Many important physical systems belong to distributed parameter systems (DPSs), which require appropriate models for optimization, decision-making, and control. The modeling of DPSs is typically challenging due to their highly time-space coupled nature. Time/space separation-based modeling methods have attracted substantial attention due to their ability to decouple spatiotemporal dynamics. From a systematic perspective, we identify and clarify a developmental trajectory underlying these methods, which, to the best of our knowledge, has not been explicitly presented in prior surveys. Specifically, these methods have progressed through a trajectory from traditional physics-based methods, to physics-data hybrid methods, then to purely data-driven methods, and more recently to deep physics-informed learning methods. Motivated by this newly identified paradigm, this paper presents a comprehensive review of time/space separation-based modeling methods over the past 15 years. Furthermore, we systematically summarize uncertainty quantification that has been overlooked in existing related surveys. Building upon these findings, we reveal potential future research directions for this class of methods. In addition, several representative application examples are provided to offer practical guidance for researchers and practitioners.

Balancing optimality and robustness is the key to solving expensive robust multiobjective optimization problems (ExRMOPs) by evolutionary algorithms. However, existing studies usually design algorithms based on either the average perspective or the worst perspective, overlooking the complementarity of these two perspectives-the former prefers optimality, whereas the latter prefers robustness. Therefore, this article proposes a Kriging-assisted evolutionary algorithm with dual perspectives and dual indicators (called KPI) to solve ExRMOPs. In KPI, we develop a dual-perspective aggregation function (DPAF) as the replaced objective to guide the evolutionary search. Specifically, in terms of each original objective, DPAF of each solution is defined as the weighted sum of the performance evaluated from the average perspective and the worst perspective. The weight used in DPAF is related to the stability level of the current population, enabling DPAF to adaptively balance optimality and robustness. In addition, we design a dual-indicator candidate selection strategy to identify high-quality candidates from the final population of the evolutionary search for expensive function evaluations. In this strategy, we first eliminate solutions with poor robust optimality by the proposed robust optimality indicator. Subsequently, based on the robust optimality indicator and a common diversity indicator, several solutions with good robust optimality and diversity are selected as candidates from the remaining solutions. Extensive experiments on two test suites and a real-world application verify the superiority of KPI.

Dynamic multi-objective optimisation (DMO) handles optimisation problems with multiple (often conflicting) objectives in varying environments. This paper proposes vector autoregressive evolution (VARE) consisting of vector autoregression (VAR) and environment-aware hypermutation (EAH) to address environmental changes in DMO. In light of mutual dependency between decision variables in Pareto-optimal solutions, VARE builds an efficient VAR model, capturing such mutual relationship while handling dense model parameterisation with dimensionality reduction, to predict the moving solutions in dynamic environments. Additionally, VARE introduces EAH to address the blindness of existing hypermutation strategies in increasing population diversity, for scenarios where predictive approaches are unsuitable, by making hypermutation aware of the significance of environmental changes in both decision and objective spaces. A seamless integration of VAR and EAH in an environment-adaptive manner makes VARE effective to handle a variety of dynamic environments and competitive with several popular DMO algorithms, as demonstrated in extensive empirical studies. Specially, the proposed algorithm is computationally much faster than popular transfer learning based approaches while producing significantly better results.

Constrained multiobjective optimization evolutionary algorithms cope with various constraints through the combination of a constraint violation evaluation (CVE) framework with a constraint handling technique. The evaluation of constraint violation is a critical problem that determines how effectively constraint information is utilized. However, this topic has received limited attention in existing research. To bridge this gap, an adaptive CVE (ACVE) framework that considers the evolutionary state is proposed in this paper. ACVE first divides solutions into multiple clusters. Each cluster is then reassigned a constraint violation value. By adjusting the number of clusters based on the evolutionary state, ACVE adaptively utilizes constraint information at different levels of granularity. This design allows ACVE to achieve a more optimal balance between constraint satisfaction and objective optimization, thereby reducing the dependency on constraint handling techniques. Extensive experiments conducted on several benchmark test suites demonstrate the effectiveness of ACVE. Based on ACVE, we develop the dual-population dynamic coevolutionary algorithm (DDCo). In experiments on multiple benchmark test suites, DDCo demonstrates superior or competitive performance compared with state-of-the-art algorithms, as evaluated using indicators such as inverted generational distance and hypervolume. Moreover, DDCo is successfully applied to optimize the charging protocols of lithium-ion batteries.

This article focuses on indoor antenna layout optimization, aiming to minimize the number of antennas while maximizing the network coverage rate by optimizing the number and locations of antennas. This optimization problem exhibits three key characteristics: 1)Expensive evaluation:Indoor scenarios often require expensive propagation models to evaluate the network coverage rate; 2)Permutation invariance:Rearranging antenna locations does not change the network coverage rate; and 3)Variable dimension:The variable number of antennas leads to variable-dimensional solutions. Although surrogate models can be adopted for existing data-driven evolutionary algorithms to replace expensive propagation models, thereby reducing evaluation costs, they overlook permutation invariance and struggle to handle variable-dimensional solutions. To this end, this article proposes a novel$d$ata-driven$e$volutionary algorithm for indoor$a$ntenna$l$ayout optimization, called DEAL. Since the indoor antenna layout consists of a set of antenna locations, DEAL leverages a neural network that operates on sets as a surrogate model. For this surrogate model, the inherent invariance of set elements to permutations is leveraged, enabling DEAL to effectively achieve permutation invariance. Furthermore, due to the variable number of set elements, DEAL can adapt to variable-dimensional solutions. DEAL also incorporates a clustering-based strategy to generate initial antenna layouts and a local search method to further improve the performance of promising solutions. Extensive experiments on eight test scenarios demonstrate that DEAL outperforms five other algorithms in terms of the network coverage rate.

Zero-shot learning (ZSL) for unknown target recognition in synthetic aperture radar (SAR) images is of great significance as the training samples of certain targets may not be available under real-world scenarios. Existing ZSL algorithms usually integrate SAR images with semantics for unknown target recognition in SAR images. However, for these ZSL algorithms, semantics can only provide very limited information and some semantics may not be helpful compared with optical images. To this end, we develop an optical image-assisted ZSL (OZSL) algorithm, which includes three loss terms. Specifically, the first loss term explores intermodal correlation between SAR images and optical images. Subsequently, the second loss term explores the interview correlation among SAR images. However, combining these two loss terms is still unable to recognize unknown SAR images. Considering this, the third loss term is designed by exploiting additional optical images to assist the above two loss terms. Afterward, these three loss terms are jointly optimized to acquire the optimal solutions of the related variables involved in them by the corresponding optimization algorithms. Finally, according to the Kullback–Leibler divergence, OZSL exploits the optimal solutions to achieve unknown SAR image recognition. The results verify the superiority of OZSL.

