Sensor networks serve as the perceptual core of industrial systems, such as the Internet of Things and smart grid, where the accuracy of individual measurements is pivotal to ensuring reliable state estimation. Nevertheless, factors, such as environmental interference and device aging, introduce measurement uncertainties to sensors, and the incompleteness of the information would propagate and accumulate across the network. Despite advances in evaluating the measurement uncertainty of individual sensors and incomplete assessments of the measurement status in sensor networks, network-level evaluation of measurement uncertainty remains an open yet practical problem. Thus, we propose a recursive framework for online evaluation of measurement uncertainties within sensor networks, demonstrated through a case study on voltage transformers in the smart grid. The proposed method incorporates a measurement model to capture interdevice dependencies and utilizes the Monte Carlo method to propagate parameter distributions considering the measurement uncertainties. Next, a recursive workflow for evaluating measurement uncertainties across the entire sensor network is proposed. Specifically, the calibrated nodes are selected based on the uncertainty propagation theory, and then the Bayesian fusion is applied to estimate the parameter distributions of extended nodes. Simulations conducted on the IEEE 30-node system, supplemented by real-world validation with power system data, indicate that the proposed method is capable of achieving high evaluation accuracy and stable uncertainty propagation.

For black-box expensive optimization problems, the limitation in the number of function evaluations prevents evolutionary algorithms (EAs) from achieving convergence. To date, surrogate models have emerged as the predominant technique to accelerate the convergence of EAs by offering numerous virtual evaluations. However, surrogate models are often criticized for their low accuracy in fitting complex objective functions and low generalizability in handling heterogeneous decision variables. In this work, we propose an alternative idea that abandons surrogate models, focusing instead on the customization of simple EAs for expensive optimization. We construct EAs by incorporating the translation, scale, and rotation invariant variation operators, which have robust generalization capabilities due to their space independent properties, and have outstanding convergence performance due to their learnable parameterized representation. Through a series of comparative experiments, this work answers two questions: Can an EA without surrogate models outperform those with surrogate models for expensive optimization? If so, can surrogate models further accelerate the convergence of such an EA?

Recently, some studies have shown that the popular large-scale multiobjective optimization problem (LSMOP) test suite cannot fairly test the performance of algorithms due to the specificity of its Pareto solution set position. Specifically, some large-scale multiobjective evolutionary algorithms (LSMOEAs) have achieved completely opposite (poor) performance on the GLSMOP test suite (the LSMOP test suite with generic Pareto solution sets). Since many real-world LSMOPs are computationally expensive, several surrogate-assisted LSMOEAs are developed and their effectiveness is verified on the LSMOP test suite. Motivated by the above, we aim to study the performance of those surrogate-assisted LSMOEAs on the GLSMOP test suite in this work. Firstly, we elaborate on the existing surrogate-assisted LSMOEAs for solving the expensive LSMOPs from different perspectives. Secondly, the basic formulation of the GLSMOP test suite and its difference from the original LSMOP test suite are shown. Finally, the performance of six surrogate-assisted LSMOEAs on the GLSMOP test suite is systematically tested. According to the experimental results, we give the current best algorithmic structure for handling expensive LSMOPs: using a decomposition-based framework, using differential evolution to search the original decision space, and fitting a scalarization function with the surrogate model. The proposed algorithmic structure is simple but is expected to guide the design of effective surrogate-assisted LSMOEAs.

Non-contact voltage measurement represents a novel but challenging area within modern smart grid systems, which mainly includes structure design for sensing the electric field and voltage estimation via the sensed information. Existing approaches regarded the two components as independent optimization tasks, where the former is low-dimensional but computationally expensive while the latter is high-dimensional but relatively cheap. In this study, we argue that the non-contact measurement task is an asynchronous optimization problem with heterogeneous features in both the decision and objective spaces. Specifically, the problem includes two subproblems to be optimized sequentially, and the result of the first subproblem affects the optimum of the second one. Consequently, we propose a new optimization paradigm called evolutionary asynchronous optimization to achieve the collaborative optimization of the two tasks, and accordingly develop a new asynchronous evolutionary algorithm to solve it. Experimental results on three laboratory cases validate that, the proposed algorithm exhibits superior performance over existing multi-stage, multi-objective, and bi-level evolutionary algorithms.

