {\text {PS}}_{t-2} -C_{t-2})\) and \( ({\text {PS}}_{t-1} -C_{t-1})\), respectively. Figure 2 depicts the process of subspace alignment strategy. According to the above-given assumption that rotation changes of PS in adjacent environments are identical, the approximate transformation matrix \( M_{t} \) of the rotation change of PS in t time step is
$$\begin{aligned} M_{t}=M_{t-1} \end{aligned}$$
(9)
Due to the complexity of the change of PS in the new environment, a single translation and rotation transformation is insufficient to describe the complex changes in PS when the size of PS is large. To address the problem, the two PS manifolds in \( t-2 \) and \( t-1 \) time steps are divided into K submanifolds by local Principal component analysis(LPCA), respectively:
The two submanifolds with the smallest distance in the different environments are paired to predict the submanifolds of the PS in the next environment using the above-mentioned translation transform (3) and rotation transform (8) (9). Thus, the prediction model with multiple translation and rotation transformations can capture the complex change ofPS in the new environment. Next, the detailed process is presented.
First, by (3), the transformation matrix of translation change for each submanifold of PS in t time step is expressed as
where \(i\in \left\{ 1,2,\ldots ,K \right\} \), \(c_{t-1}^{i} \) is the center of \( {\text {PS}}_{t-1}^{i} \), and \( p_{i} \) is the index of the submanifold in \( [{\text {PS}}_{t-2}^{1},{\text {PS}}_{t-2}^{2},\ldots {\text {PS}}_{t-2}^{K}] \) closest to \( {\text {PS}}_{t-1}^{i} \). Then according to (8) (9), the transformation matrix of rotation change for the ith submanifold of PS in t time step can be approximately expressed as
where \( Y_{t-2}^{p_{i}} \) and \( Y_{t-1}^{i} \) denote the matrices constructed by the orthogonal basis of the \( p_{i}\)th, ith partitioned subspace \( S_{t-2}^{p_{i}}\), \( S_{t-1}^{i}\) of \( ({\text {PS}}_{t-2}^{p_{i}}- C_{t-2}^{p_{i}})\) and \( ({\text {PS}}_{t-1}^{i}- C_{t-1}^{i}) \), respectively. Finally, by (11) (12), the half quasi-initial population is generated
where \( C_{t-1}^{i} \) and \( D_{t}^{i} \) are \( N_{i}\times D \) matrix, in which all elements are \(c_{t-1}^{i} \) and \(d_{t}^{i} \) in (11), respectively, and \( N_{i} \) is the size of \( {\text {PS}}_{t-1}^{i} \). Algorithm 2 provides the pseudocode for obtaining half quasi initial population.
Algorithm 2
SAS
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Generating quasi-initial population by correlation alignment
Correlation alignment
When there is a large difference in distribution between source data and target one, Obtaining the map function aligning the two distributions is greatly challenging. To address this issue, the map function aligning the statistical characteristic values, mean and covariance of the two data manifolds, is used to approximate the map function aligning the distributions between these two manifolds. The detail is as follows: first, both source data and target one are translated into where their centers coincide with the origin. This makes the means of two data manifolds both equal to zero, aligning their one statistical characteristic value, mean. Then a map matrix A aligning the covariance of distributions between these two translated data manifolds (called correlation alignment [34])is obtained by
where \( D_{s}\) and \(D_{t} \) denote the source data and target one, \( \textrm{cov} \) denotes covariance.
Generating quasi-initial population
The covariance can describe a correlation between elements in a data manifold. When one manifold distorts into another, the change in the correlation between individuals in the manifold is the key to the distortion change. Therefore, the map matrix aligning the covariance between the two manifolds can approximate the transformation matrix corresponding to the distortion change between the two manifolds. In DMOPs, if \( {\text {PS}}_{t-1} \) translates and distorts into \( {\text {PS}}_{t} \), \( {\text {PS}}_{t-1} \) and \( {\text {PS}}_{t} \) are taken as the source data and target one, respectively. First, \( {\text {PS}}_{t-1} \) and \( {\text {PS}}_{t} \) are translated into where their centers coincide with the origin. Then the map matrix \( A_{t} \) that the translated \( {\text {PS}}_{t-1} \) (denoted \(( {\text {PS}}_{t-1}-C_{t-1}) \) where \(C_{t-1} \) consists of the center of \( {\text {PS}}_{t-1}\)) distort into the translated \( {\text {PS}}_{t} \) (denoted \(( {\text {PS}}_{t}-C_{t}) \) )can be approximately obtained by (15) in correlation alignment strategy, and expressed as
However, The \( {\text {PS}}_{t} \) in the new environment can not be provided. To address this problem, the most promising individuals in the population \( {\text {QinitPop}}1_{t} \) obtained by combining the translation and rotation change patterns is used to approximate the \( {\text {PS}}_{t} \) in the new environment. Thus, the transformation matrix \( A_{t} \) can be approximately expressed as
where \( \textrm{ElitePop}_{t}\) is an optimal solution set selected from \( {\text {QinitPop}}1_{t} \) based on non-dominated sorting and degree of congestion [50]. Its size is set to 0.5N. \( C_{t-1} \) is a \( N\times D \) matrix where all elements are \( c_{t-1} \) in (4), \( \textrm{CElitePop}_{t} \) is a \( 0.5N\times D \) matrix where all elements are the center of \( \textrm{ElitePop}_{t} \). Thus, by (17), another half of quasi initial population in t time step can be generated:
The pseudocode for the CAS is presented in Algorithm 3.
Algorithm 3
CAS
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Generate initial population in new environment
The selected promising individuals from this quasi-initial population, based on their non-dominated relation and degree of crowding [50], are set to the initial population in this new environment. Figure 3 depicts the whole process of the proposed method PSCA.
Analysis of time complexity
In the algorithm SAS, clustering \( {\text {PS}}_{t-1} \)and \( {\text {PS}}_{t-2 }\) by LPCA calls for \( O(2D^{2}) \). The pairwise Euclidean distances (line 1) calculation requires O(DK) computations. Pairing the center points of PS submanifolds in adjacent environments (line 1)needs \( O(K^2) \). Obtaining the mapping matrix consumes O(2K) . The cost of calculating the center points is O(2K) .
In algorithm CAS, extracting elite solution set \( \textrm{ElitePop}_{t} \) with size of \(0.5*N\) from \( {\text {QinitPop}}1_{t} \) spends \( O(m(0.5*N)^{2}) \). Calculating the covariance matrix consumes \( O(ND^{2}) \). Computing the inverse of the covariance matrix needs \( O(D^{3}) \). Finally, extracting the optimal solutions from the quasi-initial population as the response population for the new environment calls for \( O(mN^{2}) \). Consequently, the whole-time complexity of the proposed prediction method is \( O( DN^{2}) \). If \( m> D \), then the time complexity will be \( O(mN^{2}) \). If \( D>m \) and \( D>N \), the time complexity is \( O(D^{3}) \).
Experimental design
Test instances
The DF test suite [57] is employed to assess the performance of the proposed algorithm compared to other state-of-the-art algorithms. This suite encompasses a comprehensive set of 14 test problems, carefully designed to capture diverse characteristics observed in real-world scenarios. These characteristics include time-dependent PF/PS geometries, irregular PF shapes, disconnectivity, knee points, and more. These test problems can be classified as Table 1 according to the types of the PS change given in “Types of manifold changes and “Translation change of PS”. By utilizing this test suite, all evaluated algorithms can be thoroughly and impartially assessed. The dynamics of DMOP considered in this study are characterized by a time-dependent factor, specifically represented as \( t=(1/n_{t})\lfloor \tau /\tau _{t} \rfloor \). Here, \( n_{t} \) represents the severity of the change, \( \tau _{t} \) denotes the change frequency, and \( \tau \) signifies the maximum generation. To ensure robustness and account for randomness, each tested instance is executed 10 times with a distinct random initial population.
Fig. 3
Illustration of the proposed method
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Table 1
Types and dynamic features of test problems
Type
Feature of PS change
Problems
Type 1
Translation
DF1, DF5, DF6, D13, DF14
Type 2
Rotation
DF11
Type 3
Distortion
DF8, DF10
Type 4
Translation and rotation
DF2
Type 5
Rotation and distortion
DF4, DF7, DF12
Type 6
Translation, rotation and distortion
DF3, DF9
Compared algorithms
To facilitate a comprehensive evaluation, five representative prediction algorithms, namely, PPS [13], evolutionary dynamic multiobjective optimization using a difference model(DM) [58], TL [11], knee point-based transfer learning algorithm(KT) [47], and SVR [21], are carefully selected as benchmarks for comparison with the proposed algorithm. To ensure an equitable comparison, all chosen prediction algorithms are integrated with the same static optimizer, specifically MOEM/D-DE [21, 59].
