Aligning large language models with multiple human expectations and values is crucial for ensuring that they adequately serve a variety of user needs. To this end, offline multiobjective alignment algorithms such as the Rewards-in-Context algorithm have shown strong performance and efficiency. However, inappropriate preference representations and training with imbalanced reward scores limit the performance of such algorithms. In this work, we introduce ParetoHqD that addresses the above issues by representing human preferences as preference directions in the objective space and regarding data near the Pareto front as ''high-quality'' data. For each preference, ParetoHqD follows a two-stage supervised fine-tuning process, where each stage uses an individual Pareto high-quality training set that best matches its preference direction. The experimental results have demonstrated the superiority of ParetoHqD over five baselines on two multiobjective alignment tasks.

Model merging combines expert models for multitask performance but faces challenges from parameter interference. This has sparked recent interest in controllable model merging, giving users the ability to explicitly balance performance trade-offs. Existing approaches employ a compile-then-query paradigm, performing a costly offline multi-objective optimization to enable fast, preference-aware model generation. This offline stage typically involves iterative search or dedicated training, with complexity that grows exponentially with the number of tasks. To overcome these limitations, we shift the perspective from parameter-space optimization to a direct correction of the model's final representation. Our approach models this correction as an optimal linear transformation, yielding a closed-form solution that replaces the entire offline optimization process with a single-step, architecture-agnostic computation. This solution directly incorporates user preferences, allowing a Pareto-optimal model to be generated on-the-fly with complexity that scales linearly with the number of tasks. Experimental results show our method generates a superior Pareto front with more precise preference alignment and drastically reduced computational cost.

Automatic design of intelligent robots plays a central role and has been a trending topic in embodied intelligence. A promising approach is the co-design framework, wherein evolutionary algorithms (EAs) are employed to optimize the robot’s morphology while reinforcement learning algorithms are utilized to refine its control strategies. However, the discrete morphology design space introduces significant challenges for EAs, to efficiently identify optimal morphologies. In this work, we propose a novel morphology optimization method driven by a classifier-guided diffusion model to enhance the search efficiency of EAs. Prior to the design process, a universal diffusion model is trained using a set of randomly sampled feasible morphologies, prompting that the generated structures satisfy physical constraints. In each iteration of the EA for morphology optimization, a classifier is trained using previously evaluated morphologies and is then used to condition the diffusion model to generate quality solutions. Subsequently, the generated morphology is further refined based on the voxel distribution to incorporate features from the current high-quality morphology. Extensive experiments on a large-scale benchmark for co-designing the morphology and control of voxel-based soft robots demonstrate that our method significantly improves search efficiency in the morphological design space, outperforming both traditional EAs and surrogate-assisted EAs.

The automatic design of soft robots is an intertwined process of evolving morphology and learning control. As reinforcement learning is repeatedly used to learn the control policy for each candidate robot design, the design process becomes time-consuming. So far, the common design paradigm in robotics has been based on a single task. In fact, there is control similarity between different tasks. Learning a controller with combinatorial generalization capabilities across a variety of tasks can significantly reduce the computational cost of the design process. To this end, we propose a cross-task collaborative evolutionary algorithm that constructs a universal controller capable of solving a group of tasks simultaneously. Instead of “one robot, one controller, one task" paradigm, the proposed universal controller is to learn a control policy, which can generalize to unseen morphologies. After the controller learning on easy tasks, the universal controller can be further transferred to new hard tasks. Furthermore, the knowledge transfer is incorporated in the search strategy to enhance the performance of the universal controller. The experimental results on 13 test tasks demonstrate that the proposed algorithm outperforms the SOTA design algorithms on 8 of them. Compared to these algorithms, the proposed algorithm reduces the computational cost by 55% while achieving comparable performance, particularly for unseen hard tasks.

This article proposes a method to generate accurate and concise analytic equations for device compact modeling using Physics-Assisted Kolmogorov–Arnold Networks (PKAN). The equations are directly extracted from the trained neural network architecture. PKAN uses variable activation functions informed by prior physical knowledge to model device behaviors. Similarity constraints map these trained activation functions to mathematical symbols. Sparsification techniques simplify the network structure, producing concise and explicit equations. This article also presents four approaches for physics-assisted device modeling using PKAN: (1) generating entire continuous equations without human intervention, (2) applying correlation factors to existing models without requiring knowledge of internal physical mechanisms, (3) revising specific parts of existing models, and (4) automatically extending existing models. Experimental results show that PKAN demonstrates significant accuracy improvements, achieving error reductions of 91.8%, 91.5%, 66.2%, and 83.7% for corresponding experiments, respectively. These findings demonstrate PKAN’s potential for various device modeling applications. By combining the precision of neural networks with the clarity of symbolic representation, PKAN offers a powerful tool for device modeling applications.

The core challenge of high-dimensional and expensive black-box optimization (BBO) is how to obtain better performance faster with little function evaluation cost. The essence of the problem is how to design an efficient optimization strategy tailored to the target task. This paper designs a powerful optimization framework to automatically learn the optimization strategies from the target or cheap surrogate task without human intervention. However, current methods are weak for this due to poor representation of optimization strategy. To achieve this, 1) drawing on the mechanism of genetic algorithm, we propose a deep neural network framework called B2Opt, which has a stronger representation of optimization strategies based on survival of the fittest; 2) B2Opt can utilize the cheap surrogate functions of the target task to guide the design of the efficient optimization strategies. Compared to the state-of-the-art BBO baselines, B2Opt can achieve multiple orders of magnitude performance improvement with less function evaluation cost.

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.

High-fidelity electromagnetic simulation-driven optimization are crucial in modern antenna design. However, many antenna optimization models involve multiple expensive constraints, which can be formulated as expensive constrained optimization problems (ECOPs). Currently, surrogate-assisted evolutionary algorithms are widely used to solve ECOPs. However, existing methods face significant challenges in addressing errors in constraint surrogate models and the strong conflicts among constraints and the objective, making it difficult to find feasible solutions with optimal objective value within a limited number of simulations. We propose a parallel surrogate-assisted multiple penalties search method for simulation-driven antenna design problems with expensive conflicting constrains. In the proposed method, a multi-penalty function search mechanism is designed, followed by an adaptive parallel sampling approach to collect multiple samples for the parallel simulation. The experimental results applied to the design of the three-layer filtering antenna demonstrate the effectiveness and great potential of the proposed method in addressing simulation-driven antenna design problems with expensive conflicting constrains.

Zero-shot optimization aims to achieve both generalization and performance gains on solving previously unseen black-box optimization problems over SOTA methods without task-specific tuning. Pre-trained optimization models (POMs) address this challenge by learning a general mapping from task features to optimization strategies, enabling direct deployment on new tasks. In this paper, we identify three essential components that determine the effectiveness of POMs: (1) task feature modeling, which captures structural properties of optimization problems; (2) optimization strategy representation, which defines how new candidate solutions are generated; and (3) the feature-to-strategy mapping mechanism learned during pre-training. However, existing POMs often suffer from weak feature representations, rigid strategy modeling, and unstable training. To address these limitations, we propose EPOM, an enhanced framework for pre-trained optimization. EPOM enriches task representations using a cross-attention-based tokenizer, improves strategy diversity through deformable attention, and stabilizes training by replacing non-differentiable operations with a differentiable crossover mechanism. Together, these enhancements yield better generalization, faster convergence, and more reliable performance in zero-shot black-box optimization.