Accurate cell classification is crucial but expensive for large-scale single-cell RNA sequencing (scRNA-seq) analysis. Gene selection (GS) emerges as a pivotal technique in identifying gene subsets of scRNA-seq for classification accuracy improvement and gene scale reduction. Nevertheless, the rising scale of scRNA-seq data presents challenges to existing GS methods regarding performance and computational time. Thus, we propose a surrogate-assisted evolutionary algorithm for multiobjective GS to address these deficiencies. An innovative two-phase initialization method is proposed to select sparse solutions to provide preliminary insights into gene contributions. Then, a binary competitive swarm optimizer is proposed for effective global search, where a local search method is embedded to eliminate irrelevant genes for efficiency consideration. Additionally, a surrogate model is adopted to forecast classification accuracy efficiently and substitutes part of the computationally expensive classification process. Experiments are conducted on eight large-scale scRNA-seq datasets with more than 20 000 genes. The effectiveness of the proposed GS method for scRNA-seq cell classification compared with eight state-of-the-art methods is validated. Gene expression analysis results of selected genes further validated the significance of the genes selected by the proposed method in the classification of scRNA-seq data.

In recent decades, various surrogate-assisted evolutionary algorithms (SAEAs) have been proposed to solve computationally expensive multiobjective optimization problems (EMOPs). Nevertheless, designing an SAEA to handle high-dimensional EMOPs and balance convergence, diversity, and computational complexity remains challenging. Here, we propose a two-phase SAEA (TP-SAEA), which follows the idea of convergence first and diversity second, for solving high-dimensional EMOPs. In Phase I, a surrogate-assisted problem reformulation method is proposed to fast-track the Pareto optimal set in association with some reference solutions. Specifically, the high-dimensional EMOP is reformulated into an expensive single-objective one with low-dimensional decision space. Then, the surrogate-assisted optimization is utilized to obtain well-converged solutions. In Phase II, the high-dimensional EMOP is decomposed into two subproblems to explore subregions of the decision space that can effectively promote the diversity of the solutions. The two subproblems are optimized independently via surrogate-assisted optimization, aiming to push the population towards different regions of the Pareto optimal front. Experiments are conducted on EMOPs with 100 to 500 decision variables compared with four state-of-the-art SAEAs. The proposed TP-SAEA obtains well-converged and diverse solutions with only 509 real function evaluations. Moreover, its superiority is examined in six real-world instances with up to 12,000 decision variables.

Instrument transformers (ITs) are integral components of power systems characterized by a certain level of failure probability. Monitoring the errors of ITs and promptly replacing faulty ones is important for ensuring the secure operation of power systems. Existing online calibration methods exhibit specific problem formulation and optimization algorithm selection issues, rendering them inadequate to meet practical engineering requirements. To address the above deficiencies, we first formulate an optimization problem that aligns with the practical considerations of engineering. Notably, parameter normalization and objective integration are incorporated for engineering considerations. Next, we propose a hybrid meta-heuristic algorithm that balances convergence and diversity for handling variables in the complex number field and the nonlin-earity of the formulated optimization problem. This algorithm is expected to fill the gap between mathematical local search and the meta-heuristic global search for real-world applications, considering the accuracy and reliability of the optimization results simultaneously. Numerical validation results indicate the superiority of the proposed method in terms of convergence rate and assessment precision over existing algorithms.

