In multimodal multi-objective optimization problems (MMOPs), multiple solutions on different Pareto optimal solution sets (PSs) are mapped to the same point on the Pareto front. Considering these different solutions can provide users with richer decisions, the search of multiple PSs is crucial when solving MMOPs. To this end, many multimodal multi-objective evolutionary algorithms (MMOEAs) often employ intricate mechanisms to maintain the diversity of the offspring in mating selection, but ignore to learn PSs. In this paper, a data-driven evolutionary algorithm based on inductive graph neural networks (DEA-IGNN) is proposed to solve MMOPs, which successfully learns the PSs topology by the graph structure to generate offspring with good performance. Specifically, a graph topology construction method based on Euclidean distance in the decision space is designed. It determines the neighborhood by calculating the Euclidean distance of individuals in the decision space and establishes the topological relationships to construct the graphs representing of population distribution. On this basis, a model based on inductive graph neural networks is constructed to assist offspring reproduction, which can learn unknown nodes by sampling and aggregating existing information. Moreover, a data-driven reproduction strategy is proposed to predict offspring with the good diversity and convergence, which uses the traditional variation operators to generate training data and adopts these data to train the model. The proposed DEA-IGNN is implemented and compared with eleven competitive MMOEAs on three test suites and a practical problem. The experimental results show that DEA-IGNN has good performance.

Deep neural networks (DNNs) have achieved remarkable progress in remote sensing image (RSI) analysis, yet their vulnerability to subtle adversarial perturbations poses a critical threat to safety-critical applications such as environmental monitoring. While black-box attacks have garnered attention for their practicality, existing methods face a dilemma in RSI scenarios: restricted attacks often result in compromised image quality and limited stealthiness, whereas unrestricted attacks risk degrading transferability. To address this challenge, this paper proposes the latent-space adversarial diffusion attack framework (LADA), which focuses on balancing stealthiness and transferability in adversarial attacks against RSI models. LADA employs a pre-trained diffusion model to map high-resolution RSIs into a low-dimensional latent space, enabling semantic-level perturbation optimization while avoiding pixel-wise explicit noise. Additionally, text prompts are automatically generated using a large multimodal model to guide adversarial sample synthesis, ensuring semantic consistency. To enhance perturbation search efficiency in the latent space, a hybrid strategy combining multi-scale sampling and covariance matrix adaptation evolution strategy is introduced. Extensive experiments demonstrate that LADA achieves superior performance across multiple RSI datasets, model architectures, and defense mechanisms. This paper establishes a benchmark for high-stealthiness and high-transferability adversarial attacks, advancing the secure deployment of DNNs in remote sensing applications.

We consider the black-box adversarial attack problem in the field of remote sensing images (RSIs) to reveal the vulnerabilities of various deep neural networks (DNNs), including classification and semantic segmentation models. Existing adversarial attack methods typically focus solely on maximizing attack success rates (ASRs) under given perturbation constraints, neglecting the differences between adversarial samples and clean images. We propose a batchwise amplitude spectrum normalization (BAMPN) method, which is a plug-and-play and transfer-based black-box attack strategy. Using Fourier transform, we convert RSIs from the spatial domain into the frequency domain to obtain the amplitude spectrum, which is normalized within the batch. Moreover, we use a moving average strategy to retain the historical amplitudes of the batch, enhancing input diversity. By mixing the low-level statistical features of RSIs, BAMPN reduces the differences between adversarial samples and clean images while improving attack transferability. In addition, BAMPN is applicable not only to RSIs classification tasks but also directly to semantic segmentation tasks, as it preserves the spatial semantic structure of RSIs. We conduct extensive experiments using 20 DNN models and four benchmark RSIs’ datasets, comparing our method against 14 state-of-the-art (SOTA) approaches. The results demonstrate that BAMPN achieves superior attack performance while ensuring a promising similarity between adversarial and clean samples.

This work concerns evolutionary approaches to black-box noisy optimization where the problem is accessed via noisy function evaluations. An evolution strategy (ES) algorithm is proposed, which uses a Gaussian distribution to guide the search and requires only the comparisons among solutions. The new method achieves the similar convergence rate as finite-difference based gradient methods on non-convex landscapes, while being adaptive in the sense that the convergence is ensured with any initial step-size. We further improve the method with a variance adaptation mechanisms that alleviates the need for hyper-parameter tunning and an asynchronous parallelization implementation that enables linear speedup. The persistent monitoring task is chosen as an application to investigate the effectiveness of the proposed method. The task is to schedule a team of agents to minimize the uncertainties of some targets in a changing environment defined on a grid map. We show in the single-agent case that the task can be cast into a sequence of pathfinding subproblems of which the sequential order can be modeled as a Markov decision procedure and solved by the proposed ES method. In the multi-agent case, we show that solving the single-agent subproblems using the ES method in a round-robin way can provide collision-free solutions. Numerical studies demonstrate the reduction in convergence time and the robustness against complicated environments relative to several existing evolutionary algorithms and zeroth-order gradient methods.