Black-box adversarial attacks can be categorized into transfer-based and query-based attacks. The former usually has poor transfer performance due to the mismatch between the architectures of models, while the query-based attacks require massive queries and high dimensional optimization variables. In order to solve the above problems, we propose a novel attack framework integrating the advantages of transfer- and query-based attacks, where the framework is divided into two phases: training the adversarial generator and executing the black-box attacks. In the first stage, a generator is trained by the adversarial loss function so that it can output adversarial perturbation, where the latent variables are designed as the input of the generator to reduce the dimension of the optimization variables. In the second stage, based on the trained generator, we further employ a particle swarm optimization algorithm to optimize the latent variables so that the generator can output the perturbation that can achieve a successful attack. Extensive experiments are performed on the ImageNet dataset, and the results demonstrate that the proposed framework can obtain better attack performance compared with a number of the state-of-the-art black-box adversarial attack methods. In addition, we show the flexibility of the proposed framework by extending the experiment for few-pixel attacks.

Large-scale multiobjective binary optimization problems (MBOPs) often occur in real-world applications, where the function evaluation can only be performed through computationally expensive simulations, which renders standard exact and heuristic methods ineffective. Aggregation-based surrogate-assisted multiobjective evolutionary algorithms have been developed and shown to be promising for solving such problems. They define a set of uniformly distributed weight vectors as search directions, on which all search resources are placed to perform the evolution. However, the Pareto fronts of large-scale MBOPs are discrete and nonuniform. As a result, many weight vectors are useless, thus a lot of search resources are wasted. To address this challenge, we propose a surrogate-assisted neighborhood search (SANS) for expensive large-scale multiobjective binary optimization. SANS uses only a few weight vectors to save search resources while maintaining an adequate diversity. To further utilize the limited search resources, a Q-learning-based method is designed to dynamically allocate search resources to weight vectors. Furthermore, a surrogate-assisted variable neighborhood search is developed to speed up the search without getting trapped in a local optimum prematurely. To robustly and reliably predict the quality of the found solutions, global and local surrogate models are trained by different training samples and then work collaboratively. The experimental results have demonstrated the superiority of SANS over seven state-of-the-art algorithms on the MBOPs with up to 1000 decision variables using only 500 real solution evaluations.

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.

In recent years, the research on large-scale mul-tiobjective optimization has attracted much attention. Many competitive large-scale multiobjective evolutionary algorithms have been proposed. Usually, their performance is evaluated on the widely used large-scale multiobjective test suite (i.e. LSMOP test suite). Those algorithms often exhibit a strong convergence capability on the instances of LSMOP test suite. The purpose of this study is to show our concern that the development of algorithms may be over specialized for the LSMOP test suite. We first explain some issues in the original LSMOP test suite. Then, we propose a general LSMOP test suite (termed GLSMOP), in which the Pareto set has a more general structure in the decision space. Experimental results on two test suites suggest that the performance of some large-scale multiobjective evolutionary algorithms will be deteriorated rapidly by the change of Pareto set. It also reveals the good performance of MOEAID-DE on large-scale multiobjective optimization problems.

Patch-based attacks are a major black-box attack paradigm, where there is no limit to the intensity of the perturbation. The existing patch-based attack methods focus on obtaining the optimal position, shape, and pixel values against adversarial patches, however, the generated patch looks conspicuous and makes it easy to attract people's attention. Quick response(QR) code has been widely used in various fields, such as image copyright protection, stored image information. Further, it does not get noticed when a QR code is attached to the image. Therefore, we propose a deceptive texture-based black-box adversarial attack method to address the above problem. Specifically, we use the QR code pattern as the basis of the adversarial patches. Then, we model the adversarial attack as a discrete optimization problem, where the optimization variables are designed as the center coordinates of the patch pasting locations and the pixel values. Further, an upsampling technique is introduced to reduce the dimension of the optimization variables. Finally, genetic algorithm is employed as the optimizer to obtain the optimal parameter of the patch. In order to verify the effectiveness of the proposed method, we compare a number of the state-of-the-art patch-based attack methods on the ImageNet dataset, and the experimental results show that the proposed method can effectively generate deceptive adversarial examples in both digital and physical space and obtain the best attack performance, especially for the defense models.

Many real-world optimization problems involve mixed variables, multiple conflicting objectives, and computation-ally expensive evaluations. Such problems are called expensive mixed-variable multi-objective optimization problems (EMV-MOPs). Solving EMVMOPs is challenging due to a complex search space involving mixed variables, balancing conflicts among multiple objectives, and a limited number of function evaluations. In this work, we propose an infill criterion ensemble in surrogate-assisted multi-objective evolutionary algorithm, which primarily consists of two core components considering the convergence to non-dominated front, model uncertainty, and diversity. We use two indicators, ensemble through an adaptive weighted sum, to play a crucial role in enhancing diversity, while local search with hybrid operators contributes to improving local convergence. The experimental results show that our algorithm is competitive compared to other algorithms on benchmark problems.

In practical applications, combinatorial optimization problems demonstrate intrinsic complexity, predominantly characterized by discrete decision variables and fitness evaluations that are both expensive and subject to noise. Surrogate-assisted evolutionary algorithms (SAEAs) are commonly used to solve expensive optimization problems in which expensive fitness evaluations are replaced by computationally cheaper surrogate models. The quality and quantity of training data are two crucial factors affecting surrogate models' accuracy, especially in noisy environments. Implicit and explicit averaging are two straightforward and effective noise-tolerant techniques, both entirely applicable to combinatorial optimization problems. In scenarios where fitness evaluations are subject to noise, and the allotted number of evaluations is constrained, implicit averaging tends to yield a considerable quantity of training data with diminished quality, whereas explicit averaging exhibits the opposite trend. This paper discusses which of these two noise-tolerant techniques is more suitable for embedding into SAEAs. The results of six multidimensional knapsack problems show that explicit averaging is a good choice, regardless of whether the noise type is additive or multiplicative.

Evolutionary multitasking optimization (EMTO) has attracted much attention in the community of evolutionary computation, which solves multiple tasks simultaneously by exchanging information between tasks. Multitasking traveling salesman problem (MTSP) is one of the most important combinatorial optimization problems in EMTO. However, redundant encoding and inefficient probabilistic transfer mechanisms used in most existing works may lead to negative transfer. In this paper, a new multi-population evolutionary algorithm via seed transfer (MPEA-ST) is proposed for MTSP. Firstly, combining heuristics with EMTO, a new seed encoding strategy, and seed growth mechanism are proposed to overcome redundant coding and suppress negative transfer. Moreover, a new seed selection mechanism and transfer strategy are designed to select seeds with knowledge. Finally, a new dataset construction method is developed to address the lack of MTSP benchmarks with different similarities. Experimental results show the superiority of the proposed MPEA-ST compared to other state-of-the-art methods on synthetic datasets and real-world datasets.