Table 2
Statistical results of MIGD obtained by six algorithms on test problems with different change frequency
Problem
\(\tau _{t},n_{t} \)
PPS
DM
TL
KT
SVR
PSCA
DF1
5 10
0.4282 ± 2.03E−02(+)
0.0242 ± 9.78E−04(+)
0.0254 ± 1.74E−04(+)
0.0383 ± 8.38E−04(+)
0.0165 ± 1.11E−03(+)
0.0147 ± 5.24E−04
10 10
0.0540 ± 7.50E−04(+)
0.0089 ± 1.36E−04(+)
0.0117 ± 2.18E−04(+)
0.0134 ± 3.12E−04(+)
0.0081 ± 1.97E−04(+)
0.0071 ± 1.04E−04
20 10
0.0083 ± 1.43E−04(+)
0.0052 ± 1.57E−05(+)
0.0060 ± 2.84E−05(+)
0.0061 ± 2.49E−05(+)
0.0053 ± 3.64E−05(+)
0.0050 ± 8.04E−05
DF2
5 10
0.3806 ± 3.06E−03(+)
0.0292 ± 2.03E−03(+)
0.0228 ± 1.36E−03(+)
0.0465 ± 7.03E−03(+)
0.0262 ± 2.28E−03(+)
0.0201 ± 2.35E−03
10 10
0.0876 ± 8.33E−03(+)
0.0107 ± 3.44E−04(+)
0.0107 ± 3.42E−04(+)
0.0124 ± 4.50E−04(+)
0.0092 ± 2.20E−04(+)
0.0081 ± 3.56E−04
20 10
0.0097 ± 5.95E−04(+)
0.0058 ± 3.33E−05(+)
0.0058 ± 3.42E−04(+)
0.0061 ± 1.03E−04(+)
0.0056 ± 7.88E−05(+)
0.0052 ± 2.75E−05
DF3
5 10
0.3447 ± 1.26E−03(+)
0.1872 ± 3.32E−03(+)
0.1225 ± 8.14E−03(+)
0.2106 ± 2.81E−03(+)
0.1384 ± 3.27E−03(−)
0.0974 ± 1.27E−02
10 10
0.1792 ± 1.76E−03(+)
0.1049 ± 2.18E−03(+)
0.0728 ± 6.05E−03(+)
0.1191 ± 1.79E−03(+)
0.0798 ± 2.72E−03(+)
0.0369 ± 2.64E−03
20 10
0.0678 ± 1.01E−03(+)
0.0423 ± 4.87E−04(+)
0.0387 ± 2.33E−04(+)
0.0481 ± 9.20E−04(+)
0.0402 ± 9.86E−04(+)
0.0169 ± 3.07E−04
DF4
5 10
0.2293 ± 3.84E−02(+)
0.0797 ± 3.06E−03(−)
0.0848 ± 7.86E−03(=)
0.0858 ± 1.22E−03(=)
0.0942 ± 4.91E−03(+)
0.0836 ± 2.89E−03
10 10
0.0780 ± 3.30E−03(+)
0.0677 ± 8.50E−04(−)
0.0689 ± 5.38E−03(−)
0.0697 ± 2.99E−04(−)
0.0705 ± 4.05E−04(−)
0.0748 ± 7.49E−04
20 10
0.0725 ± 1.08E−04(−)
0.0730 ± 7.78E−04(−)
0.0747 ± 4.58E−03(=)
0.0730 ± 3.11E−04(−)
0.0737 ± 8.43E−04(=)
0.0744 ± 2.18E−04
DF5
5 10
0.8923 ± 1.90E−02(+)
0.1818 ± 7.85E−03(+)
0.1923 ± 3.67E−03(+)
0.1485 ± 2.30E−03(+)
0.2064 ± 3.63E−02(+)
0.0097 ± 2.72E−04
10 10
0.1134 ± 2.96E−02(+)
0.0151 ± 8.81E−04(+)
0.0201 ± 1.86E−03(+)
0.0156 ± 2.51E−03(+)
0.0202 ± 1.92E−03(+)
0.0060 ± 8.78E−05
20 10
0.0066 ± 9.39E−05(+)
0.0054 ± 1.23E−05(+)
0.0059 ± 2.05E−05(+)
0.0058 ± 1.42E−05(+)
0.0058 ± 6.18E−05(+)
0.0046 ± 3.32E−05
DF6
5 10
4.0937 ± 1.75E−01(−)
2.6002 ± 1.07E−01(−)
1.0704 ± 5.07E−01(−)
0.8183 ± 6.11E−02(−)
2.3964 ± 1.72E−01(−)
4.2625 ± 4.94E−01
10 10
2.9876 ± 1.21E−02(−)
2.4573 ± 1.74E−01(−)
0.7057 ± 2.53E−01(−)
0.7215 ± 1.92E−01(−)
2.1668 ± 1.78E−01(−)
4.3303 ± 5.72E−01
20 10
3.2184 ± 2.12E−01(+)
2.2303 ± 3.69E−01(−)
0.8559 ± 4.76E−01(−)
0.9633 ± 5.40E−02(−)
1.9034 ± 1.62E−01(−)
2.9734 ± 4.78E−01
DF7
5 10
0.8427 ± 7.82E−03(+)
0.9742 ± 1.55E−03(+)
0.9442 ± 1.37E−01(+)
0.7210 ± 8.98E−03(+)
0.9787 ± 1.35E−02(+)
0.3842 ± 3.95E−02
10 10
0.6344 ± 1.90E−02(+)
0.8020 ± 1.54E−02(+)
0.8125 ± 1.23E−01(+)
0.5708 ± 3.21E−03(+)
0.8095 ± 1.30E−02(+)
0.3399 ± 9.04E−03
20 10
0.4653 ± 2.92E−02(+)
0.6403 ± 1.39E−02(+)
0.6256 ± 7.28E−02(+)
0.4041 ± 1.62E−02(+)
0.6366 ± 2.00E−02(+)
0.3095 ± 3.92E−02
DF8
5 10
0.3265 ± 5.72E−03(+)
0.0210 ± 3.69E−04(−)
0.0250 ± 5.67E−02(−)
0.0358 ± 2.01E−03(+)
0.0202 ± 7.83E−04(−)
0.0321 ± 3.29E−03
10 10
0.2391 ± 1.04E−02(+)
0.0164 ± 1.10E−04(−)
0.0180 ± 2.69E−03(−)
0.0241 ± 7.07E−04(+)
0.0164 ± 2.95E−04(−)
0.0210 ± 1.96E−03
20 10
0.2309 ± 3.21E−02(+)
0.0152 ± 3.70E−04(=)
0.0151 ± 4.47E−02(=)
0.0158 ± 1.75E−05(=)
0.0150 ± 3.28E−04(−)
0.0155 ± 1.93E−04
DF9
5 10
1.0226 ± 2.97E−02(+)
0.3252 ± 3.21E−02(+)
1.5335 ± 4.09E−01(+)
0.3970 ± 3.79E−03(+)
0.3731 ± 1.60E−02(+)
0.2786 ± 4.98E−02
10 10
0.3752 ± 3.11E−02(+)
0.2049 ± 1.02E−02(+)
0.2175 ± 7.14E−02(+)
0.2576 ± 3.59E−03(+)
0.1841 ± 8.41E−03(+)
0.1760 ± 1.93E−02
20 10
0.2647 ± 5.99E−04(+)
0.1665 ± 3.13E−03(+)
0.1409 ± 2.94E−02(+)
0.1981 ± 2.92E−02(+)
0.1173 ± 4.56E−03(+)
0.1140 ± 7.25E−03
DF10
5 10
0.5875 ± 4.15E−02(+)
0.1107 ± 3.81E−03(−)
0.1219 ± 8.17E−02(−)
0.2404 ± 6.18E−03(+)
0.1085 ± 4.13E−03(−)
0.2083 ± 7.52E−02
10 10
0.5155 ± 1.01E−02(+)
0.1032 ± 1.58E−03(−)
0.1130 ± 1.44E−01(−)
0.1839 ± 1.01E−03(+)
0.0995 ± 2.72E−03(−)
0.1343 ± 3.11E−02
20 10
0.4585 ± 8.83E−03(+)
0.0908 ± 7.56E−04(−)
0.0961 ± 8.46E−02(−)
0.1496 ± 2.22E−02(+)
0.0902 ± 7.20E−04(−)
0.1136 ± 4.06E−02
DF11
5 10
0.1576 ± 1.59E−03(+)
0.0814 ± 1.62E−03(+)
0.0774 ± 4.46E−03(−)
0.0845 ± 1.87E−03(+)
0.0691 ± 5.06E−04(−)
0.0800 ± 1.19E−03
10 10
0.0781 ± 8.00E−04(+)
0.0690 ± 3.13E−04(=)
0.0661 ± 4.60E−04(−)
0.0708 ± 5.55E−05(+)
0.0616 ± 5.94E−04(−)
0.0685 ± 3.11E−04
20 10
0.0644 ± 3.99E−05(+)
0.0620 ± 1.79E−04(=)
0.0606 ± 1.66E−04(−)
0.0632 ± 5.21E−04(+)
0.0578 ± 4.04E−04(−)
0.0620 ± 1.96E−04
DF12
5 10
0.6820 ± 3.39E−03(+)
0.4199 ± 3.20E−04(−)
0.2994 ± 2.50E−02(−)
0.6277 ± 1.02E−02(+)
0.3184 ± 4.13E−03(−)
0.5205 ± 3.70E−02
10 10
0.5938 ± 4.12E−02(−)
0.4044 ± 2.60E−03(−)
0.2892 ± 1.51E−02(−)
0.5343 ± 6.63E−03(−)
0.3013 ± 1.17E−03(−)
0.6030 ± 1.05E−01
20 10
0.5655 ± 5.59E−03(−)
0.3777 ± 3.78E−03(−)
0.2794 ± 1.61E−02(−)
0.5267 ± 7.16E−03(−)
0.2942 ± 1.26E−03(−)
0.5056 ± 4.02E−02
DF13
5 10
0.5851 ± 2.25E−02(+)
0.4190 ± 1.84E−04(+)