Multiobjective multifactorial optimization (MO-MFO), rooted in a multitasking environment, is an emerging paradigm wherein multiple distinct multiobjective optimization problems are solved together. This article proposes an evolutionary multitasking algorithm with learning task relationships (LTR) for MO-MFO. In the proposed algorithm, a procedure of LTR is well designed. The decision space of each task is treated as a manifold, and all decision spaces of different tasks are jointly modeled as a joint manifold. Then, through solving a generalized eigenvalue decomposition problem, the joint manifold is projected to a latent space while keeping the necessary features for all tasks and the topology of each manifold. Finally, the task relationships are represented as the joint mapping matrix, which is composed of multiple mapping functions, and they are utilized for information transfer across different decision spaces during the evolutionary process. In the empirical experiments, the performance of the proposed algorithm is verified and compared with several state-of-the-art solvers for MO-MFO on three suites of MO-MFO test problems. Empirical results demonstrate that the proposed algorithm surpasses other competitors on most test instances, and can well tackle complicated MO-MFO problems which involve distinct optimization tasks with heterogeneous decision spaces.

Expensive optimization problems arise in diverse fields, and the expensive computation in terms of function evaluation poses a serious challenge to global optimization algorithms. In this article, a simple yet effective optimization algorithm for computationally expensive optimization problems is proposed, which is called the neighborhood regression optimization algorithm. For a minimization problem, the proposed algorithm incorporates the regression technique based on a neighborhood structure to predict a descent direction. The descent direction is then adopted to generate new potential offspring around the best solution obtained so far. The proposed algorithm is compared with 12 popular algorithms on two benchmark suites with up to 30 decision variables. Empirical results demonstrate that the proposed algorithm shows clear advantages when dealing with unimodal and smooth problems, and is better than or competitive with other peer algorithms in terms of the overall performance. In addition, the proposed algorithm is efficient and keeps a good tradeoff between solution quality and running time.

Federated learning allows multiple participants to collaboratively train an efficient model without exposing data privacy. However, this distributed machine learning training method is prone to attacks from Byzantine clients, which interfere with the training of the global model by modifying the model or uploading the false gradient. In this article, we propose a novel serverless federated learning frameworkCommittee Mechanism based Federated Learning(CMFL), which can ensure the robustness of the algorithm with convergence guarantee. In CMFL, a committee system is set up to screen the uploaded local gradients. The committee system selects the local gradients rated by the elected members for the aggregation procedure through the selection strategy, and replaces the committee member through the election strategy. Based on the different considerations of model performance and defense, two opposite selection strategies are designed for the sake of both accuracy and robustness. Extensive experiments illustrate that CMFL achieves faster convergence and better accuracy than the typical Federated Learning, in the meanwhile obtaining better robustness than the traditional Byzantine-tolerant algorithms, in the manner of a decentralized approach. In addition, we theoretically analyze and prove the convergence of CMFL under different election and selection strategies, which coincides with the experimental results.

This work concerns the evolutionary approaches to distributed stochastic black-box optimization, in which each worker can individually solve an approximation of the problem with nature-inspired algorithms. We propose a distributed evolution strategy (DES) algorithm grounded on a proper modification to evolution strategies, a family of classic evolutionary algorithms, as well as a careful combination with existing distributed frameworks. On smooth and nonconvex landscapes, DES has a convergence rate competitive to existing zeroth-order methods, and can exploit the sparsity, if applicable, to match the rate of first-order methods. The DES method uses a Gaussian probability model to guide the search and avoids the numerical issue resulted from finite-difference techniques in existing zeroth-order methods. The DES method is also fully adaptive to the problem landscape, as its convergence is guaranteed with any parameter setting. We further propose two alternative sampling schemes which significantly improve the sampling efficiency while leading to similar performance. Simulation studies on several machine learning problems suggest that the proposed methods show much promise in reducing the convergence time and improving the robustness to parameter settings.