This paper presents an interactive platform to interpret multi-objective evolutionary algorithms. Sokoban level generation is selected as a showcase for its widespread use in procedural content generation. By balancing the emptiness and spatial diversity of Sokoban levels, we illustrate the improved two-archive algorithm, Two_Arch2, a well-known multi-objective evolutionary algorithm. Our web-based platform integrates Two_Arch2 into an interface that visually and interactively demonstrates the evolutionary process in real-time. Designed to bridge theoretical optimisation strategies with practical game generation applications, the interface is also accessible to both researchers and beginners to multi-objective evolutionary algorithms or procedural content generation on a website. Through dynamic visualisations and interactive gameplay demonstrations, this web-based platform also has potential as an educational tool.

Zero-shot optimization involves optimizing a target task that was not seen during training, aiming to provide the optimal solution without or with minimal adjustments to the optimizer. It is crucial to ensure reliable and robust performance in various applications. Current optimizers often struggle with zero-shot optimization and require intricate hyperparameter tuning to adapt to new tasks. To address this, we propose a Pretrained Optimization Model (POM) that leverages knowledge gained from optimizing diverse tasks, offering efficient solutions to zero-shot optimization through direct application or fine-tuning with few-shot samples. Evaluation on the BBOB benchmark and two robot control tasks demonstrates that POM outperforms state-of-the-art black-box optimization methods, especially for high-dimensional tasks. Fine-tuning POM with a small number of samples and budget yields significant performance improvements. Moreover, POM demonstrates robust generalization across diverse task distributions, dimensions, population sizes, and optimization horizons. For code implementation, see https://github.com/ninja-wm/POM/.

In the realm of daily services, the deployment of deep neural networks underscores the paramount importance of their reliability. However, the vulnerability of these networks to adversarial attacks, primarily evasion-based, poses a concerning threat to their functionality. Common methods for enhancing robustness involve heavy adversarial training or leveraging learned knowledge from clean data, both necessitating substantial computational resources. This inherent time-intensive nature severely limits the agility of large foundational models to swiftly counter adversarial perturbations. To address this challenge, this paper focuses on the \textbf{Ra}pid \textbf{P}lug-\textbf{i}n \textbf{D}efender (\textbf{RaPiD}) problem, aiming to rapidly counter adversarial perturbations without altering the deployed model. Drawing inspiration from the generalization and the universal computation ability of pre-trained transformer models, we propose a novel method termed \textbf{CeTaD} (\textbf{C}onsidering Pr\textbf{e}-trained \textbf{T}ransformers \textbf{a}s \textbf{D}efenders) for RaPiD, optimized for efficient computation. \textbf{CeTaD} strategically fine-tunes the normalization layer parameters within the defender using a limited set of clean and adversarial examples. Our evaluation centers on assessing \textbf{CeTaD}'s effectiveness, transferability, and the impact of different components in scenarios involving one-shot adversarial examples. The proposed method is capable of rapidly adapting to various attacks and different application scenarios without altering the target model and clean training data. We also explore the influence of varying training data conditions on \textbf{CeTaD}'s performance. Notably, \textbf{CeTaD} exhibits adaptability across differentiable service models and proves the potential of continuous learning.

Hypervolume (HV)-based evolutionary algorithms have been widely used to handle many-objective optimization problems. In such algorithms, HV-based environmental selection (HVES), which aims at selecting a subpopulation with the maximal HV from the current population, plays a crucial role in guiding evolution. However, the computation time of HV increases exponentially with the number of objectives, making the HVES task an expensive optimization problem. In this article, we propose an efficient surrogate-assisted greedy inclusion algorithm to deal with computationally expensive HVES tasks. It uses a lightweight surrogate model, radial basis function network, to replace the most time-consuming calculations. In addition, an$L_{1}$-norm distance-based filter is performed as a preselection operator to reduce the search space and avoid some unnecessary calculations. Considering the inevitable approximation errors of surrogate models, we also design an online sampling strategy to enhance the reliability of selected solutions. The proposed algorithm is tested on two types of datasets and compared with six state-of-the-art greedy algorithms. Experimental results show that the proposed algorithm performs excellently on most datasets.

The automatic design of soft robots characterizes as jointly optimizing structure and control. As reinforcement learning is gradually used to optimize control, the time-consuming controller training makes soft robots design an expensive optimization problem. Although surrogate-assisted evolutionary algorithms have made a remarkable achievement in dealing with expensive optimization problems, they typically suffer from challenges in constructing accurate surrogate models due to the complex mapping among structure, control, and task performance. Therefore, we propose an action inheritance-based evolutionary algorithm to accelerate the design process. Instead of training a controller, the proposed algorithm uses inherited actions to control a candidate design to complete a task and obtain its approximated performance. Inherited actions are near-optimal control policies that are partially or entirely inherited from optimized control actions of a real evaluated robot design. The action inheritance plays the role of surrogate models where its input is the structure and output is the near-optimal control actions. We also propose a random perturbation operation to estimate the error introduced by inherited control actions. The effectiveness of our proposed method is validated by evaluating it on a wide range of tasks, including locomotion and manipulation. Experimental results show that our algorithm is better than the other three state-of-the-art algorithms on most tasks when only a limited computational budget is available. Compared with the algorithm without surrogate models, our algorithm saves about half the computing cost.

Network structure evolves with time in the real world, and the discovery of changing communities in dynamic networks is an important research topic that poses challenging tasks. Most existing methods assume that no significant change occurs; namely, the difference between adjacent snapshots is slight. However, great change exists in the real world usually. The great change in the network will result in the community detection algorithms are difficulty obtaining valuable information from the previous snapshot, leading to negative transfer for the next time steps. This article focuses on dynamic community detection with substantial changes by integrating higher order knowledge from the previous snapshots to aid the subsequent snapshots. Moreover, to improve search efficiency, a higher order knowledge transfer strategy is designed to determine first-order and higher order knowledge by detecting the similarity of the adjacency matrix of snapshots. In this way, our proposal can keep the advantages of previous community detection results and transfer them to the next task. We conduct the experiments on four real-world networks, including the networks with great or minor changes. Experimental results in the low-similarity datasets demonstrate that higher order knowledge is more valuable than first-order knowledge when the network changes significantly and keeps the advantage even if handling the high-similarity datasets. Our proposal can also guide other dynamic optimization problems with great changes.