This article proposes a novel constrained multiobjective evolutionary Bayesian optimization algorithm based on decomposition (named CMOEBO/D) for expensive constrained multiobjective optimization problems (CMOPs). In CMOEBO/D, an expensive CMOP is decomposed into some approximate constrained subproblems by Gaussian process models, reference vectors, and the augmented Tchebycheff function. Then, we devise a new infill criterion (named CPoB) to evaluate the performance of solutions. Specifically, in CPoB, on each approximate constrained subproblem, any two solutions are compared based on the product of two probabilities. For each solution, the first probability (denoted as PoB) is its likelihood of outperforming the other in terms of the predicted augmented Tchebycheff function value, and the second probability (denoted as PoF) is its likelihood of satisfying all constraints. It is obvious that PoB and PoF measure the convergence and feasibility of a solution, respectively. Based on CPoB, the approximate constrained subproblems are solved via collaborative evolutionary optimization to obtain their near-optimal solutions. Furthermore, by combining the information of the database, we design a bilevel candidate selection strategy to select some of these near-optimal solutions for expensive fitness evaluations, which can make good diversity, convergence, and feasibility contributions simultaneously to the database. Extensive experiments verify the competitiveness of CMOEBO/D.

Many industrial processes are typically distributed parameter systems (DPSs) described by partial differential equations. Data-driven methods have become popular for spatiotemporal modeling of DPSs, which is crucial for system understanding, simulation, and control improvement. However, current data-driven methods rely heavily on data volume and fidelity. With limited high-fidelity data, they exhibit unsatisfactory long-term predictions for out-of-sample scenarios. To remedy this issue, a novel physics-enhanced separation framework (called PhysiT/S) is proposed. PhysiT/S is composed of a physics-enhanced spatiotemporal unit and a coefficient network. By enhancing physics utilization through the physics-enhanced spatiotemporal unit, PhysiT/S becomes less data-dependent while computationally efficient. Multi-fidelity learning of the coefficient network further alleviates the request for a large volume of high-fidelity data. As a result, PhysiT/S can accurately predict out-of-sample distributions, even when only limited high-fidelity data is available. PhysiT/S introduces a novel way of combining physics with multi-fidelity data for spatiotemporal modeling of DPSs. Extensive simulations on two benchmark DPSs and the thermal process of lithium-ion batteries demonstrate the merits of PhysiT/S. Note to Practitioners—This paper is motivated by the problem of data-driven spatiotemporal modeling of DPSs, which is also applicable to other prediction tasks for complex spatiotemporal systems. Existing data-driven approaches rely heavily on high-fidelity data, while physics-informed approaches struggle to extract coupled spatiotemporal features. These drawbacks limit their practical applications to engineering modeling. In this paper, we propose a new generic modeling framework that integrates physics with data through the theory of time/space separation. It aims to achieve accurate long-term predictions for out-of-sample scenarios by constructing spatiotemporal enhancement units and using multi-fidelity data to decrease reliance on high-fidelity datasets. We have applied the proposed method to the thermal processes of a catalytic rod and a cylindrical lithium-ion battery. Preliminary results show that this method is feasible with better predictions than others, but it is untested in production. Future research will address online modeling in practical engineering.

This paper proposes a surrogate-assisted evolutionary framework (called SELF) to solve expensive multitask optimization problems (ExMTOPs). SELF consists of two main phases: global knowledge transfer phase and local knowledge transfer phase. In the former, a multitask Gaussian process model (MTGP) is established by fusing previously evaluated solutions of multiple optimization tasks. MTGP can capture task-relevant information and the knowledge of landscapes. Then, differential evolution assisted with MTGP is proposed to preselect high-quality candidates. During the preselection, the knowledge of landscapes is transferred among multiple optimization tasks for locating promising regions quickly. In the latter, for each optimization task, Bayesian optimization is adopted to improve the quality of the best individual in the population. Moreover, the improved best individuals in the populations of multiple optimization tasks are adaptively transferred based on a transfer probability, which is computed through the task-relevant information provided by MTGP. By combining these two phases, SELF not only achieves the tradeoff between exploration and exploitation, but also utilizes the global and local knowledge transfer to improve the efficiency for solving ExMTOPs. We test SELF on seven benchmark test problems in the IEEE CEC2017 evolutionary multitask optimization competition. The results demonstrate that the performance of SELF is better than that of other seven advanced methods. In addition, we also apply SELF to deal with two real-world ExMTOPs. The designs provided by SELF exhibit the best performance among all the compared methods, verifying the potential of SELF in practical engineering applications.

This article proposes a distribution information-based Kriging-assisted evolutionary algorithm (named DISK) to tackle expensive many-objective optimization problems (EMaOPs). In DISK, we design a new Pareto dominance relationship (called DIPD) to guide the evolutionary search and candidate selection. DIPD works based on the Kriging models and incorporates the decision-space distribution information of the nondominated solutions in the database. Such distribution information can be used to assess the possibility of an unknown solution being located in/close to the decision-space promising region. Thanks to this property, DIPD is capable of preserving the predicted elitist solutions located in/close to the decision-space promising region. These solutions are very likely to possess good original Pareto optimality and are beneficial for improving the convergence of the nondominated-solution set in the database. In addition, to further ensure the diversity of the nondominated-solution set in the database, we also design an adaptive exploration strategy, which explores the objective-space unknown region farthest away from the nondominated solutions in the database once the optimization process stagnates. Furthermore, through a feasibility-first mechanism, we extend DISK to deal with constrained EMaOPs, obtaining$\textrm {DISK}^{+}$. Finally, we verify the competitiveness of DISK and$\textrm {DISK}^{+}$via extensive experiments.

This paper proposes a new parameterless constraint-handling technique, named constrained probabilistic Pareto dominance (CPPD), for expensive constrained multiobjective optimization problems (CMOPs). In CPPD, when comparing two solutions, in terms of each original objective, we design a new objective for each solution, which is the negative product of two probabilities calculated based on the predicted fitness mean values and the uncertainty information provided by Kriging models: 1) the probability that this solution satisfies all constraints, denoted as PoF, and 2) the probability that this solution is better than the other on the original objective, denoted as PoB. It is evident that for each solution, PoF and PoB indicate its feasibility and its optimality on the corresponding original objective, respectively. Then, Pareto dominance based on new objectives is executed. As a result, both competitive feasible solutions and promising infeasible solutions with good diversity can be preserved by CPPD. These two kinds of solutions can help the population to exploit the located feasible parts and to explore new feasible parts, respectively. Further, based on CPPD, we develop a Pareto-based Kriging-assisted constrained multiobjective evolutionary algorithm (called PEA) to deal with expensive CMOPs with two or three objectives. Finally, PEA is generalized to solve expensive constrained many-objective optimization problems, named PEA+. The effectiveness of CPPD, PEA, and PEA+ is verified by comprehensive experiments.