With the development of computer-aided engineering, various surrogate-assisted evolutionary algorithms (SAEAs) have been developed to solve the involved computationally expensive multiobjective optimization problems (EMOPs). With the increasing complexity of EMOPs, the number of decision variables has increased from tens to hundreds or even thousands. “Curse of Dimensionality” caused by large-scale decision space poses a great challenge to current SAEAs, which require massive function evaluations (FEs). We propose an SAEA with reference vector guided variables selection, namely RVSPSO, for solving large-scale EMOPs. Specifically, the reference vector guided variable selection strategy is proposed to select critical variables for optimization, which associates reference solutions for enhancing convergence and maintaining diversity. Meanwhile, a reference guided particle swarm optimization is proposed as the optimizer, where the velocities of particles are updated according to the directions pointing from the particles to the reference solution. This update strategy aims to accelerate the convergence rate in large-scale decision space. Moreover, radial basis function networks are used as surrogate models for efficient optimization. Experiments are conducted on large-scale EMOPs with up to 2000 decision variables. Experiment results show the proposed RVSPSO can effectively solve large-scale EMOPs to obtain well-converged and diverse solutions with limited FEs, compared with five state-of-the-art SAEAs.

With the rising number of large-scale multiobjective optimization problems (LSMOPs) from academia and industries, some multiobjective evolutionary algorithms (MOEAs) with different decision variable handling strategies have been proposed. Decision variable analysis (DVA) is widely used in large-scale optimization, aiming at identifying the connection between each decision variable and the objectives, and grouping those interacting decision variables to reduce the complexity of LSMOPs. Despite their effectiveness, existing DVA techniques require the unbearable cost of function evaluations for solving LSMOPs. We propose a reformulation-based approach for efficient DVA to address this deficiency. Then a large-scale MOEA is proposed based on reformulated DVA, namely, LERD. Specifically, the DVA process is reformulated into an optimization problem with binary decision variables, aiming to approximate different grouping results. Afterwards, each group of decision variables is used for convergence-related or diversity-related optimization. The effectiveness and efficiency of the reformulation-based DVA are validated by replacing the corresponding DVA techniques in two large-scale MOEAs. Experiments in comparison with six state-of-the-art large-scale MOEAs on LSMOPs with up to 2000 decision variables have shown the promising performance of LERD.

Multipopulation methods are highly effective in solving dynamic optimization problems. Three factors affect this significantly: 1) the exclusion mechanisms to avoid the convergence to the same peak by multiple subpopulations; 2) the resource allocation mechanism that assigns the computational resources to the subpopulations; and 3) the control mechanisms to adaptively adjust the number of subpopulations by considering the number of optima and available computational resources. In the existing exclusion mechanisms, when the distance (i.e., the distance between their best found positions) between two subpopulations becomes less than a predefined threshold, the inferior one will be removed/reinitialized. However, this leads to incapability of algorithms in covering peaks/optima that are closer than the threshold. Moreover, despite the importance of resource allocation due to the limited available computational resources between environmental changes, it has not been well studied in the literature. Finally, the number of subpopulations should be adapted to the number of optima. However, in most existing adaptive multipopulation methods, there is no predefined upper bound for generating subpopulations. Consequently, in problems with large numbers of peaks, they can generate too many subpopulations sharing limited computational resources. In this article, a multipopulation framework is proposed to address the aforementioned issues by using three adaptive approaches: 1) subpopulation generation; 2) double-layer exclusion; and 3) computational resource allocation. The experimental results demonstrate the superiority of the proposed framework over several peer approaches in solving various benchmark problems.

It has been widely recognized that the efficient training of neural networks (NNs) is crucial to classification performance. While a series of gradient-based approaches have been extensively developed, they are criticized for the ease of trapping into local optima and sensitivity to hyperparameters. Due to the high robustness and wide applicability, evolutionary algorithms (EAs) have been regarded as a promising alternative for training NNs in recent years. However, EAs suffer from the curse of dimensionality and are inefficient in training deep NNs (DNNs). By inheriting the advantages of both the gradient-based approaches and EAs, this article proposes a gradient-guided evolutionary approach to train DNNs. The proposed approach suggests a novel genetic operator to optimize the weights in the search space, where the search direction is determined by the gradient of weights. Moreover, the network sparsity is considered in the proposed approach, which highly reduces the network complexity and alleviates overfitting. Experimental results on single-layer NNs, deep-layer NNs, recurrent NNs, and convolutional NNs (CNNs) demonstrate the effectiveness of the proposed approach. In short, this work not only introduces a novel approach for training DNNs but also enhances the performance of EAs in solving large-scale optimization problems.