0.4261 ± 2.57E−02(+)
0.4105 ± 3.57E−04(+)
0.4092 ± 2.34E−03(=)
0.4052 ± 1.29E−03
10 10
0.5688 ± 2.03E−02(+)
0.4154 ± 4.59E−04(=)
0.4226 ± 1.35E−02(+)
0.4179 ± 4.56E−04(+)
0.4183 ± 1.75E−03(+)
0.4131 ± 1.04E−03
20 10
0.5023 ± 6.21E−03(+)
0.4200 ± 9.37E−04(=)
0.4260 ± 1.04E−02(+)
0.4199 ± 7.89E−05(=)
0.4236 ± 9.02E−04(+)
0.4157 ± 6.66E−04
DF14
5 10
0.0837 ± 4.97E−03(+)
0.0542 ± 1.27E−03(+)
0.0491 ± 6.94E−03(+)
0.0491 ± 6.66E−04(+)
0.0554 ± 3.91E−03(+)
0.0384 ± 2.78E−04
10 10
0.0391 ± 1.12E−04(+)
0.0369 ± 1.93E−04(+)
0.0372 ± 5.78E−04(+)
0.0371 ± 4.41E−05(+)
0.0376 ± 3.80E−04(+)
0.0352 ± 1.33E−04
20 10
0.0351 ± 1.02E−05(+)
0.0348 ± 9.93E−05(+)
0.0349 ± 1.61E−04(+)
0.0348 ± 1.61E−05(+)
0.0348 ± 1.20E−04(+)
0.0334 ± 3.31E−05
Best
1
3
5
1
8
24
Parameter settings
The size of population: \(N=100\) or 300 when \(m=2\) or 3;
The dimension of variable: \(D=10\);
The number of generations: \( \tau =50*\tau _{t} \);
Change frequency: \( \tau _{t} \) = 5, 10, 20;
Change severity: 5, 10, 20;
The number of K in PSCA: \( K=3 \);
Maximum iterations of the employed static algorithm after an environment change: MaxInter = \( \tau _{t} \);
The parameters for MOEA/D-DE is identical with reference [21].
The other parameters are consistent with the original references.
Performance indicators
In this experimental study, two metrics, namely, mean inverted generational distance (MIGD) and mean hypervolume (MHV), are adopted, as they can help deeply investigate the performance of algorithm regarding convergence and diversity from different perspectives.
1.
MIGD: The inverted generational distance (IGD) [60], which reflects the difference between true Pareto front and obtained Pareto front (PF), can measure both the convergence and diversity of the solutions. The IGD is defined as follows:
where \( \text {PF}_{t}^{*}\) and \( \text {PF}_{t} \) represent the true PF and the approximate one obtained by a static MOEA at time t, respectively, and N denotes the number of points in the true PF. As an extension of IGD, MIGD is the average IGD value of all time instants in a given run. It is defined as follows:
where \( \mu \) is the Lebesgue measure, and Ref is the reference point in the objective function space. As an extension of HV, MHV is the average HV value of all time instants in a given run.It is defined as follows:
where T is a set of time instants in a given run, and \(|{\textrm{T}}| \) is the cardinality of T.
Discussion on experimental results
Comparison with other methods
To evaluate the performance of the proposed algorithm, we compare the values of the performance metrics IGD and HV between the proposed method and those of five other selected algorithms. The results of this comparison are presented in Tables 2, 3, 4, 5, where the best value for each test instance is bolded. We employ the Wilcoxon rank sum test to determine the statistical significance of the observed differences in the results of all algorithms. In these tables, the symbols “+”, “=”, and “−” indicate that the values of the proposed method are statistically better than, equal to, or worse than those of the compared algorithm at a significance level of 5%, respectively.
1. Different Change Frequency: To investigate the performance of the proposed algorithm under different change frequencies, we conducted a series of experiments. Specifically, we computed the average MIGD and MHV values for our algorithm and compared ones over 10 runs on DF1-DF14 with \(n_{t}=10\) and \( \tau _{t} =5, 10, 20\). The results are presented in Tables 2 and 3.
Table 3
Statistical results of MHV obtained by six algorithms on test problems with different change frequency
Problem
\(\tau _{t},n_{t} \)
PPS
DM
TL
KT
SVR
PSCA
DF1
5 10
0.2068 ± 1.81E−02(+)
0.6176 ± 2.95E−03(+)
0.6170 ± 3.08E−03(+)
0.5839 ± 2.92E−01(+)
0.6325 ± 1.90E−03(+)
0.6363 ± 1.55E−03
10 10
0.5634 ± 8.32E−03(+)
0.6466 ± 5.02E−04(+)
0.6401 ± 3.23E−04(+)
0.6387 ± 2.86E−01(+)
0.6468 ± 3.72E−04(+)
0.6502 ± 5.36E−04
20 10
0.6518 ± 6.47E−04(+)
0.6536 ± 8.93E−05(+)
0.6518 ± 3.27E−04(+)
0.6520 ± 3.26E−01(+)
0.6535 ± 5.28E−04(+)
0.6542 ± 1.30E−04
DF2
5 10
0.3878 ± 3.36E−02(+)
0.8220 ± 4.57E−03(+)
0.8368 ± 4.08E−03(+)
0.8108 ± 4.68E−01(+)
0.8269 ± 2.83E−03(+)
0.8406 ± 3.86E−03
10 10
0.7467 ± 1.55E−02(+)
0.8578 ± 4.53E−04(+)
0.8579 ± 4.29E−04(+)
0.8534 ± 4.27E−01(+)
0.8603 ± 6.63E−04(+)
0.8620 ± 4.22E−04
20 10
0.8600 ± 1.05E−03(+)
0.8667 ± 6.43E−05(=)
0.8665 ± 3.33E−04(=)
0.8665 ± 1.47E−04(=)
0.8671 ± 1.21E−04(=)
0.8678 ± 1.10E−04
DF3
5 10
0.2497 ± 1.11E−02(+)
0.3776 ± 6.49E−03(+)
0.4360 ± 1.84E−03(+)
0.3539 ± 7.24E−03(+)
0.4187 ± 6.35E−03(+)
0.4855 ± 6.89E−03
10 10
0.3952 ± 3.33E−03(+)
0.4634 ± 4.18E−03(+)
0.5000 ± 2.29E−03(+)
0.4582 ± 5.42E−03(+)
0.4936 ± 1.23E−03(+)
0.5550 ± 4.12E−03
20 10
0.5194 ± 2.39E−03(+)
0.5531 ± 1.21E−03(+)
0.5556 ± 2.76E−03(+)
0.5425 ± 1.64E−03(+)
0.5539 ± 1.76E−03(+)
0.5899 ± 1.24E−03
DF4
5 10
4.7278 ± 8.91E−02(+)