Accurately modeling the electrochemical process of large-scale lithium-ion batteries (LLBs), which involves estimating the electrochemical state distributions within the process, is crucial for the design and management of LLBs. A two-dimensional (2-D) physics-based model can describe the electrochemical process of LLBs accurately. However, due to the presence of complex partial differential equations (PDEs), solving the model becomes a challenging task. This article develops a physics-informed composite network (PICN) as a surrogate model of the 2-D physics-based model. Specifically, PICN consists of four deep neural networks (DNNs) to estimate the distributions of four key electrochemical states, respectively. Since the architecture of PICN is inspired by PDE characteristics, it can achieve high accuracies with four lightweight DNNs. Additionally, by incorporating physics and data, PICN achieves accurate estimations using limited data. It can even estimate the electrochemical state distributions that may not be measured directly. Moreover, PICN presents a low-frequency information-based pretraining strategy and a two-stage loss balance strategy to address the convergence failure and loss imbalance that may arise in the training of PICN. PICN is a new attempt to model the electrochemical process of LLBs by integrating physics with data. Extensive experiments show that it is better than state-of-the-art models.

This article proposes a constrained evolutionary Bayesian optimization (CEBO) algorithm to cope with expensive constrained optimization problems with inequality constraints. The uniqueness of CEBO lies in its capability of balancing feasibility and objective improvement under a limited function evaluation budget, which is achieved by designing two strategies to obtain promising solutions. The first strategy prefers feasibility. It tends to obtain a feasible solution by utilizing the predicted value and uncertainty provided by Gaussian process (GP). The second strategy prefers objective improvement. It maintains and evolves the population of evolutionary algorithms, and selects a solution with a good objective function value and violating the constraints not too much based on the predicted value and uncertainty provided by GP at each iteration. The sequential implementation of these two strategies allows CEBO to balance feasibility and objective improvement. The effectiveness of CEBO is verified by 26 test instances and a practical application. The results demonstrate that CEBO is able to find high-quality solutions with 100 FEs.

This article introduces a novel time/space separation-based physics-informed machine learning (T/S-PIML) modeling method by making full use of the complementary strengths of the physics-informed neural network (PINN) and the time/space separation methodology. T/S-PIML is the first attempt to seamlessly integrate structural (including spatial and temporal) physical information with data for effective spatiotemporal modeling of distributed parameter systems (DPSs). With the help of the spectral method, spatial basis functions are first extracted to capture spatial physical information. Subsequently, a reduced-order system is derived to characterize the corresponding temporal physical information. Upon the structural physical information, PINN is developed for temporal modeling. Following the time/space synthesis, a small amount of sensing data is utilized to calibrate system errors. Experiments on a benchmark DPS and the thermal process of a lithium-ion battery demonstrate the effectiveness of T/S-PIML.

Some recent methods address few-shot image classification by extracting semantic information from class names and devising mechanisms for aligning vision and semantics to integrate information from both modalities. However, class names provide only limited information, which is insufficient to capture the visual details in images. As a result, such vision-semantics alignment is inherently biased, leading to suboptimal integration outcomes. In this paper, we avoid such biased vision-semantics alignment by introducing CLIP, a natural bridge between vision and semantics, and enforcing unbiased vision-vision alignment as a proxy task. Specifically, we align features encoded from the few-shot encoder and CLIP's vision encoder on the same image. This alignment is accomplished through a linear projection layer, with a training objective formulated using optimal transport-based assignment prediction. Thanks to the inherent alignment between CLIP's vision and text encoders, the few-shot encoder is indirectly aligned to CLIP's text encoder, which serves as the foundation for better vision-semantics integration. In addition, to further improve vision-semantics integration at the testing stage, we mine potential fine-grained semantic attributes of class names from large language models. Correspondingly, an online optimization module is designed to adaptively integrate the semantic attributes and visual information extracted from images. Extensive results on four datasets demonstrate that our method outperforms state-of-the-art methods. The code is available at https://github.com/zhuolingli/FewVS.

Multiview fusion algorithms have been widely applied in synthetic aperture radar automatic target recognition (SAR ATR). However, they usually ignore semantic information, which results in limited recognition performance. Therefore, how to effectively integrate semantic information into multiview fusion algorithms and explore the correlation between multiple views and semantic information is a crucial and urgent problem. To address this problem, we develop a semantics-assisted multiview fusion (SMVF) algorithm, which includes three loss terms, i.e., view-specific loss term, semantics-regularized loss term, and view-semantics-coupled loss term. To be specific, the view-specific and semantics-regularized loss terms convert multiple views and semantic information into their corresponding sparse codes, respectively. The view-semantics-coupled loss term constrains the sparse codes of multiple views and semantic information to explore their correlation. Finally, these three loss terms are jointly optimized to acquire the optimal sparse codes to calculate the reconstruction errors for recognition, by which SMVF not only effectively exploits semantic information, but also explores the correlation between multiple views and semantic information. Extensive experiments demonstrate that SMVF achieves high recognition accuracies (i.e., 99.7%, 91.6%, and 97.2%) under three scenarios (i.e., EOC-1, EOC-2, and SAR-ACD), which are better than other advanced algorithms.

Recently, finetuning-based methods have shown great performance in few-shot object detection. These methods employ pure convolutional structures for feature extraction, and then finetune the last layers of the base detector on novel classes. However, most of them have the following two issues: 1) the feature extraction processes of support and query sets lack mutual promotion, and 2) the features extracted by pure convolutional structures have low discriminability for confused classes, which could lead to false positive results. To overcome these two issues, we propose a support-query mutual promotion and classification correction network (SMPCCNet). First, we build a support-query mutual promotion module. In this module, we perform class attention operations with query features to produce enhanced support features, and use the dense kernels generated by the enhanced support features to obtain support-relevant query features, which achieves the mutual promotion of support and query features. In addition, we design a classification correction branch with a hybrid attention module. The hybrid attention module uses spatial and channel attentions to generate features that focus on the positions and types of detected objects, respectively, which can better discriminate confused classes and correct false positive results. Extensive experiments demonstrate the superiority of SMPCCNet over state-of-the-art methods.

Human motion prediction is to predict future human states based on the observed human states. However, current research ignores the semantic correlations between body parts (joints and bones) in the observed human states and motion time; thus, the prediction accuracy is limited. To address this issue, we propose a novel semantic correlation attention-based multiorder multiscale feature fusion network (SCAFF), which includes an encoder and a decoder. In the encoder, a multiorder difference calculation module (MODC) is designed to calculate the multiorder difference information of joint and bone attributes in the observed human states. Then, multiple semantic correlation attention-based graph calculation operators (SCA-GCOs) are stacked to extract the multiscale features of the multiorder difference information. Each SCA-GCO captures joint and bone dependencies of the multiorder difference information, refines them with a semantic correlation attention module (SCAM), and captures temporal dynamics of the refined joint and bone dependencies as the output features. Note that SCAM learns a semantic attention mask describing the semantic correlations between body parts and motion time for feature refinement. Afterward, multiple multiorder feature fusion modules (MOFFs) and multiscale feature fusion modules (MSFFs) are designed to fuse the multiscale features of the multiorder difference information extracted by multiple SCA-GCOs, thus obtaining the motion features of the observed human states. Based on the obtained motion features, the decoder recurrently recruits a composite gated recurrent module (CGRM) and multilayer perceptrons (MLPs) to predict future human states. As far as we know, this is the first attempt to consider the semantic correlations between body parts and motion time in human motion prediction. The results on public datasets demonstrate that SCAFF outperforms existing models.