Offspring generation plays an important role in evolutionary multiobjective optimization. However, generating promising candidate solutions effectively in high-dimensional spaces is particularly challenging. To address this issue, we propose an adaptive offspring generation method for large-scale multiobjective optimization. First, a preselection strategy is proposed to select a balanced parent population, and then these parent solutions are used to construct direction vectors in the decision spaces for reproducing promising offspring solutions. Specifically, two kinds of direction vectors are adaptively used to generate offspring solutions. The first kind takes advantage of the dominated solutions to generate offspring solutions toward the Pareto optimal set (PS) for convergence enhancement, while the other kind uses those nondominated solutions to spread the solutions over the PS for diversity maintenance. The proposed offspring generation method can be embedded in many existing multiobjective evolutionary algorithms (EAs) for large-scale multiobjective optimization. Experiments are conducted to reveal the mechanism of our proposed adaptive reproduction strategy and validate its effectiveness. Experimental results on some large-scale multiobjective optimization problems have demonstrated the competitive performance of our proposed algorithm in comparison with five state-of-the-art large-scale EAs.

Large-scale multiobjective optimization problems (LSMOPs) are challenging for existing approaches due to the complexity of objective functions and the massive volume of decision space. Some large-scale multiobjective evolutionary algorithms (LSMOEAs) have recently been proposed, which have shown their effectiveness in solving some benchmarks and real-world applications. They merely focus on handling the massive volume of decision space and ignore the complexity of LSMOPs in terms of objective functions. The complexity issue is also important since the complexity grows along with the increment in the number of decision variables. Our previous study proposed a framework to accelerate evolutionary large-scale multiobjective optimization via problem reformulation for handling large-scale decision variables. Here, we investigate the effectiveness of LSMOF combined with decomposition-based MOEA (MOEA/D), aiming to handle the complexity of LSMOPs in both the decision and objective spaces. Specifically, MOEA/D is embedded in LSMOF via two different strategies, and the proposed algorithm is tested on various benchmark LSMOPs. Experimental results indicate the encouraging performance improvement benefited from the solution of the complexity issue in large-scale multiobjective optimization.

Recent advancements in deep neural networks have made remarkable leap-forwards in dense image prediction. However, the issue of feature alignment remains as neglected by most existing approaches for simplicity. Direct pixel addition between upsampled and local features leads to feature maps with misaligned contexts that, in turn, translate to mis-classifications in prediction, especially on object boundaries. In this paper, we propose a feature alignment module that learns transformation offsets of pixels to contextually align upsampled higher-level features; and another feature selection module to emphasize the lower-level features with rich spatial details. We then integrate these two modules in a top-down pyramidal architecture and present the Feature-aligned Pyramid Network (FaPN). Extensive experimental evaluations on four dense prediction tasks and four datasets have demonstrated the efficacy of FaPN, yielding an overall improvement of 1.2 - 2.6 points in AP / mIoU over FPN when paired with Faster / Mask R-CNN. In particular, our FaPN achieves the state-of-the-art of 56.7% mIoU on ADE20K when integrated within Mask-Former. The code is available from https://github.com/EMI-Group/FaPN.

Recently, increasing works have been proposed to drive evolutionary algorithms using machine-learning models. Usually, the performance of such model-based evolutionary algorithms is highly dependent on the training qualities of the adopted models. Since it usually requires a certain amount of data (i.e., the candidate solutions generated by the algorithms) for model training, the performance deteriorates rapidly with the increase of the problem scales due to the curse of dimensionality. To address this issue, we propose a multiobjective evolutionary algorithm driven by the generative adversarial networks (GANs). At each generation of the proposed algorithm, the parent solutions are first classified into real and fake samples to train the GANs; then the offspring solutions are sampled by the trained GANs. Thanks to the powerful generative ability of the GANs, our proposed algorithm is capable of generating promising offspring solutions in high-dimensional decision space with limited training data. The proposed algorithm is tested on ten benchmark problems with up to 200 decision variables. The experimental results on these test problems demonstrate the effectiveness of the proposed algorithm.