Multi-objective feature selection aims to find a set of feature subsets that achieves a trade-off between two objectives, i.e., reducing the number of selected features and improving the classification performance. However, these two objectives might not be always conflicting during the optimization process and have varying difficulties in optimization. Such characteristics pose a great challenge to existing multi-objective evolutionary approaches, which often treat two objectives equally. Specifically, a large number of feature subsets with few features may appear in the population and compete for survival opportunities with promising feature subsets located in not fully explored regions, leading to poor performance. To this end, we propose a two-archive evolutionary feature selection algorithm for multi-objective feature selection. In the proposed method, all individuals are equally allocated into two independent archives. A two-archive based solution generation strategy is proposed, where a dynamic dimensionality reduction operator is used to exploit small features subsets while a diversity-based mutation operator is utilized to find feature subsets with better classification performance. Moreover, a novel environmental selection scheme is proposed, which aims to improve the survival probability of promising feature subsets by providing different selection environments. Experimental results on 23 datasets demonstrate that the proposed algorithm is superior to the other five state-of-the-art algorithms.

In dealing with expensive constrained multi-objective optimization problems using surrogate-assisted evolutionary algorithms, it is a great challenge to reduce the negative impact caused by the approximate errors of surrogate models for constraints. To address this issue, we propose a Kriging-assisted evolutionary algorithm with two search modes to adaptively reduce the utilization frequency of surrogate models for constraints. To be more specific, an adaptively switching strategy analyzing the correlation between the objective optimization direction and constraint satisfaction direction is designed to determine whether to build the constraint surrogate models to assist the current evolutionary search. Accordingly, the proposed algorithm contains two search modes: 1) unconstrained surrogate-assisted search mode and 2) constrained surrogate-assisted search mode. In the first search mode, an existing surrogate-assisted evolutionary algorithm without considering constraint is introduced, which rapidly drives the population to move to the feasible region(s) while avoiding the negative effects of the constraint surrogate models. In the second search mode, a novel Kriging-assisted constrained multi-objective optimization algorithm is designed for locating constrained Pareto front in the feasible region. In addition, a data selection strategy is proposed to improve the efficiency and quality of surrogate models for constraint functions. The proposed method has been tested on numerous instances from three popular benchmark test suites. The experimental results demonstrate that the performance of the proposed algorithm outperforms other state-of-the-art methods.

Recently, a number of evolutionary algorithms (EAs) have been proposed for large-scale multiobjective optimization. Among them, due to the high efficiency, the problem reformulation-based large-scale multiobjective optimization framework (LSMOF) has shown to be promising. By associating the weight variables with a set of specific bi-directional vectors representing search directions, LSMOF reformulates the original large-scale multiobjective optimization problem (LSMOP) into a low-dimensional single-objective optimization problem (SOP). For the reformulated SOP, the weight variables are as the decision vector and the hypervolume is as the objective. In many recently proposed competitive EAs for large-scale multiobjective optimization, the promising search directions are also specified similar to the bi-directional vectors of LSMOF. The aim of this paper is to point out some challenges in the construction of bi-directional vectors in LSMOF. We first verify that the lower and upper boundary points from which the bi-directional vectors are generated are crucial to the good performance of LSMOF on the LSMOP test suite. Then, for demonstrating how LSMOF performs when the Pareto set (PS) is changed, a new test suite is designed. The experimental results show that the performance of LSMOF will deteriorate rapidly on the problems with the new PS.

Differentiable architecture search is now one of mainstream methods to design the structure of neural networks. It makes the neural architecture search efficient through parameter sharing and differentiable search. However, there are still many significant challenges for designing the optimal architecture, including the phenomenon of skip connection aggregation, excessive memory usage, and large discretization errors. To address these issues, we propose a novel approach called Gate-DARTS where a sparse gating network is introduced to route each input sample to the best$k$operators. This can reduce the memory requirements of the algorithm and break the residual structure. For the issue of discretization error, we then propose a auxiliary loss to enlarge the difference between different operators. We conduct comprehensive experiments on DARTS-like search space, and Gate-DARTS achieves 97.45% test accuracy on CIFAR10 with 0.23 GPU-days, 83.82% on CIFAR100 with 0.28 GPU-days. Our code has been made available at https://github.com/HandingWangXDGroup/Gate-DARTS.

Adversarial training (AT) is one of the most effective ways for deep neural network models to resist adversarial examples. However, there is still a significant gap between robust training accuracy and testing accuracy. Although recent studies have shown that data augmentation can effectively reduce this gap, most methods heavily rely on generating large amounts of training data without considering which features are beneficial for model robustness, making them inefficient. To address the above issue, we propose a two-stage AT algorithm for image data that adopts different data augmentation strategies during the training process to improve model robustness. In the first stage, we focus on the convergence of the algorithm, which uses structure and texture information to guide AT. In the second stage, we introduce a strategy that randomly fuses the data features to generate diverse adversarial examples for AT. We compare our proposed algorithm with five state-of-the-art algorithms on three models, and the experimental results achieve the best robust accuracy under all evaluation metrics on the CIFAR10 dataset, demonstrating the superiority of our method.

A number of real-world multiobjective optimization problems (MOPs) are driven by the data from experiments or computational simulations. In some cases, no new data can be sampled during the optimization process and only a certain amount of data can be sampled before optimization starts. Such problems are known as offline data-driven MOPs. Although multiple surrogate models approximating each objective function are able to replace the real fitness evaluations in evolutionary algorithms (EAs), their approximation errors are easily accumulated and therefore, mislead the solution ranking. To mitigate this issue, a new surrogate-assisted indicator-based EA for solving offline data-driven multiobjective problems is proposed. The proposed algorithm adopts an indicator-based selection EA as the baseline optimizer due to its selection robustness to the approximation errors of surrogate models. Both the Kriging models and radial basis function networks (RBFNs) are employed as surrogate models. An adaptive model selection mechanism is designed to choose the right type of models according to a maximum acceptable approximation error that is less likely to mislead the indicator-based search. The main idea is that when the uncertainty of the Kriging models exceeds the acceptable error, the proposed algorithm selects RBFNs as the surrogate models. The results comparing with state-of-the-art algorithms on benchmark problems with up to ten objectives indicate that the proposed algorithm is effective on offline data-driven optimization problems with up to 20 and 30 decision variables.

Real-world industrial engineering optimization problems often have a large number of decision variables. Most existing large-scale evolutionary algorithms (EAs) need a large number of function evaluations to achieve high-quality solutions. However, the function evaluations can be computationally intensive for many of these problems, particularly, which makes large-scale expensive optimization challenging. To address this challenge, surrogate-assisted EAs based on the divide-and-conquer strategy have been proposed and shown to be promising. Following this line of research, we propose a surrogate-assisted differential evolution algorithm with adaptive multisubspace search for large-scale expensive optimization to take full advantage of the population and the surrogate mechanism. The proposed algorithm constructs multisubspace based on principal component analysis and random decision variable selection, and searches adaptively in the constructed subspaces with three search strategies. The experimental results on a set of large-scale expensive test problems have demonstrated its superiority over three state-of-the-art algorithms on the optimization problems with up to 1000 decision variables.