5.5752 ± 5.64E−03(+)
5.5767 ± 6.79E−03(−)
–
5.5092 ± 8.91E−03(+)
5.6633 ± 2.81E−03
10 10
5.2402 ± 8.14E−03(+)
5.7039 ± 3.70E−03(=)
5.7048 ± 2.85E−03(=)
5.7002 ± 2.88E−03(=)
5.7028 ± 3.95E−03(=)
5.7321 ± 9.72E−04
20 10
5.3396 ± 1.46E−03(+)
5.7566 ± 3.72E−04(+)
5.7587 ± 2.88E−04(=)
5.7575 ± 2.88E−03(+)
5.7574 ± 4.66E−04(+)
5.7625 ± 8.21E−04
DF5
5 10
0.3448 ± 2.64E−02(+)
0.5411 ± 4.09E−03(+)
0.5151 ± 5.69E−03(+)
0.5537 ± 5.17E−03(+)
0.5004 ± 4.41E−03(+)
0.6923 ± 9.73E−04
10 10
0.6131 ± 2.09E−02(+)
0.6779 ± 6.32E−03(+)
0.6493 ± 3.37E−04(+)
0.6832 ± 2.14E−03(+)
0.6717 ± 2.11E−03(+)
0.6985 ± 2.57E−04
20 10
0.6968 ± 2.80E−04(+)
0.6988 ± 1.44E−04(=)
0.6969 ± 3.50E−04(+)
0.6968 ± 8.94E−05(+)
0.6970 ± 1.39E−04(+)
0.7013 ± 7.27E−05
DF6
5 10
0.196 ± 1.34E−02(−)
0.5121 ± 1.28E−02(−)
0.4090 ± 2.49E−02(−)
0.3937 ± 2.27E−01(−)
0.2205 ± 4.55E−02(−)
0.0510 ± 2.79E−02
10 10
0.2567 ± 2.48E−02(−)
0.5683 ± 1.07E−02(−)
0.4763 ± 2.76E−02(−)
0.4664 ± 2.69E−01(−)
0.2881 ± 1.21E−02(−)
0.1721 ± 9.48E−02
20 10
0.2777 ± 1.95E−02(−)
0.5886 ± 8.09E−03(−)
0.4805 ± 2.90E−02(−)
–
0.3147 ± 1.29E−02(−)
0.2171 ± 9.78E−02
DF7
5 10
1.7341 ± 8.10E−02(+)
2.5479 ± 1.92E−02(+)
2.5339 ± 1.26E−02(+)
–
2.5082 ± 5.33E−02(=)
2.4964 ± 1.47E−01
10 10
2.3704 ± 4.85E−02(+)
2.6135 ± 1.12E−02(+)
2.6031 ± 1.31E−02(+)
–
2.6018 ± 3.82E−02(+)
2.6347 ± 5.51E−02
20 10
2.7133 ± 4.16E−02(+)
2.6492 ± 1.18E−02(+)
2.6365 ± 1.32E−02(+)
–
2.6423 ± 2.04E−03(+)
2.6916 ± 7.01E−02
DF8
5 10
0.4087 ± 3.42E−02(+)
0.7377 ± 2.79E−04(=)
0.7360 ± 2.69E−04(=)
0.7350 ± 3.68E−01(=)
0.7387 ± 7.01E−04(=)
0.7359 ± 4.52E−04
10 10
0.5241 ± 3.26E−02(+)
0.7408 ± 6.60E−05(=)
0.7400 ± 3.70E−04(=)
0.7406 ± 4.28E−01(=)
0.7415 ± 1.44E−04(=)
0.7400 ± 1.47E−04
20 10
0.5506 ± 3.04E−02(+)
0.7423 ± 5.70E−05(=)
0.7420 ± 6.71E−05(=)
0.7423 ± 4.29E−01(+)
0.7426 ± 5.00E−05(=)
0.7420 ± 6.36E−05
DF9
5 10
0.2798 ± 1.42E−02(+)
0.5186 ± 2.07E−02(+)
0.4600 ± 2.53E−02(+)
0.4566 ± 1.47E−02(+)
0.5193 ± 1.63E−02(+)
0.6349 ± 1.37E−02
10 10
0.5315 ± 2.00E−02(+)
0.6942 ± 7.39E−03(+)
0.6639 ± 7.44E−02(+)
0.6568 ± 2.06E−02(+)
0.7103 ± 2.44E−03(+)
0.7801 ± 1.26E−02
20 10
0.7096 ± 2.90E−02(+)
0.7770 ± 8.84E−03(+)
0.7692 ± 3.87E−03(+)
0.8009 ± 2.36E−02(+)
0.8432 ± 7.14E−03(+)
0.8856 ± 8.92E−03
DF10
5 10
0.3174 ± 1.08E−01(+)
0.9919 ± 2.22E−03(−)
0.9814 ± 4.94E−03(−)
–
0.9914 ± 4.02E−03(−)
0.9238 ± 2.01E−02
10 10
0.4644 ± 1.88E−01(+)
0.9995 ± 2.72E−03(−)
0.9939 ± 4.98E−03(−)
–
1.0026 ± 7.11E−03(−)
0.8561 ± 2.52E−01
20 10
0.4536 ± 2.23E−01(+)
1.0053 ± 7.09E−04(−)
1.0046 ± 6.02E−04(−)
–
1.0059 ± 4.30E−04(−)
0.9893 ± 1.13E−02
DF11
5 10
0.7398 ± 1.78E−02(+)
0.9508 ± 1.73E−03(=)
0.9613 ± 2.75E−03(−)
0.9349 ± 4.67E−01(+)
0.9750 ± 1.74E−03(−)
0.9521 ± 3.99E−03
10 10
0.9404 ± 1.33E−03(+)
0.9798 ± 3.86E−04(=)
0.9837 ± 3.90E−04(=)
0.9709 ± 4.85E−01(+)
0.9915 ± 3.41E−04(−)
0.9789 ± 8.27E−04
20 10
0.9706 ± 5.16E−04(+)
0.9940 ± 3.29E−04(=)
0.9957 ± 4.97E−04(=)
0.9917 ± 5.73E−01(=)
1.0017 ± 2.72E−04(=)
0.9939 ± 4.11E−04
DF12
5 10
6.5456 ± 1.77E−01(+)
8.6180 ± 3.67E−03(−)
8.7146 ± 3.31E−03(−)
–
8.7122 ± 7.46E−03(−)
8.0325 ± 2.45E−01
10 10
7.0188 ± 1.73E−01(+)
8.6480 ± 3.39E−03(−)
8.7769 ± 4.32E−03(−)
–
8.7765 ± 2.80E−03(−)
8.3742 ± 1.97E−01
20 10
7.6268 ± 1.65E−01(+)
8.6861 ± 5.85E−03(−)
8.8119 ± 5.34E−03(−)
–
8.8066 ± 1.14E−03(−)
8.3748 ± 9.91E−02
DF13
5 10
2.8381 ± 3.21E−02(+)
3.0258 ± 1.72E−02(+)
2.9158 ± 2.51E−03(+)
3.0864 ± 9.47E−03(+)
3.0060 ± 1.38E−02(+)
3.1697 ± 5.89E−03
10 10
2.9881 ± 3.74E−02(+)
3.1669 ± 7.08E−03(+)
3.1619 ± 3.59E−03(+)
3.1716 ± 2.18E−03(+)
3.1738 ± 2.53E−03(+)
3.2097 ± 2.24E−03
20 10
3.1209 ± 1.15E−02(+)
3.2138 ± 2.67E−03(+)
3.2017 ± 1.61E−03(+)
3.2114 ± 2.34E−03(+)
3.2133 ± 9.23E−03(+)
3.2234 ± 2.42E−03
DF14
5 10
0.3998 ± 1.77E−02(+)
0.4119 ± 7.16E−03(+)
0.4196 ± 6.01E−03(+)
0.4199 ± 3.19E−03(+)
0.4033 ± 1.02E−03(+)
0.4299 ± 1.36E−04
10 10
0.4250 ± 2.53E−04(+)
0.4291 ± 1.46E−04(+)
0.4288 ± 2.15E−04(+)
0.4292 ± 1.07E−04(+)
0.4294 ± 2.12E−04(+)
0.4311 ± 1.45E−04
20 10
0.4263 ± 2.06E−04(+)
0.4305 ± 1.89E−04(=)
0.4309 ± 3.15E−04(=)
0.4308 ± 1.58E−04(=)
0.4307 ± 4.77E−04(=)
0.4313 ± 1.32E−04
Best
0
5
4
0
8
25
Table 4
Statistical results of MIGD obtained by six algorithms on test problems with different change severity
Problem
\(\tau _{t},n_{t} \)
PPS
DM
TL
KT
SVR
PSCA
DF1
10 5
0.0631 ± 1.87E−04(+)
0.0116 ± 2.82E−04(+)
0.0160 ± 2.59E−04(+)
0.0173 ± 5.52E−04(+)
0.0088 ± 3.27E−04(−)
0.0111 ± 2.84E−04
10 10
0.0540 ± 7.50E−04(+)
0.0089 ± 1.36E−04(+)
0.0117 ± 2.18E−04(+)
0.0134 ± 3.12E−04(+)
0.0081 ± 1.97E−04(+)
0.0071 ± 1.04E−04
10 20
0.0539 ± 1.32E−03(+)
0.0079 ± 9.91E−05(+)
0.0091 ± 2.22E−04(+)
0.0116 ± 4.37E−04(+)
0.0084 ± 1.81E−04(+)
0.0065 ± 4.49E−05
DF2
10 5
0.0783 ± 5.26E−04(+)
0.0147 ± 1.97E−04(+)
0.0146 ± 3.06E−04(+)
0.0187 ± 3.70E−04(+)
0.0096 ± 8.50E−04(−)
0.0114 ± 2.70E−04
10 10
0.0876 ± 8.33E−03(+)
0.0107 ± 3.44E−04(+)
0.0107 ± 3.42E−04(+)
0.0124 ± 4.50E−04(+)
0.0092 ± 2.20E−04(+)
0.0081 ± 3.56E−04
10 20
0.0728 ± 2.97E−03(+)