The covariance matrix adaptation evolution strategy (CMA-ES) is a powerful evolutionary algorithm for single-objective real-valued optimization. However, the time and space complexity may preclude its use in high-dimensional decision space. Recent studies suggest that putting sparse or low-rank constraints on the structure of the covariance matrix can improve the efficiency of CMA-ES in handling large-scale problems. Following this idea, this paper proposes a search direction adaptation evolution strategy (SDA-ES) which achieves linear time and space complexity. SDA-ES models the covariance matrix with an identity matrix and multiple search directions, and uses a heuristic to update the search directions in a way similar to the principal component analysis. We also generalize the traditional 1/5th success rule to adapt the mutation strength which exhibits the derandomization property. Numerical comparisons with nine state-of-the-art algorithms are carried out on 31 test problems. The experimental results have shown that SDA-ES is invariant under search-space rotational transformations, and is scalable with respect to the number of variables. It also achieves competitive performance on generic black-box problems, demonstrating its effectiveness in keeping a good tradeoff between solution quality and computational efficiency.

Decomposition-based multiobjective evolutionary algorithms (MOEAs) are a class of popular methods for solving the multiobjective optimization problems (MOPs), and have been widely studied in numerical experiments and successfully applied in practice. However, we know little about these algorithms from the theoretical aspect. In this paper, we present running time analysis of a simple MOEA with mutation and crossover based on the MOEA/D framework (MOEA/D-C) on five pseudo-Boolean functions. Our rigorous theoretical analysis shows that by properly setting the number of subproblems, the upper bounds of expected running time of MOEA/D-C obtaining a set of solutions to cover the Pareto fronts (PFs) of these problems are apparently lower than those of the one with mutation-only (MOEA/D-M). Moreover, to effectively obtain a set of solutions to cover the PFs of these problem, MOEA/D-C only needs to decompose these MOPs into several subproblems with a set of simple weight vectors while MOEA/D-M needs to find Ω(n) optimally decomposed weight vectors. This result suggests that the use of crossover in decomposition-based MOEA can simplify the setting of weight vectors for different problems and make the algorithm more efficient. This paper provides some insights into the working principles of MOEA/D and explains why some existing decomposition-based MOEAs work well in computational experiments.

This work provides an efficient sampling method for the covariance matrix adaptation evolution strategy (CMA-ES) in large-scale settings. In contract to the Gaussian sampling in CMA-ES, the proposed method generates mutation vectors from a mixture model, which facilitates exploiting the rich variable correlations of the problem landscape within a limited time budget. We analyze the probability distribution of this mixture model and show that it approximates the Gaussian distribution of CMA-ES with a controllable accuracy. We use this sampling method, coupled with a novel method for mutation strength adaptation, to formulate the mixture model-based evolution strategy (MMES)-a CMA-ES variant for large-scale optimization. The numerical simulations show that, while significantly reducing the time complexity of CMA-ES, MMES preserves the rotational invariance, is scalable to high dimensional problems, and is competitive against the state-of-the-arts in performing global optimization.

Constrained multiobjective optimization abounds in practical applications and is gaining growing attention in the evolutionary computation community. Artificial test problems are critical to the progress in this research area. Nevertheless, many of them lack important characteristics, such as scalability and variable dependencies, which may be essential in benchmarking modern evolutionary algorithms. This article first proposes a new framework for constrained test problem construction. This framework splits a decision vector into position and distance variables and forces their optimal values to lie on a nonlinear hypersurface such that the interdependencies can be introduced among the position ones and among the distance ones individually. In this framework, two kinds of constraints are designed to introduce convergence-hardness and diversity-hardness, respectively. The first kind introduces infeasible barriers in approaching the optima, and at the same time, makes the position and distance variables interrelate with each other. The second kind restricts the feasible optimal regions such that different shapes of Pareto fronts can be obtained. Based on this framework, we construct 16 scalable and constrained test problems covering a variety of difficulties. Then, in the second part of this article, we evaluate the performance of some state of the art on the proposed test problems, showing that they are quite challenging and there is room for further enhancement of the existing algorithms. Finally, we discuss in detail the source of difficulties presented in these new problems.

Decomposition-based multiobjective evolutionary algorithms (MOEAs) are a class of popular methods for solving multiobjective optimization problems (MOPs), and have been widely studied in numerical experiments and successfully applied in practice. However, we know little about these algorithms from the theoretical aspect. In this paper, we present a running time analysis of a simple MOEA with crossover based on the MOEA/D framework (MOEA/D-C) on four discrete optimization problems. Our rigorous theoretical analysis shows that the MOEA/D-C can obtain a set of Pareto optimal solutions to cover the Pareto front of these problems in expected running time apparently lower than the one without crossover. Moreover, the MOEA/D-C only needs to decompose an MOP into a few scalar optimization subproblems according to several simple weight vectors. This result suggests that the use of crossover in decomposition-based MOEA can simplify the setting of weight vector for different problems and make the algorithm more efficient. This study theoretically explains why some decomposition-based MOEAs work well in computational experiments and provides insights in design of MOEAs for MOPs in future research.