Under certain smoothness assumptions, the Pareto set of a continuous multiobjective optimization problem is a piecewise continuous manifold in the decision space, which can be derived from the Karush-Kuhn-Tucker condition. Despite that a number of multiobjective evolutionary algorithms (MOEAs) have been proposed, their performance on multiobjective optimization problems with complicated Pareto sets (MOP-cPS) is still unsatisfying. In this article, we adopt the concept of manifold and propose a manifold learning-inspired mating strategy to enhance the diversity maintenance in MOEAs for solving MOP-cPS efficiently. In the proposed strategy, all of the individuals are first clustered into different manifolds according to their distribution in the objective space, and then the mating reproduction is restricted among individuals in the same manifold. Moreover, we embed the proposed mating strategy in three representative MOEAs and compare the embedded MOEAs with their original versions using the assortative genetic operators on a variety of MOP-cPS. The experimental results demonstrate the significant performance improvements benefitting from the proposed mating restriction strategy.

Constrained multiobjective optimization problems (CMOPs) widely exist in real-world applications, and they are challenging for conventional evolutionary algorithms (EAs) due to the existence of multiple constraints and objectives. When the number of objectives or decision variables is scaled up in CMOPs, the performance of EAs may degenerate dramatically and may fail to obtain any feasible solutions. To address this issue, we propose a paired offspring generation-based multiobjective EA for constrained large-scale optimization. The general idea is to emphasize the role of offspring generation in reproducing some promising feasible or useful infeasible offspring solutions. We first adopt a small set of reference vectors for constructing several subpopulations with a fixed number of neighborhood solutions. Then, a pairing strategy is adopted to determine some pairwise parent solutions for offspring generation. Consequently, the pairwise parent solutions, which could be infeasible, may guide the generation of well-converged solutions to cross the infeasible region(s) effectively. The proposed algorithm is evaluated on CMOPs with up to 1000 decision variables and ten objectives. Moreover, each component in the proposed algorithm is examined in terms of its effect on the overall algorithmic performance. Experimental results on a variety of existing and our tailored test problems demonstrate the effectiveness of the proposed algorithm in constrained large-scale multiobjective optimization.

Only a small number of function evaluations can be afforded in many real-world multiobjective optimization problems (MOPs) where the function evaluations are economically/computationally expensive. Such problems pose great challenges to most existing multiobjective evolutionary algorithms (EAs), which require a large number of function evaluations for optimization. Surrogate-assisted EAs (SAEAs) have been employed to solve expensive MOPs. Specifically, a certain number of expensive function evaluations are used to build computationally cheap surrogate models for assisting the optimization process without conducting expensive function evaluations. The infill sampling criteria in most existing SAEAs take all requirements on convergence, diversity, and model uncertainty into account, which is, however, not the most efficient in exploiting the limited computational budget. Thus, this article proposes a Kriging-assisted two-archive EA for expensive many-objective optimization. The proposed algorithm uses one influential point-insensitive model to approximate each objective function. Moreover, an adaptive infill criterion that identifies the most important requirement on convergence, diversity, or uncertainty is proposed to determine an appropriate sampling strategy for reevaluations using the expensive objective functions. The experimental results on a set of expensive multi/many-objective test problems have demonstrated its superiority over five state-of-the-art SAEAs.