0.0085 ± 8.10E−05(+)
0.0091 ± 3.79E−04(+)
0.0103 ± 2.28E−04(+)
0.0089 ± 1.50E−04(+)
0.0072 ± 6.98E−05
DF3
10 5
0.1866 ± 4.89E−04(+)
0.1153 ± 1.22E−03(+)
0.0834 ± 5.46E−03(+)
0.1243 ± 6.22E−03(+)
0.0820 ± 2.06E−03(+)
0.0431 ± 2.62E−03
10 10
0.1792 ± 1.76E−03(+)
0.1049 ± 2.18E−03(+)
0.0728 ± 6.05E−03(+)
0.1191 ± 1.79E−03(+)
0.0798 ± 2.72E−03(+)
0.0369 ± 2.64E−03
10 20
0.1973 ± 3.47E−03(+)
0.1255 ± 2.51E−03(+)
0.0804 ± 6.04E−03(+)
0.1280 ± 2.75E−03(+)
0.0877 ± 2.36E−03(+)
0.0429 ± 2.42E−03
DF4
10 5
0.0749 ± 7.74E−04(−)
0.0694 ± 6.71E−04(−)
0.0710 ± 6.40E−03(−)
0.0679 ± 1.48E−03(−)
0.0710 ± 3.05E−04(−)
0.0744 ± 1.12E−03
10 10
0.0780 ± 3.30E−03(+)
0.0677 ± 8.50E−04(−)
0.0689 ± 5.38E−03(−)
0.0697 ± 2.99E−04(−)
0.0705 ± 4.05E−04(−)
0.0748 ± 7.49E−04
10 20
0.0966 ± 1.30E−03(+)
0.0874 ± 2.34E−03(−)
0.0887 ± 3.79E−03(−)
0.0856 ± 1.25E−04(−)
0.0882 ± 1.36E−03(−)
0.0942 ± 2.84E−04
DF5
10 5
0.0514 ± 3.26E−03(+)
0.0162 ± 6.14E−04(+)
0.0178 ± 6.20E−03(+)
0.0186 ± 5.88E−03(+)
0.0225 ± 3.30E−03(+)
0.0074 ± 1.74E−04
10 10
0.1134 ± 2.96E−02(+)
0.0151 ± 8.81E−04(+)
0.0201 ± 1.86E−03(+)
0.0156 ± 2.51E−03(+)
0.0202 ± 1.92E−03(+)
0.0060 ± 8.78E−05
10 20
0.0204 ± 5.69E−03(+)
0.0083 ± 2.26E−06(+)
0.0086 ± 8.81E−04(+)
0.009 ± 5.12E−05(+)
0.0099 ± 2.21E−04(+)
0.0058 ± 5.04E−05
DF6
10 5
3.9945 ± 6.26E−02(+)
3.1640 ± 3.75E−02(+)
0.5787 ± 6.86E−01(−)
0.5882 ± 1.03E−02(−)
2.5026 ± 1.78E−01(+)
2.2542 ± 8.86E−01
10 10
2.9876 ± 1.21E−02(−)
2.4573 ± 1.74E−01(−)
0.7057 ± 2.53E−01(−)
0.7215 ± 1.92E−01(−)
2.1668 ± 1.78E−01(−)
4.3303 ± 5.72E−01
10 20
2.3249 ± 1.57E−01(−)
1.0143 ± 3.17E−01(−)
0.9393 ± 6.29E−01(−)
0.4564 ± 5.38E−02(−)
0.5644 ± 1.54E−01(−)
4.1683 ± 8.89E−01
DF7
10 5
0.8394 ± 2.69E−02(+)
0.9652 ± 1.10E−02(+)
0.9551 ± 1.61E−01(+)
0.6140 ± 3.47E−03(+)
0.9893 ± 9.01E−03(+)
0.3619 ± 3.18E−02
10 10
0.6344 ± 1.90E−02(+)
0.8020 ± 1.54E−02(+)
0.8125 ± 1.23E−01(+)
0.5708 ± 3.21E−03(+)
0.8095 ± 1.30E−02(+)
0.3399 ± 9.04E−03
10 20
0.7307 ± 1.07E−02(+)
0.9042 ± 1.61E−02(+)
0.9175 ± 7.31E−02(+)
0.7373 ± 2.77E−02(+)
0.9181 ± 5.14E−03(+)
0.4868 ± 2.14E−02
DF8
10 5
0.1815 ± 1.88E−02(−)
0.0181 ± 5.68E−04(−)
0.0199 ± 4.58E−02(−)
0.0233 ± 1.59E−03(+)
0.0170 ± 3.41E−04(−)
0.0214 ± 2.08E−03
10 10
0.2391 ± 1.04E−02(+)
0.0164 ± 1.10E−04(−)
0.0180 ± 2.69E−03(−)
0.0241 ± 7.07E−04(+)
0.0164 ± 2.95E−04(−)
0.0210 ± 1.96E−03
10 20
0.2152 ± 3.04E−04(=)
0.0175 ± 2.80E−04(−)
0.0183 ± 7.15E−02(−)
0.0218 ± 4.80E−04(=)
0.0167 ± 5.29E−04(−)
0.0211 ± 2.18E−03
DF9
10 5
0.3927 ± 1.91E−02(+)
0.2043 ± 6.63E−03(−)
1.5440 ± 8.52E−02(+)
0.2208 ± 1.24E−02(−)
0.1943 ± 1.58E−02(−)
0.2927 ± 4.36E−02
10 10
0.3752 ± 3.11E−02(+)
0.2049 ± 1.02E−02(+)
0.2175 ± 7.14E−02(+)
0.2576 ± 3.59E−03(+)
0.1841 ± 8.41E−03(+)
0.1760 ± 1.93E−02
10 20
0.4126 ± 6.92E−04(+)
0.2361 ± 9.26E−03(+)
1.8856 ± 7.41E−02(+)
0.3222 ± 8.14E−03(+)
0.2280 ± 1.86E−02(+)
0.1178 ± 7.17E−03
DF10
10 5
0.6483 ± 4.95E−02(+)
0.1016 ± 1.23E−03(−)
0.1228 ± 8.71E−02(−)
0.1599 ± 8.44E−03(+)
0.1042 ± 2.98E−03(−)
0.1742 ± 2.75E−02
10 10
0.5155 ± 1.01E−02(+)
0.1032 ± 1.58E−03(−)
0.1130 ± 1.44E−01(−)
0.1839 ± 1.01E−03(+)
0.0995 ± 2.72E−03(−)
0.1343 ± 3.11E−02
10 20
0.5963 ± 9.50E−02(+)
0.1219 ± 2.83E−03(−)
0.1375 ± 8.33E−02(−)
0.2505 ± 1.63E−02(−)
0.1190 ± 5.29E−03(−)
0.3433 ± 3.18E−02
DF11
10 5
0.0756 ± 8.67E−05(+)
0.0695 ± 1.34E−04(+)
0.0679 ± 8.41E−04(=)
0.0706 ± 4.99E−04(+)
0.0614 ± 1.54E−04(−)
0.0688 ± 2.49E−04
10 10
0.0781 ± 8.00E−04(+)
0.0690 ± 3.13E−04(=)
0.0661 ± 4.60E−04(−)
0.0708 ± 5.55E−05(+)
0.0616 ± 5.94E−04(−)
0.0685 ± 3.11E−04
10 20
0.0755 ± 1.06E−04(+)
0.0657 ± 3.69E−04(+)
0.0630 ± 9.36E−04(=)
0.0678 ± 1.77E−04(+)
0.0605 ± 3.59E−04(−)
0.0662 ± 3.75E−04
DF12
10 5
0.7430 ± 4.47E−03(+)
0.3918 ± 1.47E−03(−)
0.3222 ± 1.36E−02(−)
0.5096 ± 7.63E−03(−)
0.3402 ± 1.01E−02(−)
0.6563 ± 5.19E−02
10 10
0.5938 ± 4.12E−02(−)
0.4044 ± 2.60E−03(−)
0.2892 ± 1.51E−02(−)
0.5343 ± 6.63E−03(−)
0.3013 ± 1.17E−03(−)
0.6030 ± 1.05E−01
Table 2 demonstrates the superior performance of the proposed algorithm compared to that of the other five algorithms on the majority of the test instances. This suggests that our method is more effective in tracking the change in PS when the environment changes. One possible explanation for this superior performance is that our proposed method excels at capturing the changing pattern of the PS manifold, in contrast to the other five algorithms when the environment undergoes change. However, it is important to note that there is a small subset of test instances where the proposed method underperforms compared to that of the other algorithms. This may be attributed to the nature of the PS changes in these specific instances, which may not align with the conditions under which our proposed method is used. As given in “Proposed method”, the condition is that the translation and rotation change patterns of PS under adjacent environments are identical or similar, and the distortion degree of PS in the new environment is slight.