The goal of constrained multiobjective evolutionary optimization is to obtain a set of well-converged and well-distributed feasible solutions. To achieve this goal, a delicate tradeoff must be struck among feasibility, diversity, and convergence. However, balancing these three elements simultaneously through a single tradeoff model is nontrivial, mainly because the significance of each element varies in different evolutionary phases. As an alternative approach, we adapt distinct tradeoff models in various phases and introduce a novel algorithm named adaptive tradeoff model with reference points (ATM-R). In the infeasible phase, ATM-R takes the tradeoff between diversity and feasibility into account, aiming to move the population toward feasible regions from diverse search directions. In the semi-feasible phase, ATM-R promotes the transition from "the tradeoff between feasibility and diversity" to "the tradeoff between diversity and convergence." This transition is instrumental in discovering an adequate number of feasible regions and accelerating the search for feasible Pareto optima in succession. In the feasible phase, ATM-R places an emphasis on balancing diversity and convergence to obtain a set of feasible solutions that are both well-converged and well-distributed. It is worth noting that the merits of reference points are leveraged in ATM-R to accomplish these tradeoff models. Also, in ATM-R, a multiphase mating selection strategy is developed to generate promising solutions beneficial to different evolutionary phases. Systemic experiments on a diverse set of benchmark test functions and real-world problems demonstrate that ATM-R is effective. When compared to eight state-of-the-art constrained multiobjective optimization evolutionary algorithms, ATM-R consistently demonstrates its competitive performance.

Dynamic constrained multiobjective optimization problems (DCMOPs) abound in real-world applications and gain increasing attention in the evolutionary computation community. To evaluate the capability of an algorithm in solving DCMOPs, artificial test problems play a fundamental role. Nevertheless, some characteristics of real-world scenarios are not fully considered in the previous test suites, such as time-varying size, location and shape of feasible regions, the controllable change severity, as well as small feasible regions. Therefore, we develop the generators of objective functions and constraints to facilitate the systematic design of DCMOPs, and then a novel test suite consisting of nine benchmarks, termed as DCP, is put forward. To solve these problems, a dynamic constrained multiobjective evolutionary algorithm with a two-stage diversity compensation strategy (TDCEA) is proposed. Some initial individuals are randomly generated to replace historical ones in the first stage, improving the global diversity. In the second stage, the increment between center points of Pareto sets in the past two environments is calculated and employed to adaptively disturb solutions, forming an initial population with good diversity for the new environment. Intensive experiments show that the proposed test problems enable a good understanding of strengths and weaknesses of algorithms, and TDCEA outperforms other state-of-the-art comparative ones, achieving promising performance in tackling DCMOPs.

The charging process design is crucial for optimizing the performance of lithium-ion batteries by identifying protocols that meet diverse demands. The main challenges include: 1) the high costs of battery experiments; 2) the multiple user preferences associated with the demands; and 3) the intricate high-dimensional search space of charging protocols. In light of this, this article presents a Gaussian process-accelerated multiobjective evolutionary design method for effective charging process design. To resolve the first concern, an electrochemical-thermal-aging model is constructed to evaluate charging protocols precisely, substituting the need for expensive battery experiments. Besides, the Gaussian process is applied to accelerate the evaluation process further. Regarding the second issue, a Gaussian process-accelerated two-archive evolutionary algorithm (GPA-TAEA) is developed to efficiently search for a set of optimal charging protocols that satisfy multiple user preferences. To address the third challenge, differential evaluation—an evolutionary algorithm proven effective for large-scale optimization—is employed to enhance the search process. The simulation results demonstrate that: 1) the proposed method effectively reduces the time required for charging process design, yielding a collection of optimal charging protocols that includes 530 solutions within 1000 simulations; 2) compared with five other multiobjective optimization algorithms, GPA-TAEA exhibits superior convergence and diversity; and 3) compared with fixed preference-based methods, GPA-TAEA demonstrates greater efficiency, saving 54% in simulation evaluations and 70% in time when considering seven user preferences.

This article studies a multiagile earth observation satellite system, in which a group of satellites provides observation services to acquire images of targets on the earth’s surface. In this system, the large-scale multiobjective task scheduling problem is studied by jointly optimizing the task assignment scheme and the observation window allocation to maximize the total profit of all executed tasks on all satellites and the loading balance among satellites. Note, however, that it is challenging to solve this problem since the task assignment scheme and the observation window allocation are tightly coupled. Therefore, a bilevel optimization problem is formulated, where the tasks are assigned at the upper level and the observation windows are allocated at the lower level. In this way, the observation windows are allocated based on the given task assignment scheme, thus decoupling the task assignment scheme and the observation window allocation. Furthermore, the observation windows can be allocated in parallel on different satellites to improve computational efficiency. Subsequently, a bilevel evolutionary algorithm is proposed. Specifically, at the upper level, an initialization strategy is devised to efficiently generate feasible task assignment schemes by constructing the candidate satellite set for each task, and then a constrained multiobjective evolutionary algorithm is adopted to optimize the task assignment schemes. In addition, at the lower level, for each task assignment scheme, a greedy strategy is proposed to allocate the observation windows to as many tasks as possible on each satellite and a local search method is suggested to further improve the observation window allocation. Experiments on a diverse set of instances involving up to 1000 tasks demonstrate that the proposed algorithm exhibits better or at least competitive performance against other compared algorithms on each instance.

This article devises a two-phase Kriging-assisted evolutionary algorithm (named TEA) to tackle expensive constrained multiobjective optimization problems (CMOPs). In the first phase, only objectives are considered, which can help the population to cross infeasible obstacles and to evolve toward the unconstrained Pareto front. Since the unconstrained Pareto front is in front of the feasible region in the objective space, the first phase can find some feasible solutions during the evolution. In the second phase, both objectives and constraints are considered. In this article, we also propose two transition conditions to judge whether the search should be switched from the first phase to the second phase, by making use of the candidates evaluated by the original objectives and constraints in the first phase. These two transition conditions aim at maintaining some high-quality feasible solutions when the first phase ends, which is able to motivate the population to converge toward the constrained Pareto front with good diversity in the second phase. Furthermore, in both phases, we design a new Pareto dominance relationship (called PDPD) by incorporating the probability distribution information derived from the Kriging models. PDPD is further generalized to handle constraints in expensive CMOPs, Constrained PDPD (CPDPD), which provides high credibility for the comparison between two individuals with respect to both objectives and constraints. Finally, three benchmark test suites and a real-world application confirm the superiority of TEA.