Many real-world optimization tasks suffer from noise. So far, the research on noise-tolerant optimization algorithms is still restricted to low-dimensional problems with less than 100 decision variables. In reality, many problems are high dimensional. Cooperative coevolutionary (CC) algorithms based on a divide-and-conquer strategy are promising in solving complex high-dimensional problems. However, noisy fitness evaluations pose a challenge in problem decomposition for CC. The state-of-the-art grouping methods, such as differential grouping (DG) and recursive DG, are unable to work properly in noisy environments. Because it is impossible to distinguish whether the change of one variable’s difference value is caused by noise or the perturbation of its interacting variables. As a result, every pair of variables will be identified as nonseparable in these methods. In this article, we study how to group decision variables with the covariance matrix adaptation evolution strategy (CMA-ES) in noisy environments and subsequently propose a landscape-aware grouping (LAG) method. Instead of detecting pairwise interacting variables, we directly identify a nonseparable subcomponent. To this end, we propose to use two convergence features: 1) variable convergence time and 2) accumulative path, to describe variables’ fitness landscapes; then, variables are clustered according to these two features. Numerical experiments show that LAG can more effectively identify interactive decision variables in the presence of multiplicative noise than the DG and some of its variants. Up to 500 dimensions, the performance of CC CMA-ES with landscape-aware grouping (CC-CMAES-LAG) is competitive compared with existing CC algorithms and uncertainty-handling CMA-ES (UH-CMA-ES).

Dynamic time-linkage optimization problems (DTPs) are special dynamic optimization problems (DOPs) with the time-linkage property. The environment of DTPs changes not only over time but also depends on the previous applied solutions. DTPs are hardly solved by existing dynamic evolutionary algorithms because they ignore the time-linkage property. In fact, they can be viewed as multiple decision-making problems and solved by reinforcement learning (RL). However, only some discrete DTPs are solved by RL-based evolutionary optimization algorithms with the assumption of observable objective functions. In this work, we propose a dynamic evolutionary optimization algorithm using surrogate-assisted$Q$-learning for continuous black-box DTPs. To observe the states of black-box DTPs, the state extraction and prediction methods are applied after the search process at each time step. Based on the learned information, a surrogate-assisted$Q$-learning is introduced to evaluate and select candidate solutions in the continuous decision space in a long-term consideration. We evaluate the components of our proposed algorithm on various benchmark problems to study their behaviors. Results of comparative experiments indicate that the proposed algorithm outperforms other compared algorithms and performs robustly on DTPs with up to 30 decision variables and different dynamic changes.

Optimization problems whose evaluations of the objective and constraints involve costly numerical simulations or physical experiments are referred to as expensive constrained optimization (ECO) problems. Such problems can be solved by evolutionary algorithms (EAs) in conjunction with computationally cheap surrogates that separately approximate the expensive objective and constraint functions. During the process of the ECO, the interested regions of surrogate models for the objective and constraints usually have a small overlap only. Specifically, the surrogate model for the objective function should focus on the prediction accuracy in the promising region, while the models for constraint functions should concentrate on the accuracy at the boundary of the feasible region. However, most existing methods neglect such differences and train those different models using the same training data, barely resulting in satisfactory performance. Therefore, we propose a general framework for solving expensive optimization problems with inequality constraints. In the proposed framework, the objective and constraints are separately trained with two different sets of training data to enhance the prediction accuracy and reliability in the interested regions. A novel infill sampling criterion is tailored to decide whether potentially better or more uncertain solutions should be sampled. Moreover, a new strategy, termed search intensity adjustment, is designed for adjusting the number of search generations on new surrogate models. We attempt to embed three competitive constrained EAs into our framework to verify its generality. The experimental results obtained on numerous benchmark functions from CEC2006, CEC2010, and CEC2017 have demonstrated the superiority of our approach over three state-of-the-art surrogate-assisted EAs.

Recent studies show that deep neural networks are vulnerable to adversarial attacks in the form of subtle perturbations to the input image, which leads the model to output wrong prediction. Such an attack can easily succeed by the existing white-box attack methods, where the perturbation is calculated based on the gradient of the target network. Unfortunately, the gradient is often unavailable in the real-world scenarios, which makes the black-box adversarial attack problems practical and challenging. In fact, they can be formulated as high-dimensional black-box optimization problems at the pixel level. Although evolutionary algorithms are well known for solving black-box optimization problems, they cannot efficiently deal with the high-dimensional decision space. Therefore, we propose an approximated gradient sign method using differential evolution (DE) for solving black-box adversarial attack problems. Unlike most existing methods, it is novel that the proposed method searches the gradient sign rather than the perturbation by a DE algorithm. Also, we transform the pixel-based decision space into a dimension-reduced decision space by combining the pixel differences from the input image to neighbor images, and two different techniques for selecting neighbor images are introduced to build the transferred decision space. In addition, six variants of the proposed method are designed according to the different neighborhood selection and optimization search strategies. Finally, the performance of the proposed method is compared with a number of the state-of-the-art adversarial attack algorithms on CIFAR-10 and ImageNet datasets. The experimental results suggest that the proposed method shows superior performance for solving black-box adversarial attack problems, especially nontargeted attack problems.

Various evolutionary algorithms (EAs) have been proposed to address feature selection (FS) problems, in which a large number of fitness evaluations are needed. With the rapid growth of data scales, the fitness evaluation becomes time consuming, which makes FS problems expensive optimization problems. Surrogate-assisted EAs (SAEAs) have been widely used to solve expensive optimization problems. However, the SAEAs still face difficulties in solving expensive FS problems due to their high-dimensional discrete decision variables. To address this issue, we propose an SAEA with parallel random grouping for expensive FS problems, in which three main components consist. First, a constraint-based sampling strategy is proposed, which considers the influence of the constraint boundary and the number of selected features. Second, a high-dimensional FS problem is randomly divided into several low-dimensional subproblems. Surrogate models are then constructed in these low-dimensional decision spaces. After that, all the subproblems are optimized in parallel. The process of random grouping and parallel optimization continues until the termination condition is met. Finally, a final solution is chosen from the best solution in the historical search and the best solution in the last population using a random, distance-, or voting-based method. Experimental results show that the proposed algorithm generally outperforms traditional, ensemble, and evolutionary FS methods on 14 datasets with up to 10 000 features, especially when the required number of real fitness evaluations is limited.

In many practical applications, data enrichment can generate a large amount of accurate data to alleviate the problem of data scarcity. In order to make the fake data generated in data enrichment as close to the real data as possible, the data enriching model must be tuned to meet the loss requirements of multiple objectives in different scenarios, which makes it a multi-scenario many-objective optimization problem. However, due to the curse of the dimensionality of the scenario space and the objective space, the existing many-objective evolutionary algorithms cannot solve the problem in data enrichment well. To effectively handle this problem, we propose an adaptive formulation-based multi-objective evolutionary algorithm, where the aggregation function is used to reduce the dimension of the scenario space to one and the multiple objectives into three objectives through the adaptive formulation of the original problem. In this way, a multi-scenario many-objective problem is converted into a multi-objective problem which could be solved by existing multi-objective evolutionary algorithms. The proposed algorithm is applied to the practical data enrichment problem to solve the multi-scenario many-objective optimization problem and compared with NSGA-III. The experimental results demonstrate the remarkable superiority of the proposed algorithm over NSGA-III.