Compared with that of the other five algorithms, the proposed method achieves the best performance on all cases of DF1, DF2, DF3, DF5, DF7, DF9, DF13, and DF14 in terms of MIGD. These test problems encompass various change patterns of PS. In DF1, DF5, DF13, and DF14, the PS undergoes solely a change in location. In DF2, both translation and rotation changes occur. Nevertheless, DF3 and DF9 involve translation changes, along with rotation and minor distortion changes in the PS. These results indicate that our proposed method not only excels at tracking PS change due to translation but also successfully captures changes involving rotation and slight distortion when the environment experiences change. However, the proposed method falls short in performance compared to that of one or more of the other algorithms on all cases of DF4, DF6, DF8, DF10, DF11, and DF12. The vast majority of these test problems are characterized by nonlinear changes or small distortion changes in the PS as the environment changes. MOEA/D-SVR outperforms other algorithms on DF8, DF10, and DF11, potentially due to its inherent advantages in nonlinear prediction. TL outperforms the other algorithms on DF12, where the PF remains unchanged. This may be attributed to the proficiency of TL in dealing with DMOPs featuring static PF. Additionally, on DF6, the performance of our algorithm significantly deteriorates, which can potentially be ascribed to the input error stemming from the utilization of MOEA/D-DE with uniform weight vectors. It is worth noting that MOEA/D with uniform weight vectors proves unsuitable for addressing multiobjective problems (MOPs) characterized by a long-tailed PF, such as DF6. Consequently, the proposed prediction model is susceptible to significant input error, contributing to its inferior performance.
The performance evaluation presented in Table 3 demonstrates that the proposed PSCA algorithm consistently outperforms the other five algorithms on a majority of the 42 tested instances, as measured by MHV values. These numerical findings align closely with those reported in Table 2 for MIGD, reinforcing the reliability of the results.
To evaluate the performance of the algorithms following each environmental shift, we present the plots illustrating the average IGD values over 10 runs for both PSCA and the five compared algorithms at \( n_{t} = 10 \) and \( \tau _{t} = 10 \) in Fig. 4. Remarkably, the proposed algorithm consistently exhibits a smoother and lower curve than that of its counterparts on most benchmark test suites. This notable trend signifies PSCA’s superiority in effectively tracking a dynamic PS or PF with enhanced stability compared to the comparison algorithms. In addition, we offer a graphical representation of the tracking capabilities of the algorithms by depicting their obtained PF of DF1, DF3, DF5, and DF7 at \( n_{t} = 10 \) and \( \tau _{ t} = 10 \) over a randomly selected set of 20 time instants in Fig. 5. A careful examination of Fig. 5 reveals the good adaptability of PSCA to environmental changes, particularly evident in its performance on DF1 and DF5, surpassing the other algorithms.
2. Different Change Severity: To assess the performance of the proposed method and the other algorithms under varying change severity, we analyzed the MIGD and MHV values obtained by all methods on test problems with \(\tau _{t}=10\) and \(n_{t}=5,10,20\). The results are summarized in Tables 4 and 5. From Table 4, it becomes evident that the proposed algorithm outperforms the other five compared algorithms in terms of MIGD on 21 out of 42 test instances. PPS, DM, TL, KT, and SVR achieve the best performance in 0, 3, 5, 3, and 10 instances, respectively, out of the total 42 instances. This result indicates the superiority of the proposed algorithm over its counterparts on the benchmark test suite with varying change severity. Furthermore, for DF1, DF2, DF5, DF9, DF11, DF12, DF13, and DF14, the MIGD values obtained by the proposed algorithm decrease as the value of \(n_{t}\) increases. This trend suggests that as the change severity becomes more pronounced, the dynamic pattern of the problem becomes more complex. However, it is noteworthy that the proposed algorithm demonstrates relatively stable performance on the remaining benchmark test problems. Consequently, the proposed method exhibits the ability to effectively capture the changing regularities of the PS under dynamic environments with varying change severity.
Table 5 reveals the good performance of PSCA, surpassing five other algorithms across 22 out of 42 instances with respect to MHV values. Importantly, the numerical outcomes align closely with the MIGD values presented in Table 4. The MIGD value serves as a vital indicator of the comparative convergence exhibited by the solutions derived from all algorithms. Meanwhile, the MHV value offers valuable insights into the remarkable diversity and distribution characteristics of the algorithm. In other words, the proposed prediction model based on joint subspace and correlation alignment can effectively capture the rotation and small distortion change patterns in the PS manifold for predicting the PS under the new environment.
Table 5
Statistical results of MHV obtained by six algorithms on test problems with different change severity
Problem
\(\tau _{t},n_{t} \)
PPS
DM
TL
KT
SVR
PSCA
DF1
10 5
0.5590 ± 9.97E−03(+)
0.6406 ± 1.30E−03(+)
0.6313 ± 2.20E−03(+)
0.6153 ± 3.08E−01(+)
0.6456 ± 3.14E−03(−)
0.6414 ± 1.38E−03
10 10
0.5634 ± 8.32E−03(+)
0.6466 ± 5.02E−04(+)
0.6401 ± 3.23E−04(+)
0.6387 ± 2.86E−01(+)
0.6468 ± 3.72E−04(+)
0.6502 ± 5.36E−04
10 20
0.5124 ± 9.87E−03(+)
0.5897 ± 2.48E−04(+)
0.5874 ± 2.95E−04(+)
0.5856 ± 5.43E−04(+)
0.5883 ± 2.12E−04(+)
0.5929 ± 1.39E−04
DF2
10 5
0.7469 ± 5.57E−03(+)
0.8508 ± 7.53E−04(+)
0.8474 ± 1.25E−03(+)
0.8465 ± 7.67E−04(+)
0.8587 ± 2.24E−03(=)
0.8550 ± 1.68E−03
10 10
0.7467 ± 1.55E−02(+)