Some recent methods address few-shot classification by integrating visual and semantic prototypes. However, they usually ignore the difference in feature structure between the visual and semantic modalities, which leads to limited performance improvements. In this paper, we propose a novel method, called bimodal integrator (BMI), to better integrate visual and semantic prototypes. In BMI, we first construct a latent space for each modality via a variational autoencoder, and then align the semantic latent space to the visual latent space. Through this semantics-to-vision alignment, the semantic modality is mapped to the visual latent space and has the same feature structure as the visual modality. As a result, the visual and semantic prototypes can be better integrated. In addition, based on the multivariate Gaussian distribution and the prompt engineering, a data augmentation scheme is designed to ensure the accuracy of modality alignment during the training process. Experimental results demonstrate that BMI significantly improves few-shot classification, making simple baselines outperform the most advanced methods on miniImageNet and tieredImageNet datasets.

Recent years have witnessed the success of encoder‐decoder structure‐based approaches in lung region segmentation of chest X‐ray (CXR) images. However, accurate lung region segmentation is still challenging due to the following three issues: (1) inaccurate lung region segmentation boundaries, (2) existence of lesion‐related artifacts (e.g., opacity and pneumonia), and (3) lack of the ability to utilize multiscale information. To address these issues, we propose an edge‐assisted computing and mask attention based network (called EAM‐Net), which consists of an encoder‐decoder network, an edge‐assisted computing module, and multiple mask attention modules. Based on the encoder‐decoder structure, an edge‐assisted computing module is first proposed, which integrates the feature maps of the shallow encoding layers for edge prediction, and uses the edge evidence map as a strong cue to guide the lung region segmentation, thereby refining the lung region segmentation boundaries. We further design a mask attention module after each decoding layer, which employs a mask attention operation to make the model focus on lung regions while suppressing the lesion‐related artifacts. Besides, a multiscale aggregation loss is proposed to optimize EAM‐Net. Extensive experiments on the JSRT, Shenzhen, and Montgomery datasets demonstrate that EAM‐Net outperforms existing state‐of‐the‐art lung region segmentation methods.

This article studies an intelligent reflecting surface (IRS)-aided communication system under the time-varying channels and stochastic data arrivals. In this system, we jointly optimize the phase-shift coefficient and the transmit power in sequential time slots to maximize the long-term energy consumption for all mobile devices while ensuring queue stability. Due to the dynamic environment, it is challenging to ensure queue stability. In addition, making real-time decisions in each short time slot also needs to be considered. To this end, we propose a method (called LETO) that combines Lyapunov optimization with evolutionary transfer optimization (ETO) to solve the above optimization problem. LETO first adopts Lyapunov optimization to decouple the long-term stochastic optimization problem into deterministic optimization problems in sequential time slots. As a result, it can ensure queue stability since the deterministic optimization problem in each time slot does not involve future information. After that, LETO develops an evolutionary transfer method to solve the optimization problem in each time slot. Specifically, we first define a metric to identify the optimization problems in past time slots similar to that in the current time slot, and then transfer their optimal solutions to construct a high-quality initial population in the current time slot. Since ETO effectively accelerates the search, we can make real-time decisions in each short time slot. Experimental studies verify the effectiveness of LETO by comparison with other algorithms.

This article presents a new constraint-handling technique (CHT), called shift-based penalty (ShiP), for solving constrained multiobjective optimization problems. In ShiP, infeasible solutions are first shifted according to the distributions of their neighboring feasible solutions. The degree of shift is adaptively controlled by the proportion of feasible solutions in the current parent and offspring populations. Then, the shifted infeasible solutions are penalized based on their constraint violations. This two-step process can encourage infeasible solutions to approach/enter the feasible region from diverse directions in the early stage of evolution, and guide diverse feasible solutions toward the Pareto optimal solutions in the later stage of evolution. Moreover, ShiP can achieve an adaptive transition from both diversity and feasibility in the early stage of evolution to both diversity and convergence in the later stage of evolution. ShiP is flexible and can be embedded into three well-known multiobjective optimization frameworks. Experiments on benchmark test problems demonstrate that ShiP is highly competitive with other representative CHTs. Further, based on ShiP, we propose an archive-assisted constrained multiobjective evolutionary algorithm (CMOEA), called ShiP+, which outperforms two other state-of-the-art CMOEAs. Finally, ShiP is applied to the vehicle scheduling of the urban bus line successfully.

Bilevel metaheuristics have been widely used for bilevel optimization. However, recent studies have indicated that most bilevel metaheuristics are inefficient since they perform the lower-level optimization task for each upper-level solution independently and neglect the relationship among lower-level optimization tasks. In this article, we develop a bilevel metaheuristic with the collaborations among lower-level optimization tasks. Specifically, a population is evolved to solve the lower-level optimization tasks for all upper-level solutions collaboratively at each generation. In the population, each solution is associated with a lower-level optimization task. In such a way, all lower-level optimization tasks can be solved in a single run. To capture the individual features of different lower-level optimization tasks, we construct a lower-level search distribution for each lower-level optimization task based on all solutions in the population. In addition, an information-sharing mechanism is proposed to share good solutions among lower-level optimization tasks. Experiments on two sets of test problems and three practical applications demonstrate that our proposed algorithm performs better than other bilevel metaheuristics in comparison.

Although many methods have been proposed to solve expensive optimization problems (EOPs), they often consume hundreds of function evaluations (FEs) to find the optimal solution, which is unacceptable when facing highly EOPs. To reduce the number of FEs, we incorporate the population distribution into the well-known expected improvement (EI); thus, a new infill criterion called evolutionary EI (EEI) is proposed. In EEI, the covariance matrix adaptation evolution strategy is used to provide the population distribution. Compared with the original EI, EEI focuses more on promising regions provided by the population distribution, thus, reducing the FEs wasted in unpromising regions. By employing EEI as the infill criterion of Bayesian optimization, a new algorithm called EEI-BO is designed. Moreover, we also introduce an extended version of EEI-BO, called EEI-BO+, to handle multitask EOPs. To verify the effectiveness of EEI-BO, it is used to solve 10−, 20−, and 30-D test problems by using only 40, 50, and 60 FEs, respectively. The results show that EEI-BO is able to obtain high-quality solutions by consuming limited FEs. In addition, we apply EEI-BO to deal with the lightweight and crashworthiness design of the side body of an automobile. The results demonstrate that EEI-BO performs well on solving it. Furthermore, the performance of EEI-BO+is investigated by nine test problems. The results show that it has the capability to solve multitask EOPs with fast convergence speed.