Diversity preservation is a crucial technique in multiobjective evolutionary algorithms (MOEAs), which aims at evolving the population toward the Pareto front (PF) with a uniform distribution and a good extensity. In spite of many diversity preservation approaches in existing MOEAs, most of them encounter difficulties in tackling complex PFs. This article gives a detail introduction to existing diversity preservation approaches, as well as a revelation of the limitations of them. To address the limitations of existing diversity preservation approaches, this article proposes a multistage MOEA for better diversity performance. The proposed MOEA divides the optimization process into multiple stages according to the population in each generation, and updates the population by different steady-state selection schemes in different stages. According to the experimental results on 21 benchmark problems, the proposed MOEA exhibits better diversity performance than 11 existing MOEAs.

Due to the curse of dimensionality, two main issues remain challenging for applying evolutionary algorithms (EAs) to large-scale multiobjective optimization. The first issue is how to improve the efficiency of EAs for reducing computation cost. The second one is how to improve the diversity maintenance of EAs to avoid local optima. Nevertheless, these two issues are somehow conflicting with each other, and thus it is crucial to strike a balance between them in practice. Thereby, we propose an iterated problem reformulation based EA for large-scale multiobjective optimization, where the problem reformulation based method and the decomposition based method are used iteratively to address the aforementioned issues. The proposed method is compared with several state-of-the-art EAs on a variety of large-scale multiobjective optimization problems. Experimental results demonstrate the effectiveness of our proposed iterated method in large-scale multiobjective optimization.

A variety of evolutionary algorithms have been proposed for many-objective optimization in recent years. However, the difficulties in balancing the convergence and diversity of the population and selecting promising parents for offspring reproduction remain. In this paper, we propose a subregion division-based evolutionary algorithm with an effective mating selection strategy, termed SdEA, for many-objective optimization. In SdEA, a subregion division approach is proposed to divide the objective space into different subregions for balancing the diversity and convergence of the population. Besides, an effective mating selection strategy is proposed to enhance the diversity of the mating pool solutions, aimed at enhancing the selection probability of solutions in the sparse subregions. The proposed SdEA is compared with five state-of-the-art many-objective evolutionary algorithms on 23 test problems from DTLZ, WFG, and MaF test suites. Experimental results on these problems demonstrate that the proposed algorithm is competitive in solving many-objective problems. Furthermore, the proposed mating selection strategy is embedded in several evolutionary algorithms and experimental results demonstrate its effectiveness on improving the performance of the embedded algorithms.

In evolutionary multiobjective optimization, the Pareto front (PF) is approximated by using a set of representative candidate solutions with good convergence and diversity. However, most existing multiobjective evolutionary algorithms (MOEAs) have general difficulty in the approximation of PFs with complicated geometries. To address this issue, we propose a generic front modeling method for evolutionary multiobjective optimization, where the shape of the nondominated front is estimated by training a generalized simplex model. On the basis of the estimated front, we further develop an MOEA, where both the mating selection and environmental selection are driven by the approximate nondominated fronts modeled during the optimization process. For performance assessment, the proposed algorithm is compared with several state-of-the-art evolutionary algorithms on a wide range of benchmark problems with various types of PFs and different numbers of objectives. Experimental results demonstrate that the proposed algorithm performs consistently on a variety of multiobjective optimization problems.

Ratio error (RE) estimation of the voltage transformers (VTs) plays an important role in modern power delivery systems. Existing RE estimation methods mainly focus on periodical calibration but ignore the time-varying property. Consequently, it is difficult to efficiently estimate the state of the VTs in real time. To address this issue, we formulate a time-varying RE estimation (TREE) problem into a large-scale multiobjective optimization problem, where the multiple objectives and inequality constraints are formulated by statistical and physical rules extracted from the power delivery systems. Furthermore, a set of TREE problems from different substations is systematically formulated into a benchmark test suite for characterizing their different properties. The formulation of these TREE problems not only transfers an expensive RE estimation task to a relatively cheaper optimization problem but also promotes the research in large-scale multiobjective optimization by providing a real-world benchmark test suite with complex variable interactions and correlations to different objectives. To the best of our knowledge, this is the first time to formulate a real-world problem into a benchmark test suite for large-scale multiobjective optimization, and it is also the first work proposing to solve TREE problems via evolutionary multiobjective optimization.