Many real-world optimization problems have dynamic multiple objectives and constrains, such problems are called dynamic multi-objective optimization problems (DMOPs). Although many dynamic multi-objective evolutionary algorithms (DMOEAs) have been proposed to solve DMOPs, how to effectively track the optimal solutions in dynamic environments is still a major challenge for dynamic multi-objective optimization. Two classical DMOEAs, population prediction strategy (PPS) and transfer learning based DMOEA (Tr-DMOEA) are validated to have great performance because they integrate machine learning mechanism for optimization. However, there are still some disadvantages in both algorithms. In this paper, we propose a combined algorithm to make up the respective disadvantages of PPS and Tr-DMOEA. Our algorithm retains the prediction method of PPS considering sufficient historical information. Then, we improve the prediction strategy in Tr-DMOEA to further modify the solutions provided by PPS. These modified solutions finally construct the initial population for optimization in the new environment. The experiment results indicate that our algorithm has the overall best performance comparing with PPS and Tr-DMOEA on the test problems.

Multi-depot Cumulative Capacited Vehicle Routing Problem (MDCCVRP) is a relatively new research field in Vehicle Routing Problems (VRP), which is composed of several traditional VRP variants. This model is usually applicable to the logistics and transportation problems after the disaster. A decomposition-based Max-Min ant system (DMMAS) algorithm is proposed to solve MDCCVRP in this paper. First of all, a new indicator which measures the nearness between two routes for multi-depot problems is proposed. The original problem is decomposed into a series of smaller subproblems, and then, after the optimization phase, a specific pheromone communication rule between the master problem and subproblems is adopted to guide the search direction of ants. Finally, when the search gets into a halt, a mechanism of perturbation is used to get rid of the local optimality. The algorithm is tested on many benchmark problems and the experimental results show that our algorithm can effectively improve the performance in most cases compared to several state-of-the-art evolutionary algorithms.

A real-world multiobjective optimization problem (MOP) usually has differently scaled objectives. Objective space normalization has been widely used in multiobjective optimization evolutionary algorithms (MOEAs). Without objective space normalization, most of the MOEAs may fail to obtain uniformly distributed and well-converged solutions on MOPs with differently scaled objectives. Objective space normalization requires information on the Pareto front (PF) range, which can be acquired from the ideal and nadir points. Since the ideal and nadir points of a real-world MOP are usually not knowna priori, many recently proposed MOEAs tend to estimate and update the two points adaptively during the evolutionary process. Different methods to estimate ideal and nadir points have been proposed in the literature. Due to inaccurate estimation of the two points (i.e., inaccurate estimation of the PF range), objective space normalization may deteriorate the performance of an MOEA. Different methods have also been proposed to alleviate the negative effects of inaccurate estimation. This article presents a comprehensive survey of objective space normalization methods, including ideal point estimation methods, nadir point estimation methods, and different methods based on the utilization of the estimated PF range.

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.

The Pareto sets of many real-world multi-objective optimization problems in engineering and computer fields are high-dimensional but sparse. Such multi-objective optimization problems are called large-scale sparse multi-objective optimization problems. A sparse evolutionary algorithm has also been raised and verified effective on benchmark problems. However, in practical applications, it needs a large number of expensive function evaluations. Although surrogate-assisted evolutionary algorithms are common solutions to deal with expensive optimization problems within limited computation resources and especially Kriging-assisted evolutionary algorithms are widely used, they cannot cope with large-scale expensive sparse multi-objective optimization problems due to the inaccurate surrogate models on high-dimensional problems. Therefore, we first propose a feature selection operator based on non-dominated sorting to choose the non-zero decision variables in the Pareto set. Then, the dimension of the original problem is reduced and a Kriging-assisted multi-objective evolutionary algorithm is employed to solve the reformulated problem. Finally, the selected zero decision variables are added to the obtained optimal solutions as the result of the original problem. The experimental results on benchmark problems show that our proposed algorithm outperforms the existing algorithms.

In the last decades, evolutionary algorithms (EAs) have demonstrated strong search capabilities in solving multi-objective optimization problems (MOPs). To improve the search performance of EAs, as problems seldom exist in isolation, transferring knowledge from related problems have attracted considerable attentions in recent years. In this paper, we present a preliminary study to enhance existing evolutionary algorithms (MOEAs) by transferring knowledge from the process of solving the single objectives involved in a given MOP of interest. As the single objectives are the objectives of the MOP, they naturally share great similarity with the given MOP, which thus could yield useful traits for enhancing the problem-solving of the MOP. To the best of our knowledge, this work severs as the first attempt to improve evolutionary multi-objective optimization via transferring knowledge from single objective problems. To evaluate the performance of the proposed method, empirical studies using a popular MOEA, i.e., NSGAII, on commonly used multi-objective benchmarks are conducted. The obtained results confirmed the efficacy of the proposed method in terms of both convergence speed and solution quality.

Many real-world optimization problems can be solved by using the data-driven approach only, simply because no analytic objective functions are available for evaluating candidate solutions. In this paper, we address a class of expensive data-driven constrained multiobjective combinatorial optimization problems, where the objectives and constraints can be calculated only on the basis of a large amount of data. To solve this class of problems, we propose using random forests (RFs) and radial basis function networks as surrogates to approximate both objective and constraint functions. In addition, logistic regression models are introduced to rectify the surrogate-assisted fitness evaluations and a stochastic ranking selection is adopted to further reduce the influences of the approximated constraint functions. Three variants of the proposed algorithm are empirically evaluated on multiobjective knapsack benchmark problems and two real-world trauma system design problems. Experimental results demonstrate that the variant using RF models as the surrogates is effective and efficient in solving data-driven constrained multiobjective combinatorial optimization problems.

Convolutional neural networks (CNNs) have shown remarkable performance in various real-world applications. Unfortunately, the promising performance of CNNs can be achieved only when their architectures are optimally constructed. The architectures of state-of-the-art CNNs are typically handcrafted with extensive expertise in both CNNs and the investigated data, which consequently hampers the widespread adoption of CNNs for less experienced users. Evolutionary deep learning (EDL) is able to automatically design the best CNN architectures without much expertise. However, the existing EDL algorithms generally evaluate the fitness of a new architecture by training from scratch, resulting in the prohibitive computational cost even operated on high-performance computers. In this paper, an end-to-end offline performance predictor based on the random forest is proposed to accelerate the fitness evaluation in EDL. The proposed performance predictor shows the promising performance in term of the classification accuracy and the consumed computational resources when compared with 18 state-of-the-art peer competitors by integrating into an existing EDL algorithm as a case study. The proposed performance predictor is also compared with the other two representatives of existing performance predictors. The experimental results show the proposed performance predictor not only significantly speeds up the fitness evaluations but also achieves the best prediction among the peer performance predictors.