This article investigates a wireless-powered mobile edge computing (MEC) system, where the service provider (SP) provides the device owner (DO) with both computing resources and energy to execute tasks from Internet-of-Things devices. In this system, SP first sets the prices of computing resources and energy whereas DO then makes the optimal response according to the given prices. In order to jointly optimize the prices of computing resources and energy, we formulate a bilevel optimization problem (BOP), in which the upper level generates the prices of computing resources and energy for SP and then under the given prices, the lower level optimizes the mode selection, broadcast power, and computing resource allocation for DO. This BOP is difficult to address due to the mixed variables at the lower level. To this end, we first derive the relationships between the optimal broadcast power and the mode selection and between the optimal computing resource allocation and the mode selection. After that, it is only necessary to consider the discrete variables (i.e., mode selection) at the lower level. Note, however, that the transformed BOP is still difficult to solve because of the extremely large search space. To solve the transformed BOP, we propose a divide-and-conquer bilevel optimization algorithm (called DACBO). Based on device status, task information, and available resources, DACBO first groups tasks into three independent small-size sets. Afterward, analytical methods are devised for the first two sets. As for the last one, we develop a nested bilevel optimization algorithm that uses differential evolution and variable neighborhood search (VNS) at the upper and lower levels, respectively. In addition, a greedy method is developed to quickly construct a good initial solution for VNS. The effectiveness of DACBO is verified on a set of instances by comparing with other algorithms.

A mixed-integer programming (MIP) problem contains both constraints and integer restrictions. Integer restrictions divide the feasible region defined by constraints into multiple discontinuous feasible parts. In particular, the number of discontinuous feasible parts will drastically increase with the increase of the number of integer decision variables and/or the size of the candidate set of each integer decision variable. Due to the fact that the optimal solution is located in one of the discontinuous feasible parts, it is a challenging task to solve a MIP problem. This article presents a cutting and repulsion-based evolutionary framework (called CaR) to solve MIP problems. CaR includes two main strategies: 1) the cutting strategy and 2) the repulsion strategy. In the cutting strategy, an additional constraint is constructed based on the objective function value of the best individual found so far, the aim of which is to continuously cut unpromising discontinuous feasible parts. As a result, the probability of the population entering a wrong discontinuous feasible part can be decreased. In addition, in the repulsion strategy, once it has been detected that the population has converged to a discontinuous feasible part, the population will be reinitialized. Moreover, a repulsion function is designed to repulse the previously explored discontinuous feasible parts. Overall, the cutting strategy can significantly reduce the number of discontinuous feasible parts and the repulsion strategy can probe the remaining discontinuous feasible parts. Sixteen test problems developed in this article and two real-world cases are used to verify the effectiveness of CaR. The results demonstrate that CaR performs well in solving MIP problems.

As an effective optimization tool for expensive optimization problems (EOPs), surrogate-assisted evolutionary algorithms (SAEAs) have been widely studied in recent years. However, most current SAEAs are designed for continuous/ combinatorial EOPs, which are not suitable for mixed-variable EOPs. This article focuses on one kind of mixed-variable EOP: EOPs with continuous and categorical variables (EOPCCVs). A multisurrogate-assisted ant colony optimization algorithm (MiSACO) is proposed to solve EOPCCVs. MiSACO contains two main strategies: 1) multisurrogate-assisted selection and 2) surrogate-assisted local search. In the former, the radial basis function (RBF) and least-squares boosting tree (LSBT) are employed as the surrogate models. Afterward, three selection operators (i.e., RBF-based selection, LSBT-based selection, and random selection) are devised to select three solutions from the offspring solutions generated by ACO, with the aim of coping with different types of EOPCCVs robustly and preventing the algorithm from being misled by inaccurate surrogate models. In the latter, sequence quadratic optimization, coupled with RBF, is utilized to refine the continuous variables of the best solution found so far. By combining these two strategies, MiSACO can solve EOPCCVs with limited function evaluations. Three sets of test problems and two real-world cases are used to verify the effectiveness of MiSACO. The results demonstrate that MiSACO performs well in solving EOPCCVs.

This article proposes a robust image-sequence-based framework to deal with two challenges of visual place recognition in changing environments: 1) viewpoint variations and 2) environmental condition variations. Our framework includes two main parts. The first part is to calculate the distance between two images from a reference image sequence and a query image sequence. In this part, we remove the deep features of nonoverlap contents in these two images and utilize the remaining deep features to calculate the distance. As the deep features of nonoverlap contents are caused by viewpoint variations, removing these deep features can improve the robustness to viewpoint variations. Based on the first part, in the second part, we first calculate the distances of all pairs of images from a reference image sequence and a query image sequence, and obtain a distance matrix. Afterward, we design two convolutional operators to retrieve the distance submatrix with the minimum diagonal distribution. The minimum diagonal distribution contains more environmental information, which is insensitive to environmental condition variations. The experimental results suggest that our framework exhibits better performance than several state-of-the-art methods. Moreover, the analysis of runtime shows that our framework has the potential to satisfy real-time demands.

A considerable number of surrogate-assisted evolutionary algorithms (SAEAs) have been developed to solve expensive optimization problems (EOPs) with continuous objective functions. However, in the real-world applications, we may face EOPs with discontinuous objective functions, which are also called EOPs with discontinuous responses (EOPDRs). Indeed, EOPDRs pose a great challenge to current SAEAs. In this article, a surrogate-assisted differential evolution (DE) algorithm with region division is proposed, named ReDSADE. ReDSADE includes three main strategies: 1) the region division strategy; 2) the Kriging-based search; and 3) the radial basis function (RBF)-based local search. In the region division strategy, we define a new distance measure, called the objective-decision distance. Based on this distance, the evaluated solutions are partitioned into several clusters, and several support vector machine (SVM) classifiers are trained to classify them. These SVM classifiers divide the decision space into several subregions, with the aim of making the objective function continuous in them. In the Kriging-based search, a Kriging model is established in each subregion and combined with DE to search for the optimal solution. In the RBF-based local search, DE is coupled with RBF to search around the best solution found so far, thus accelerating the convergence. By combining these three strategies, ReDSADE is able to solve EOPDRs with limited function evaluations. Three sets of test problems and a real-world application are utilized to verify the effectiveness of ReDSADE. The results demonstrate that ReDSADE exhibits good convergence accuracy and convergence speed.

Spatiotemporal modeling is critical to the simulation, optimization, and control of the thermal process of a lithium-ion battery, which is a typical kind of distributed parameter system (DPS). Data-driven spatiotemporal modeling methods are of practical interest to construct an analytical model of the thermal process of a lithium-ion battery, since they only need some sampled data rather than the structure descriptions or parameters of a DPS. How to sample data optimally for data-driven spatiotemporal modeling is still an open question. In this article, with the aim of minimizing the spatiotemporal modeling error, we propose a novel evolutionary algorithm to optimally place sensors for data sampling. First, an objective function that can quantify both the spatial error and the temporal error is designed. Additionally, a novel differential evolution algorithm with two kinds of encoding mechanisms (called DETEM) is proposed to optimize the objective function. Numerical simulations and experimental studies have shown that the proposed method is competitive. Besides, both the objective function and DETEM are critical to the proposed method. In summary, the proposed method provides an effective way to obtain the optimal sensor placement for spatiotemporal modeling of the thermal process of a lithium-ion battery.