0.8578 ± 4.53E−04(+)
0.8579 ± 4.29E−04(+)
0.8534 ± 4.27E−01(+)
0.8603 ± 6.63E−04(+)
0.8620 ± 4.22E−04
10 20
0.7599 ± 8.04E−03(+)
0.8603 ± 4.23E−04(+)
0.8606 ± 4.30E−04(+)
0.8592 ± 4.96E−01(+)
0.8601 ± 4.50E−04(+)
0.8645 ± 3.15E−04
DF3
10 5
0.3957 ± 5.30E−03(+)
0.4626 ± 3.04E−03(+)
0.4846 ± 3.19E−03(+)
0.4375 ± 4.18E−03(+)
0.4952 ± 5.63E−04(+)
0.5459 ± 6.82E−03
10 10
0.3952 ± 3.33E−03(+)
0.4634 ± 4.18E−03(+)
0.5000 ± 2.29E−03(+)
0.4582 ± 5.42E−03(+)
0.4936 ± 1.23E−03(+)
0.5550 ± 4.12E−03
10 20
0.318 ± 6.62E−03(+)
0.3864 ± 4.78E−03(+)
0.4389 ± 1.92E−03(+)
0.3840 ± 1.04E−03(+)
0.4290 ± 4.05E−03(+)
0.4823 ± 1.70E−03
DF4
10 5
5.3270 ± 4.06E−03(+)
5.5837 ± 4.87E−03(=)
5.5869 ± 5.79E−03(=)
5.5784 ± 3.22E−03(+)
5.5854 ± 3.38E−04(=)
5.6029 ± 2.62E−03
10 10
5.2402 ± 8.14E−03(+)
5.7039 ± 3.70E−03(=)
5.7048 ± 2.85E−03(=)
5.7002 ± 2.88E−03(=)
5.7028 ± 3.95E−03(=)
5.7321 ± 9.72E−04
10 20
7.0379 ± 4.96E−03(+)
7.2803 ± 3.48E−03(=)
7.2879 ± 3.64E−03(=)
7.2778 ± 5.02E−03(=)
7.2726 ± 3.72E−03(=)
7.3111 ± 3.55E−03
DF5
10 5
0.6214 ± 2.32E−02(+)
0.6822 ± 1.88E−03(+)
0.6681 ± 2.40E−04(+)
0.6826 ± 1.66E−03(+)
0.6742 ± 2.30E−04(+)
0.6968 ± 1.34E−04
10 10
0.6131 ± 2.09E−02(+)
0.6779 ± 6.32E−03(+)
0.6493 ± 3.37E−04(+)
0.6832 ± 2.14E−03(+)
0.6717 ± 2.11E−03(+)
0.6985 ± 2.57E−04
10 20
0.6770 ± 2.57E−03(+)
0.6923 ± 1.07E−03(+)
0.6926 ± 3.46E−04(+)
0.6904 ± 1.45E−03(+)
0.6902 ± 4.65E−04(+)
0.6983 ± 1.15E−04
DF6
10 5
0.2451 ± 1.43E−02(−)
0.4012 ± 2.65E−02(−)
0.4397 ± 1.80E−02(−)
–
0.2644 ± 5.08E−02(−)
0.2037 ± 1.65E−01
10 10
0.2567 ± 2.48E−02(−)
0.5683 ± 1.07E−02(−)
0.4763 ± 2.76E−02(−)
0.4664 ± 2.69E−01(−)
0.2881 ± 1.21E−02(−)
0.1721 ± 9.48E−02
10 20
0.3777 ± 1.62E−02(−)
0.7621 ± 8.49E−03(−)
0.6050 ± 3.78E−02(−)
0.5367 ± 2.68E−01(−)
0.6390 ± 1.83E−02(−)
0.2185 ± 7.48E−02
DF7
10 5
2.3236 ± 1.00E−01(+)
3.2761 ± 5.19E−02(+)
3.1788 ± 1.60E−02(+)
–
3.3731 ± 1.13E−02(+)
2.7043 ± 1.87E−01
10 10
2.3704 ± 4.85E−02(+)
2.6135 ± 1.12E−02(+)
2.6031 ± 1.31E−02(+)
–
2.6018 ± 3.82E−02(+)
2.6347 ± 5.51E−02
10 20
1.7077 ± 1.10E−02(+)
1.7710 ± 9.60E−03(+)
1.7576 ± 8.79E−03(+)
–
1.7577 ± 2.27E−03(+)
1.8461 ± 2.44E−03
DF8
10 5
0.5320 ± 5.92E−02(+)
0.7440 ± 1.82E−05(=)
0.7427 ± 3.72E−04(=)
–
0.7443 ± 2.99E−04(=)
0.7430 ± 1.79E−04
10 10
0.5241 ± 3.26E−02(+)
0.7408 ± 6.60E−05(=)
0.7400 ± 3.70E−04(=)
0.7406 ± 4.28E−01(=)
0.7415 ± 1.44E−04(=)
0.7400 ± 1.47E−04
10 20
0.4917 ± 6.46E−02(+)
0.7459 ± 3.74E−05(=)
0.7457 ± 3.73E−05(=)
–
0.7467 ± 1.19E−04(=)
0.7452 ± 1.28E−04
DF9
10 5
0.5422 ± 2.96E−02(+)
0.6870 ± 2.21E−02(−)
0.5868 ± 3.33E−02(+)
0.6685 ± 2.42E−02(−)
0.7300 ± 9.18E−03(−)
0.6577 ± 2.44E−02
10 10
0.5315 ± 2.00E−02(+)
0.6942 ± 7.39E−03(+)
0.6639 ± 7.44E−02(+)
0.6568 ± 2.06E−02(+)
0.7103 ± 2.44E−03(+)
0.7801 ± 1.26E−02
10 20
0.4906 ± 3.28E−02(+)
0.6818 ± 1.34E−02(+)
0.5170 ± 3.31E−02(+)
0.6437 ± 2.20E−02(+)
0.6796 ± 1.88E−03(+)
0.8614 ± 1.18E−02
DF10
10 5
0.3856 ± 1.14E−01(+)
0.9453 ± 1.05E−03(−)
0.9329 ± 3.72E−03(−)
–
0.9469 ± 5.41E−03(−)
0.9114 ± 1.21E−02
10 10
0.4644 ± 1.88E−01(+)
0.9995 ± 2.72E−03(−)
0.9939 ± 4.98E−03(−)
–
1.0026 ± 7.11E−03(−)
0.8561 ± 2.52E−01
10 20
0.4042 ± 7.33E−02(+)
0.8315 ± 7.62E−04(−)
0.8339 ± 6.15E−04(−)
–
0.8346 ± 9.57E−04(−)
0.7731 ± 2.11E−02
DF11
10 5
0.9445 ± 2.93E−03(+)
0.9646 ± 7.29E−04(=)
0.9652 ± 4.82E−04(=)
0.9559 ± 5.52E−01(+)
0.9800 ± 9.66E−04(−)
0.9655 ± 1.09E−03
10 10
0.9404 ± 1.33E−03(+)
0.9798 ± 3.86E−04(=)
0.9837 ± 3.90E−04(=)
0.9709 ± 4.85E−01(+)
0.9915 ± 3.41E−04(−)
0.9789 ± 8.27E−04
10 20
0.9865 ± 2.06E−03(+)
1.0157 ± 6.88E−04(=)
1.0220 ± 5.08E−04(=)
1.0107 ± 5.05E−01(=)
1.0255 ± 2.60E−04(−)
1.0152 ± 5.17E−04
DF12
10 5
6.1865 ± 2.18E−01(+)
8.1700 ± 3.24E−03(−)
8.2592 ± 4.08E−03(−)
–
8.2577 ± 1.47E−03(−)
7.6587 ± 3.07E−01
10 10
7.0188 ± 1.73E−01(+)
8.6480 ± 3.39E−03(−)
8.7769 ± 4.32E−03(−)
–
8.7765 ± 2.80E−03(−)
8.3742 ± 1.97E−01
10 20
8.0164 ± 1.80E−01(+)
8.9078 ± 4.56E−03(−)
8.9851 ± 4.45E−03(−)
–
8.9668 ± 3.48E−03(−)
8.8341 ± 3.52E−02
DF13
10 5
3.0225 ± 2.68E−02(+)
3.1813 ± 2.78E−03(+)
3.1757 ± 1.59E−03(+)
3.1797 ± 3.94E−03(+)
3.1747 ± 9.35E−03(+)
3.2095 ± 3.49E−03
10 10
2.9881 ± 3.74E−02(+)
3.1669 ± 7.08E−03(+)
3.1619 ± 3.59E−03(+)
3.1716 ± 2.18E−03(+)
3.1738 ± 2.53E−03(+)
3.2097 ± 2.24E−03
10 20
3.0470 ± 4.39E−02(+)
3.2108 ± 3.82E−03(+)
3.2002 ± 4.61E−03(+)
3.2116 ± 2.53E−03(+)
3.2006 ± 1.52E−03(+)
3.2312 ± 8.98E−04
DF14
10 5
0.4293 ± 2.00E−04(+)
0.4352 ± 1.88E−04(=)
0.4351 ± 4.17E−04(=)
0.4353 ± 1.44E−04(=)
0.4348 ± 1.02E−04(=)
0.4359 ± 1.46E−04
10 10
0.4250 ± 2.53E−04(+)
0.4291 ± 1.46E−04(+)
0.4288 ± 2.15E−04(+)
0.4292 ± 1.07E−04(+)
0.4294 ± 2.12E−04(+)
0.4311 ± 1.45E−04
10 20
0.3869 ± 2.42E−04(−)
0.3681 ± 7.66E−05(=)
0.3678 ± 1.84E−05(=)
0.3678 ± 7.45E−05(=)
0.3676 ± 2.60E−04(=)
0.3684 ± 1.55E−04
Best
1
2
4
0
13
22
Effect of K on the proposed method
In this subsection, we analyze the effect of varying K on the proposed method using test instances DF1, DF2, DF10, DF12, DF13, and DF14. Figure 6 illustrates the changing trend of the MIGD values for these test instances as K increases. It is evident from Fig. 6 that the MIGD values for DF1, DF2, DF13, and DF14 exhibit minimal or slight fluctuations as K increases. Conversely, the MIGD values for DF10 and DF12 display significant fluctuations with increasing K. This disparity can be attributed to the distinct change patterns of the PS manifold on the various test functions. Among these test instances, DF1, DF13, and DF14 solely experience translation changes in the PS over time. Hence, the divided K submanifolds of the PS demonstrate consistent changes, resulting in minimal impact on the performance of the proposed method. In the case of DF2, the PS undergoes both translation and rotation changes, leading to a minimal or slight effect of K on the method’s performance. However, the PS of DF10 and DF12 display distortion changes in addition to translation and rotation changes, making the number of K significantly influential on the performance of the proposed algorithm.