For an image retrieval system, the metric to measure the similarity between target images and queries greatly affects its retrieval performance. However, most of the existing metrics are based on single distance metric, which has been shown lack of robustness for different kinds of queries. In this work, we view the metric learning task as an optimization problem for a robust combination of existing metrics, where the objective function is data-driven rather than analytical. Our contribution is two-fold. Firstly, considering the robustness of image retrieval systems, we formulate the optimization problem as a multiobjective optimization with both the average and worst retrieval performance (precision) for different kinds of queries as objectives. Secondly, we apply a popular multi-objective evolutionary algorithm, NSGA-II, to search the optimal combined metric. With the experiment on two different datasets for image retrieval, we find that the proposed algorithm can find combined metrics with better and more robust retrieval performance than other existing single metrics.

Cooperative coevolutionary (CC) algorithms decompose a problem into several subcomponents and optimize them separately. Such a divide-and-conquer strategy makes CC algorithms potentially well suited for large-scale optimization. However, decomposition may be inaccurate, resulting in a wrong division of the interacting decision variables into different subcomponents and thereby a loss of important information about the topology of the overall fitness landscape. In this paper, we suggest an idea that concurrently searches for multiple optima and uses them as informative representatives to be exchanged among subcomponents for compensation. To this end, we incorporate a multimodal optimization procedure into each subcomponent, which is adaptively triggered by the status of subcomponent optimizers. In addition, a nondominance-based selection scheme is proposed to adaptively select one complete solution for evaluation from the ones that are constructed by combining informative representatives from each subcomponent with a given solution. The performance of the proposed algorithm has been demonstrated by comparing five popular CC algorithms on a set of selected problems that are recognized to be hard for traditional CC algorithms. The superior performance of the proposed algorithm is further confirmed by a comprehensive study that compares 17 state-of-the-art CC algorithms and other metaheuristic algorithms on 20 1000-dimensional benchmark functions.

Most evolutionary optimization algorithms assume that the evaluation of the objective and constraint functions is straightforward. In solving many real-world optimization problems, however, such objective functions may not exist. Instead, computationally expensive numerical simulations or costly physical experiments must be performed for fitness evaluations. In more extreme cases, only historical data are available for performing optimization and no new data can be generated during optimization. Solving evolutionary optimization problems driven by data collected in simulations, physical experiments, production processes, or daily life are termed data-driven evolutionary optimization. In this paper, we provide a taxonomy of different data driven evolutionary optimization problems, discuss main challenges in data-driven evolutionary optimization with respect to the nature and amount of data, and the availability of new data during optimization. Real-world application examples are given to illustrate different model management strategies for different categories of data-driven optimization problems.

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 solving many real-world optimization problems, neither mathematical functions nor numerical simulations are available for evaluating the quality of candidate solutions. Instead, surrogate models must be built based on historical data to approximate the objective functions and no new data will be available during the optimization process. Such problems are known as offline data-driven optimization problems. Since the surrogate models solely depend on the given historical data, the optimization algorithm is able to search only in a very limited decision space during offline data-driven optimization. This paper proposes a new offline data-driven evolutionary algorithm to make the full use of the offline data to guide the search. To this end, a surrogate management strategy based on ensemble learning techniques developed in machine learning is adopted, which builds a large number of surrogate models before optimization and adaptively selects a small yet diverse subset of them during the optimization to achieve the best local approximation accuracy and reduce the computational complexity. Our experimental results on the benchmark problems and a transonic airfoil design example show that the proposed algorithm is able to handle offline data-driven optimization problems with up to 100 decision variables.

For multi-scenario airfoil shape optimization problems, an evaluation of a single airfoil is based on its full-scenario drag landscape. To obtain the full-scenario drag landscape, a large number of computational fluid dynamic simulations for different operating conditions must be conducted. Since a single computational fluid dynamic simulation is often time-consuming, evaluations for multi-scenario airfoil shape optimization will be computationally highly intensive. Although surrogate-assisted evolutionary algorithms have been widely applied to expensive optimization problems, existing surrogate-assisted evolutionary algorithms cannot be directly applied to multi-scenario airfoil shape optimization due to the lack of training data. Instead of using surrogate models to directly approximate the multi-scenario evaluations, we employ a hierarchical surrogate model consisting of a K-nearest neighbors classifier and a Kriging model to approximate the full-scenario drag landscape for each candidate design during the optimization. Then, the fitness of the candidate design is evaluated based on the approximated drag landscape to reduce the computational cost. The proposed hierarchical surrogate model is embedded in the covariance matrix adaptation evolution strategy and applied to the RAE2822 airfoil design problem. Our experimental results show that the proposed algorithm is able to obtain an airfoil design with limited computational cost that perform well in different operating conditions.

Many real-world optimization problems involve computationally intensive numerical simulations to accurately evaluate the quality of solutions. Usually, the fidelity of the simulations can be controlled using certain parameters and there is a tradeoff between simulation fidelity and computational cost, i.e., the higher the fidelity, the more complex the simulation will be. To reduce the computational time in simulation-driven optimization, it is a common practice to use multiple fidelity levels in search for the optimal solution. So far, not much work has been done in evolutionary optimization that considers multiple fidelity levels in fitness evaluations. In this paper, we aim to develop test suites that are able to capture some important characteristics in real-world multifidelity optimization, thereby offering a useful benchmark for developing evolutionary algorithms for multifidelity optimization. To demonstrate the usefulness of the proposed test suite, three strategies for adapting the fidelity level of the test problems during optimization are suggested and embedded in a particle swarm optimization (PSO) algorithm. Our simulation results indicate that the use of changing fidelity is able to enhance the performance and reduce the computational cost of the PSO, which is desired in solving expensive optimization problems.

The mapping relation between decision variables and objective functions is complicated in multi-objective optimization problems. Dimension reduction-based memetic optimization strategy was proposed to decompose a multi-objective optimization problem into several easier subproblems in decision subspaces by detecting the correlation between decision variables and objective functions. In this work, the process of optimizing the original problem by separately searching the decision space of the subproblems is termed decomposed search. We embed the decomposed search strategy in existing multi-objective evolutionary algorithms to improve their performance. However, it is highly time-consuming to detect the mapping relation and select solutions for decomposed search. To improve the computational efficiency of the strategy, we adopt nonlinear correlation information entropy to measure the correlation between the decision variables and objective functions and suggest a probabilistic similarity measurement to select solutions for the decomposed search, which is shown to be effective by experimental results. Finally, the correlation detection and solution selection strategies proposed in this paper are embedded in both Pareto- and non-Pareto-based multi-objective evolutionary algorithms to compare them with existing ones. Our experimental results demonstrate that the proposed strategies have significantly improved the computational efficiency at the expense of slightly degraded performance.

Function evaluations (FEs) of many real-world optimization problems are time or resource consuming, posing a serious challenge to the application of evolutionary algorithms (EAs) to solve these problems. To address this challenge, the research on surrogate-assisted EAs has attracted increasing attention from both academia and industry over the past decades. However, most existing surrogate-assisted EAs (SAEAs) either still require thousands of expensive FEs to obtain acceptable solutions, or are only applied to very low-dimensional problems. In this paper, a novel surrogate-assisted particle swarm optimization (PSO) inspired from committee-based active learning (CAL) is proposed. In the proposed algorithm, a global model management strategy inspired from CAL is developed, which searches for the best and most uncertain solutions according to a surrogate ensemble using a PSO algorithm and evaluates these solutions using the expensive objective function. In addition, a local surrogate model is built around the best solution obtained so far. Then, a PSO algorithm searches on the local surrogate to find its optimum and evaluates it. The evolutionary search using the global model management strategy switches to the local search once no further improvement can be observed, and vice versa. This iterative search process continues until the computational budget is exhausted. Experimental results comparing the proposed algorithm with a few state-of-the-art SAEAs on both benchmark problems up to 30 decision variables as well as an airfoil design problem demonstrate that the proposed algorithm is able to achieve better or competitive solutions with a limited budget of hundreds of exact FEs.