Fusing multi-modality medical images, such as magnetic resonance (MR) imaging and positron emission tomography (PET), can provide various anatomical and functional information about the human body. However, PET data is not always available for several reasons, such as high cost, radiation hazard, and other limitations. This paper proposes a 3D end-to-end synthesis network called Bidirectional Mapping Generative Adversarial Networks (BMGAN). Image contexts and latent vectors are effectively used for brain MR-to-PET synthesis. Specifically, a bidirectional mapping mechanism is designed to embed the semantic information of PET images into the high-dimensional latent space. Moreover, the 3D Dense-UNet generator architecture and the hybrid loss functions are further constructed to improve the visual quality of cross-modality synthetic images. The most appealing part is that the proposed method can synthesize perceptually realistic PET images while preserving the diverse brain structures of different subjects. Experimental results demonstrate that the performance of the proposed method outperforms other competitive methods in terms of quantitative measures, qualitative displays, and evaluation metrics for classification.

A mixed-integer programming (MIP) problem contains not only constraints but also integer restrictions. Integer restrictions divide the feasible region defined by constraints into multiple discontinuous feasible parts with different sizes. Several popular methods (e.g., rounding and truncation) have been proposed to deal with integer restrictions. Although it is easy for these methods to generate an integer, they tend to converge to an integer which is located in a feasible part with a big size. If the optimal solution is not in this feasible part, they are very likely to converge to a local optimal solution due to the loss of diversity of the population. To overcome this shortcoming, a biobjective optimization-based two-phase method is proposed in this article. In the first phase, a measure function is designed to compute the degree that a solution violates integer restrictions. By employing this measure function as the second objective function and removing integer restrictions, a MIP problem is transformed into a constrained biobjective optimization problem (CBOP). It can be proven that the Pareto optimal solution of the transformed CBOP which satisfies integer restrictions is the optimal solution of the original MIP problem. To solve the transformed CBOP, a new comparison rule is designed. After the first phase, the population can approach the Pareto optimal solution which satisfies integer restrictions. Then, the second phase is implemented to enhance the convergence precision and obtain the optimal solution. In addition, we design 12 test problems to verify the effectiveness of the proposed method. The results demonstrate that the proposed method shows better performance against five state-of-the-art evolutionary algorithms for MIP.

Recently, many model quantization approaches have been investigated to reduce the model size and improve the inference speed of convolutional neural networks (CNNs). However, these approaches usually inevitably lead to a decrease in classification accuracy. To address this problem, this paper proposes a mixed precision quantization method combined with channel expansion of CNNs by using a multi-objective genetic algorithm, called MOGAQNN. In MOGAQNN, each individual in the population is used to encode a mixed precision quantization policy and a channel expansion policy. During the evolution process, the two polices are optimized simultaneously by the non-dominated sorting genetic algorithm II (NSGA-II). Finally, we choose the best individual in the last population and evaluate its performance on the test set as the final performance. The experimental results of five popular CNNs on two benchmark datasets demonstrate that MOGAQNN can greatly reduce the model size and improve the classification accuracy at the same time.

Human motion prediction is essential in human-robot interaction. Current research mostly considers the joint dependencies but ignores the bone dependencies and their relationship in the human skeleton, thus limiting the prediction accuracy. To address this issue, we represent the human skeleton as a directed acyclic graph with joints as vertexes and bones as directed edges. Then, we propose a novel directed acyclic graph neural network (DA-GNN) that follows the encoder-decoder structure. The encoder is stacked by multiple encoder blocks, each of which includes a directed acyclic graph computational operator (DA-GCO) to update joint and bone attributes based on the relationship between joint and bone dependencies in the observed human states, and a temporal update operator (TUO) to update the temporal dynamics of joints and bones in the same observation. After progressively implementing the above update process, the encoder outputs the final update result, fed into the decoder. The decoder includes a directed acyclic graph-based gated recurrent unit (DAG-GRU) and a multi-layered perceptron (MLP) to predict future human states sequentially. To the best of our knowledge, this is the first time to introduce the relationship between bone and joint dependencies in human motion prediction. Our experimental evaluations on two datasets, CMU Mocap and Human 3.6m, prove that DA-GNN outperforms current models. Finally, we showcase the efficacy of DA-GNN in a realistic HRI scenario.

Solving constrained multiobjective optimization problems (CMOPs) is a challenging task since it is necessary to optimize several conflicting objective functions and handle various constraints simultaneously. A promising way to solve CMOPs is to integrate multiobjective evolutionary algorithms (MOEAs) with constraint-handling techniques, and the resultant algorithms are called constrained MOEAs (CMOEAs). At present, many attempts have been made to combine dominance-based and decomposition-based MOEAs with diverse constraint-handling techniques together. However, for another main branch of MOEAs, i.e., indicator-based MOEAs, almost no effort has been devoted to extending them for solving CMOPs. In this article, we make the first study on the possibility and rationality of combining indicator-based MOEAs with constraint-handling techniques together. Afterward, we develop an indicator-based CMOEA framework which can combine indicator-based MOEAs with constraint-handling techniques conveniently. Based on the proposed framework, nine indicator-based CMOEAs are developed. Systemic experiments have been conducted on 19 widely used constrained multiobjective optimization test functions to identify the characteristics of these nine indicator-based CMOEAs. The experimental results suggest that both indicator-based MOEAs and constraint-handing techniques play very important roles in the performance of indicator-based CMOEAs. Some practical suggestions are also given about how to select appropriate indicator-based CMOEAs. Besides, we select a superior approach from these nine indicator-based CMOEAs and compare its performance with five state-of-the-art CMOEAs. The comparison results suggest that the selected indicator-based CMOEA can obtain quite competitive performance. It is thus believed that this article would encourage researchers to pay more attention to indicator-based CMOEAs in the future.

Among the constraint-handling techniques (CHTs) in constrained multiobjective optimization, constrained dominance principle (CDP) is simple, flexible, nonparametric, and easy to be embedded into multiobjective evolutionary algorithms. However, CDP always prefers constraints to objectives, which tends to cause premature convergence. To overcome this drawback, we propose a new CHT on the basis of CDP. In our CHT, the fitness function of each solution is defined as the weighted sum of two rankings: one is the solution's ranking based on CDP and the other is the solution's ranking based on Pareto dominance (i.e., without considering any constraints). It is evident that these two rankings favor the “feasibility” and “optimality” of each solution, respectively. More importantly, the weight employed in the fitness function is related to the proportion of feasible solutions in the current population, enabling the population to adaptively balance constraints and objectives in the evolutionary process. The effectiveness of our CHT is evaluated on three test suites and the experimental results demonstrate that our CHT shows better or highly competitive performance compared with other representative CHTs.