Surrogate-assisted evolutionary algorithms have played an important role in expensive optimization where a small number of real-objective function evaluations are allowed. Usually, the surrogate models are used for the same purpose, e.g., to approximate the real-objective function or the aggregation fitness function. However, there is little work on surrogate-assisted optimization by model fusion, i.e., different surrogate models are fused for different purposes to improve the performance of the algorithm. In this work, we propose a surrogate-assisted approach by model fusion for solving expensive many-objective optimization problems, in which the Kriging assisted objective function approximation method is fused with the classifier assisted approach. The proposed algorithm is compared with some state-of-the-art surrogate-assisted algorithms on DTLZ problems and a real-world problem, and some encouraging results have been achieved by our proposed model fusion based approach.

Many real-world optimization problems are challenging because the evaluation of solutions is computationally expensive. As a result, the number of function evaluations is limited. Surrogate-assisted evolutionary algorithms are promising approaches to tackle this kind of problems. However, their performance highly depends on the number of objectives. Thus, they may not be suitable for many-objective optimization. This paper proposes a novel hybrid algorithm for computationally expensive many-objective optimization, called C-M-EA. The proposed approach combines two surrogate-assisted evolutionary algorithms during the search process. We compare the performance of the proposed approach with seven multi-objective evolutionary algorithms. Our experimental results show that our approach is competitive for solving computationally expensive many-objective optimization problems.

In this paper, we propose a framework to accelerate the computational efficiency of evolutionary algorithms on large-scale multiobjective optimization. The main idea is to track the Pareto optimal set (PS) directly via problem reformulation. To begin with, the algorithm obtains a set of reference directions in the decision space and associates them with a set of weight variables for locating the PS. Afterwards, the original large-scale multiobjective optimization problem is reformulated into a low-dimensional single-objective optimization problem. In the reformulated problem, the decision space is reconstructed by the weight variables and the objective space is reduced by an indicator function. Thanks to the low dimensionality of the weight variables and reduced objective space, a set of quasi-optimal solutions can be obtained efficiently. Finally, a multiobjective evolutionary algorithm is used to spread the quasi-optimal solutions over the approximate Pareto optimal front evenly. Experiments have been conducted on a variety of large-scale multiobjective problems with up to 5000 decision variables. Four different types of representative algorithms are embedded into the proposed framework and compared with their original versions, respectively. Furthermore, the proposed framework has been compared with two state-of-the-art algorithms for large-scale multiobjective optimization. The experimental results have demonstrated the significant improvement benefited from the framework in terms of its performance and computational efficiency in large-scale multiobjective optimization.

Surrogate-assisted evolutionary algorithms (SAEAs) have been developed mainly for solving expensive optimization problems where only a small number of real fitness evaluations are allowed. Most existing SAEAs are designed for solving low-dimensional single or multiobjective optimization problems, which are not well suited for many-objective optimization. This paper proposes a surrogate-assisted many-objective evolutionary algorithm that uses an artificial neural network to predict the dominance relationship between candidate solutions and reference solutions instead of approximating the objective values separately. The uncertainty information in prediction is taken into account together with the dominance relationship to select promising solutions to be evaluated using the real objective functions. Our simulation results demonstrate that the proposed algorithm outperforms the state-of-the-art evolutionary algorithms on a set of many-objective optimization test problems.

In this paper, we propose a sampling approach of reference points used for performance metrics of multi-objective evolutionary algorithms. Traditional reference point sampling methods, such as the Das and Dennis method, usually sample the reference points via a set of uniformly distributed weight vectors generated on an ideal hyper-plane in objective space, which however often ignore the geometric shape of a specific Pareto front. Therefore, we propose a novel reference point sampling approach by taking the specific shape of the Pareto optimal front to be tackled into account for measuring the performance of multi-objective evolutionary algorithms. The performance of the proposed reference point sampling method against the other two state-of-the-art sampling methods is tested on six test instances in various conditions, which clearly demonstrate the effectiveness and superiority of the proposed sampling method.