Evolution strategies have been demonstrated to have the strong ability to roughly train deep neural networks and well accomplish reinforcement learning tasks. However, existing evolution strategies designed specially for deep reinforcement learning only involve the plain variants which can not realize the adaptation of mutation strength or other advanced techniques. The research of applying advanced and effective evolution strategies to reinforcement learning in an efficient way is still a gap. To this end, this paper proposes a restart-based rank-1 evolution strategy for reinforcement learning. When training the neural network, it adapts the mutation strength and updates the principal search direction in a way similar to the momentum method, which is an ameliorated version of stochastic gradient ascent. Besides, two mechanisms, i.e., the adaptation of the number of elitists and the restart procedure, are integrated to deal with the issue of local optima. Experimental results on classic control problems and Atari games show that the proposed algorithm is superior to or competitive with state-of-the-art algorithms for reinforcement learning, demonstrating the effectiveness of the proposed algorithm.

The many-objective evolutionary algorithms generally make use of a set of well-spread reference vectors to increase the selection pressure toward the Pareto front in high-dimensional objective space. However, few studies have been reported on how to generate new solutions toward the Pareto set (PS) in the decision space with the help of these reference vectors. To fill this gap, we develop a novel reproduction operator based on the differential evolution. The main idea is using the evolution paths to depict the population movement and predict its tendency. These evolution paths are used to create potential solutions, and thus, accelerate the convergence toward the PS. Furthermore, a self-adaptive mechanism is introduced to adapt related parameters automatically. This operator is implemented in two well-known many-objective evolutionary algorithm frameworks. The experimental results on 20 widely used benchmark problems show that the proposed operator is able to strengthen the performance of the original algorithms in handling many-objective optimization problems.

Bilevel optimization refers to a challenging optimization problem which contains two levels of optimization problems. The task of bilevel optimization is to find the optimum of the upper-level problem, subject to the optimality of the corresponding lower-level problem. This nested nature introduces many difficulties such as nonconvexity and disconnectedness, and poses great challenges to traditional optimization methods. Using evolutionary algorithms in bilevel optimization has been demonstrated to be very promising in recent years. However, these algorithms suffer from low efficiency since they usually require a huge number of function evaluations. This paper proposes a bilevel covariance matrix adaptation evolution strategy to handle bilevel optimization problems. A search distribution sharing mechanism is designed so that we can extract a priori knowledge of the lower-level problem from the upper-level optimizer, which significantly reduces the number of function evaluations. We also propose a refinement-based elite preservation mechanism to trace the elite and avoid inaccurate solutions. Comparisons with five state-of-the-art algorithms on 22 benchmark problems and two real-world applications are carried out to test the performance of the proposed approach. The experimental results have shown the effectiveness of the proposed approach in keeping a good tradeoff between solution quality and computational efficiency.

Decomposition-based many-objective evolutionary algorithms generally decompose the objective space into multiple subregions with the help of a set of reference vectors. The resulting subregions are fixed since the reference vectors are usually predefined. When the optimization problem has a complicated Pareto front (PF), this decomposition may decrease the algorithm performance. To deal with this problem, this paper proposes a dynamical decomposition strategy. Instead of using predefined reference vectors, solution themselves are used as reference vectors. Thus, they are adapted to the shape of PF automatically. Besides, the subregions are produced one by one through successively bipartitioning the objective space. The resulting subregions are not fixed but dynamically determined by the population solutions as well as the subregions produced previously. Based on this strategy, a solution ranking method, named dynamical-decomposition-based ranking method (DDR), is proposed which can be employed in the mating selection and environmental selection in commonly used algorithm frameworks. Compared with those in the other decomposition-based algorithms, DDR has the following properties: 1) no predefined reference vectors are required; 2) less parameters are involved; and 3) the ranking results can not only be utilized directly to select solutions but also serve as a secondary criterion in traditional Pareto-based algorithms. In this paper, DDR is equipped in two algorithm frameworks for handling many-objective optimization problems. Comparisons with five state-of-the-art algorithms on 31 widely used test problems are carried out to test the performance of the proposed approach. The experimental results have shown the effectiveness of the proposed approach in keeping a good tradeoff between convergence and diversity.