Furthermore, we investigate the effect of change frequency on the sensitivity of K in the proposed method. We fix the parameter \( n_{t} \) to 10 and vary \( \tau _{t} \) to 5, 10, and 20. The corresponding figure demonstrates that the curve for \( \tau _{t}=5 \) exhibits the largest fluctuations, followed by the curve for \( \tau _{t}=20\), while the curve for \( \tau _{t}=20 \) exhibits the least fluctuation. This indicates that the proposed PSCA displays greater sensitivity to the number of clusters when the environment undergoes more frequent changes. Moreover, the performance of the algorithm improves as K increases within a certain range (from 1 to 4). This can be attributed to the fact that a small number of clusters hinders the ability of our method to capture the multiple directions of population evolution during environmental changes. As Kcontinues to increase beyond a certain threshold, the influence of K on the performance of the algorithm diminishes and becomes more subdued. This is likely because the algorithm already captures the majority of individuals’ movement directions in the population when the value of K reaches a certain level, rendering further increases in K less impactful.
Ablation study
Fig. 4
Average IGD values versus the time scale with \( n_{t} \) = 10 and \( \tau _{t}=10\)
Fig. 5
Obtained POFs on DF1, DF3, DF5 and DF7 at randomly selected 20 instants with \( n_{t} \) = 10 and \( \tau _{t}=10\)
×
×
In the proposed method, the subspace alignment strategy (SAS) and correlation alignment strategy (CAS) are combined. The experimental results demonstrate the promising performance achieved by this combination. However, it remains unclear what specific roles each strategy plays in the resolution of DMOPs. To ascertain the effectiveness of these two strategies, we conducted an ablation experiment.
As detailed in “Proposed method”, both SAS and CAS were employed to generate one-half of the quasi-initial population under the new environment. Therefore, in this ablation study, either SAS or CAS is used to produce half of the quasi-initial population, while the remaining half is randomly generated. This resulted in the establishment of two distinct variants: SAS-MOEA/D-DE and CAS-MOEA/D-DE. Then, the MIGD results obtained from these variants, along with the original algorithm PSCA and the baseline algorithm MOEA/D-DE on test problems with \( n_{t} =10\) and \( \tau _{t}=10 \), are presented in Table 6. The analysis of Table 6 reveals that both SAS and CAS exhibit performance improvement compared to the baseline algorithm MOEA/D-DE on 12 out of 14 test problems. Notably, the combination of SAS and SAS demonstrates even better performance improvements than that of any independent strategy for most test problems. This indicates that the proposed strategies effectively and efficiently capture the changing pattern of the PS under dynamic environments. However, it is worth noting that our strategies lose their effectiveness on DF6 and DF12. This may be attributed to their change type, which may not be suitable for the conditions under which these strategies are employed.
From Table 6, SAS outperforms CAS. This may be attributed to the fact that capturing the rotation change of PS is easier than capturing the distortion change of PS. Additionally, we observe that on DF1, DF2, DF5, DF13, and DF14, the performance of PSCA is only slightly better than that of SAS, implying that the CAS in the original algorithm PSCA has a minimal impact on the performance of the original algorithm on these test problems. This may be because CAS is designed to handle distortion changes, while the PS of these test problems only undergoes translation or rotation changes.
Time cost comparison
In addition to the indicators IGD and HV, which measure the approximation and coverage level of the PF obtained by the algorithm against the true PF, the runtime is also a crucial metric for evaluating the performance of the algorithm. Therefore, we have depicted the average runtime of each algorithm on all test instances in Fig. 7. From the graph, our algorithm exhibits only a slight increase in runtime compared to that of DM, while significantly outperforming the other algorithms, particularly SVR and TL. This indicates that our proposed algorithm does not incur a significant computational cost on the optimizer. Conversely, the remaining algorithms require more time to solve these problems, with TL demanding the highest computational resources.
Fig. 6
Obtained MIGD values versus cluster number with different change frequencies on six test problems with different change pattern of PS
Table 6
Average MIGD obtained by PSCA and its variants with \( n_{t} \) = 10 and \( \tau _{t}=10\)
Problem
\(\tau _{t},n_{t} \)
MOEA/D-DE
CAS
SAS
PSCA
DF1
10 10
0.0165
0.0107 (34.99%)
0.0073 (55.56%)
0.0071 (56.86%)
DF2
10 10
0.0175
0.0108 (38.41%)
0.0085 (51.4%)
0.0081 (53.81%)
DF3
10 10
0.1493
0.1283 (14.04%)
0.0540 (63.79%)
0.0369 (75.28%)
DF4
10 10
0.0838
0.0818 (2.44%)
0.0758 (9.63%)
0.0748 (10.79%)
DF5
10 10
0.0151
0.0095 (36.97%)
0.0062 (58.89%)
0.0060 (60.19%)
DF6
10 10
1.0945
1.3404 (\(-\)22.47%)
1.7035 (\(-\)55.64%)
4.3303 (\(-\)295.65%)
DF7
10 10
0.5712
0.4837 (15.31%)
0.3482 (39.04%)
0.3399 (40.49%)
DF8
10 10
0.0273
0.0227 (16.8%)
0.0215 (21.38%)
0.0210 (23.03%)
DF9
10 10
0.1787
0.1445 (19.12%)
0.1347 (24.6%)
0.1760 (1.49%)
DF10
10 10
0.1927
0.1887 (2.08%)
0.1909 (0.94%)
0.1343 (30.31%)
DF11
10 10
0.0718
0.0682 (4.95%)
0.0689 (3.96%)
0.0685 (4.54%)
DF12
10 10
0.4607
0.5072 (\(-\)10.08%)
0.4647 (\(-\)0.85%)
0.6030 (\(-\)30.88%)
DF13
10 10
0.4273
0.4188 (1.99%)
0.4140 (3.12%)
0.4131 (3.33%)
DF14
10 10
0.0384
0.0364 (5.32%)
0.0355 (7.78%)
0.0352 (8.44%)
×
Conclusion
In general, the evolution of the Pareto set (PS) involves changes in both its location and manifold, with the manifold changes further decomposed into rotation and distortion changes. However, existing methods primarily focus on the location changes of the PS, disregarding the manifold changes, leading to a reduction in algorithm performance. To address this limitation, we propose a prediction method based on joint subspace and correlation alignment. This method employs a subspace alignment strategy to capture the rotation change pattern in the PS manifold, generating half of the quasi-initial population in the new environment. The other half is obtained through the correlation alignment strategy, which captures distortion changes in the PS manifold. The optimal solutions are then extracted from this quasi-initial population based on nondominated relations and the degree of crowding, serving as the initial population in the new environment. Consequently, the proposed method effectively captures both rotation and distortion changes in the PS manifold, resulting in improved convergence of the responding population. Furthermore, the combination of these two strategies enhances the diversity of the responding population.
To evaluate the performance of our proposed method, we compare it with five state-of-the-art algorithms on fourteen benchmark problems. The results unequivocally demonstrate that our algorithm outperforms all other tested methods on the vast majority of problems exhibiting similar location and manifold change patterns of the PS in adjacent environments. This highlights the exceptional capability of our algorithm in accurately tracking the varying PS or Pareto front (PF) during environmental shifts. However, for problems with significantly different locations or manifold change patterns of the PS in adjacent environments, PSCA yields suboptimal results. This suggests that our algorithm excels in addressing DMOPs with similar change patterns of the PS in adjacent environments but may not be suitable for DMOPs with greatly different location or manifold change patterns of the PS in adjacent environments. Therefore, our future research interests lie in addressing DMOPs with stochastic changes in the PS. Furthermore, we are dedicated to applying PSCA to practical applications such as cascade hydraulic controller tuning problem for a parallel robot platform [62], disassembly line balancing [63] and workflow scheduling [64].
Fig. 7
Average run time for six algorithms on all test instances
×
Acknowledgements
This work was supported by the National Natural Science Foundation of China(no. 11761023), the Youth Science and Technology Talents Cultivating Object of Guizhou Province (no. QJHKY[2022]301), Key Laboratory of Evolutionary Artificial Intelligence in Guizhou (Qian Jiaoji [2022] no. 059) and the Key Talent Program in digital economy of Guizhou Province
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