Maintaining diversity is one important aim of multiobjective optimization. However, diversity for manyobjective optimization problems is less straightforward to define than for multiobjective optimization problems. Inspired by measures for biodiversity, we propose a new diversity metric for many-objective optimization, which is an accumulation of the dissimilarity in the population, where an Lp-norm-based (p<;1) distance is adopted to measure the dissimilarity of solutions. Empirical results demonstrate our proposed metric can more accurately assess the diversity of solutions in various situations. We compare the diversity of the solutions obtained by four popular many-objective evolutionary algorithms using the proposed diversity metric on a large number of benchmark problems with two to ten objectives. The behaviors of different diversity maintenance methodologies in those algorithms are discussed in depth based on the experimental results. Finally, we show that the proposed diversity measure can also be employed for enhancing diversity maintenance or reference set generation in many-objective optimization.

Abstract It is hard to obtain the entire solution set of a many-objective optimization problem (MaOP) by multi-objective evolutionary algorithms (MOEAs) because of the difficulties brought by the large number of objectives. However, the redundancy of objectives exists in some problems with correlated objectives (linearly or nonlinearly). Objective reduction can be used to decrease the difficulties of some MaOPs. In this paper, we propose a novel objective reduction approach based on nonlinear correlation information entropy (NCIE). It uses the NCIE matrix to measure the linear and nonlinear correlation between objectives and a simple method to select the most conflicting objectives during the execution of MOEAs. We embed our approach into both Pareto-based and indicator-based MOEAs to analyze the impact of our reduction method on the performance of these algorithms. The results show that our approach significantly improves the performance of Pareto-based MOEAs on both reducible and irreducible MaOPs, but does not much help the performance of indicator-based MOEAs.

Regularity models have been used in dealing with noise-free multiobjective optimization problems. This paper studies the behavior of a regularity model in noisy environments and argues that it is very suitable for noisy multiobjective optimization. We propose to embed the regularity model in an existing multiobjective evolutionary algorithm for tackling noises. The proposed algorithm works well in terms of both convergence and diversity. In our experimental studies, we have compared several state-of-the-art of algorithms with our proposed algorithm on benchmark problems with different levels of noises. The experimental results showed the effectiveness of the regularity model on noisy problems, but a degenerated performance on some noisy-free problems.

Most existing work on evolutionary optimization assumes that there are analytic functions for evaluating the objectives and constraints. In the real world, however, the objective or constraint values of many optimization problems can be evaluated solely based on data and solving such optimization problems is often known as data-driven optimization. In this paper, we divide data-driven optimization problems into two categories, i.e., offline and online data-driven optimization, and discuss the main challenges involved therein. An evolutionary algorithm is then presented to optimize the design of a trauma system, which is a typical offline data-driven multiobjective optimization problem, where the objectives and constraints can be evaluated using incidents only. As each single function evaluation involves a large amount of patient data, we develop a multifidelity surrogate-management strategy to reduce the computation time of the evolutionary optimization. The main idea is to adaptively tune the approximation fidelity by clustering the original data into different numbers of clusters and a regression model is constructed to estimate the required minimum fidelity. Experimental results show that the proposed algorithm is able to save up to 90% of computation time without much sacrifice of the solution quality.

There can be a complicated mapping relation between decision variables and objective functions in multi-objective optimization problems (MOPs). It is uncommon that decision variables influence objective functions equally. Decision variables act differently in different objective functions. Hence, often, the mapping relation is unbalanced, which causes some redundancy during the search in a decision space. In response to this scenario, we propose a novel memetic (multi-objective) optimization strategy based on dimension reduction in decision space (DRMOS). DRMOS firstly analyzes the mapping relation between decision variables and objective functions. Then, it reduces the dimension of the search space by dividing the decision space into several subspaces according to the obtained relation. Finally, it improves the population by the memetic local search strategies in these decision subspaces separately. Further, DRMOS has good portability to other multi-objective evolutionary algorithms (MOEAs); that is, it is easily compatible with existing MOEAs. In order to evaluate its performance, we embed DRMOS in several state of the art MOEAs to facilitate our experiments. The results show that DRMOS has the advantage in terms of convergence speed, diversity maintenance, and portability when solving MOPs with an unbalanced mapping relation between decision variables and objective functions.

Many-objective optimization problems (ManyOPs) refer, usually, to those multiobjective problems (MOPs) with more than three objectives. Their large numbers of objectives pose challenges to multiobjective evolutionary algorithms (MOEAs) in terms of convergence, diversity, and complexity. Most existing MOEAs can only perform well in one of those three aspects. In view of this, we aim to design a more balanced MOEA on ManyOPs in all three aspects at the same time. Among the existing MOEAs, the two-archive algorithm (Two_Arch) is a low-complexity algorithm with two archives focusing on convergence and diversity separately. Inspired by the idea of Two_Arch, we propose a significantly improved two-archive algorithm (i.e., Two_Arch2) for ManyOPs in this paper. In our Two_Arch2, we assign different selection principles (indicator-based and Pareto-based) to the two archives. In addition, we design a new Lp-norm-based (p <; 1) diversity maintenance scheme for ManyOPs in Two_Arch2. In order to evaluate the performance of Two_Arch2 on ManyOPs, we have compared it with several MOEAs on a wide range of benchmark problems with different numbers of objectives. The experimental results show that Two_Arch2 can cope with ManyOPs (up to 20 objectives) with satisfactory convergence, diversity, and complexity.

Nondominated sorting plays an important role in Pareto-based multiobjective evolutionary algorithms (MOEAs). When faced with many-objective optimization problems multiobjective optimization problems (MOPs) with more than three objectives, the number of comparisons needed in nondominated sorting becomes very large. In view of this, a new corner sort is proposed in this paper. Corner sort first adopts a fast and simple method to obtain a nondominated solution from the corner solutions, and then uses the nondominated solution to ignore the solutions dominated by it to save comparisons. Obtaining the nondominated solutions requires much fewer objective comparisons in corner sort. In order to evaluate its performance, several state-of-the-art nondominated sorts are compared with our corner sort on three kinds of artificial solution sets of MOPs and the solution sets generated from MOEAs on benchmark problems. On one hand, the experiments on artificial solution sets show the performance on the solution sets with different distributions. On the other hand, the experiments on the solution sets generated from MOEAs show the influence that different sorts bring to MOEAs. The results show that corner sort performs well, especially on many-objective optimization problems. Corner sort uses fewer comparisons than others.