Pareto dominance-based multiobjective optimization has been successfully applied to constrained evolutionary optimization during the last two decades. However, as another famous multiobjective optimization framework, decomposition-based multiobjective optimization has not received sufficient attention from constrained evolutionary optimization. In this paper, we make use of decomposition-based multiobjective optimization to solve constrained optimization problems (COPs). In our method, first of all, a COP is transformed into a biobjective optimization problem (BOP). Afterward, the transformed BOP is decomposed into a number of scalar optimization subproblems. After generating an offspring for each subproblem by differential evolution, the weighted sum method is utilized for selection. In addition, to make decomposition-based multiobjective optimization suit the characteristics of constrained evolutionary optimization, weight vectors are elaborately adjusted. Moreover, for some extremely complicated COPs, a restart strategy is introduced to help the population jump out of a local optimum in the infeasible region. Extensive experiments on three sets of benchmark test functions, namely, 24 test functions from IEEE CEC2006, 36 test functions from IEEE CEC2010, and 56 test functions from IEEE CEC2017, have demonstrated that the proposed method shows better or at least competitive performance against other state-of-the-art methods.

Psoriasis is a chronic skin disease which affects hundreds of millions of people around the world. This disease cannot be fully cured and requires lifelong caring. If the deterioration of Psoriasis is not detected and properly treated in time, it could cause serious complications or even lead to a life threat. Therefore, a quantitative measurement that can track the Psoriasis severity is necessary. Currently, PASI (Psoriasis Area and Severity Index) is the most frequently used measurement in clinical practices. However, PASI has the following disadvantages: (1) Time consuming: calculating PASI usually takes more than 30 minutes which poses a heavy burden on dermatologists; and (2) Inconsistency: due to the complexity of PASI calculation, different or even the same dermatologist could give different scores for the same case. To overcome these drawbacks, we propose PSENet which applies deep neural networks to estimate Psoriasis severity based on skin lesion images. Different from typical deep learning frameworks for image processing, PSENet has the following characteristics: (1) PSENet introduces a score refine module which is able to capture the visual features of skin at both coarse and fine-grained granularities; (2) PSENet uses siamese structure in training and accepts pairwise inputs, which reduces the dependency on large amount of training data; and (3) PSENet can not only estimate the severity, but also locate the skin lesion regions from the input image. To train and evaluate PSENet, we work with professional dermatologists from a top hospital and spend years in building a golden dataset. The experimental results show that PSENet can achieve the mean absolute error of 2.21 and the accuracy of 77.87% in pair comparison, outperforming baseline methods. Overall, PSENet not only relieves dermatologists from the dull PASI calculation but also enables patients to track Psoriasis severity in a much more convenient manner.

MOTIVATION: Flux balance analysis (FBA) based bilevel optimization has been a great success in redesigning metabolic networks for biochemical overproduction. To date, many computational approaches have been developed to solve the resulting bilevel optimization problems. However, most of them are of limited use due to biased optimality principle, poor scalability with the size of metabolic networks, potential numeric issues or low quantity of design solutions in a single run. RESULTS: Here, we have employed a network interdiction model free of growth optimality assumptions, a special case of bilevel optimization, for computational strain design and have developed a hybrid Benders algorithm (HBA) that deals with complicating binary variables in the model, thereby achieving high efficiency without numeric issues in search of best design strategies. More importantly, HBA can list solutions that meet users' production requirements during the search, making it possible to obtain numerous design strategies at a small runtime overhead (typically ∼1 h, e.g. studied in this article). AVAILABILITY AND IMPLEMENTATION: Source code implemented in the MATALAB Cobratoolbox is freely available at https://github.com/chang88ye/NIHBA. CONTACT: [email protected] or [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

We study a mobile edge computing system assisted by multiple unmanned aerial vehicles (UAVs), where the UAVs act as edge servers to provide computing services for Internet of Things devices. Our goal is to minimize the energy consumption of this system by planning the trajectories of UAVs. This problem is difficult to address because when planning the trajectories, we need to consider not only the order of stop points (SPs), but also their deployment (including the number and locations) and the association between UAVs and SPs. To tackle this problem, we present an energy-efficient trajectory planning algorithm (TPA) which comprises three phases. In the first phase, a differential evolution algorithm with a variable population size is adopted to update the number and locations of SPs at the same time. In the second phase, the k -means clustering algorithm is employed to group the given SPs into a set of clusters, where the number of clusters is equal to that of UAVs and each cluster contains all SPs visited by the same UAV. In the third phase, to quickly generate the trajectories of UAVs, we propose a low-complexity greedy method to construct the order of SPs in each cluster. Compared with other algorithms, the effectiveness of TPA is verified on a set of instances at different scales.

This paper studies a multiuser cooperative mobile edge computing offloading (called CoMECO) system in a multiuser interference environment, in which delay-sensitive tasks may be executed on local devices, cooperative devices, or the primary MEC server. In this system, we jointly optimize the offloading decision and computation resource allocation for minimizing the total energy consumption of all mobile users under the delay constraint. If this problem is solved directly, the offloading decision and computation resource allocation are generally generated separately at the same time. Note, however, that they are closely coupled. Therefore, under this condition, their dependency is not well considered, thus leading to poor performance. We transform this problem into a bilevel optimization problem, in which the offloading decision is generated in the upper level, and then the optimal allocation of computation resources is obtained in the lower level based on the given offloading decision. In this way, the dependency between the offloading decision and computation resource allocation can be fully taken into account. Subsequently, a bilevel optimization approach, called BiJOR, is proposed. In BiJOR, candidate modes are first pruned to reduce the number of infeasible offloading decisions. Afterward, the upper-level optimization problem is solved by ant colony system (ACS). Furthermore, a sorting strategy is incorporated into ACS to construct feasible offloading decisions with a higher probability and a local search operator is designed in ACS to accelerate the convergence. For the lower-level optimization problem, it is solved by the monotonic optimization method. In addition, BiJOR is extended to deal with a complex scenario with the channel selection. Extensive experiments are carried out to investigate the performance of BiJOR on two sets of instances with up to 400 mobile users. The experimental results demonstrate the effectiveness of BiJOR and the superiority of the CoMECO system.

This article establishes a new multiunmanned aerial vehicle (multi-UAV)-enabled mobile edge computing (MEC) system, where a number of unmanned aerial vehicles (UAVs) are deployed as flying edge clouds for large-scale mobile users. In this system, we need to optimize the deployment of UAVs, by considering their number and locations. At the same time, to provide good services for all mobile users, it is necessary to optimize task scheduling. Specifically, for each mobile user, we need to determine whether its task is executed locally or on a UAV (i.e., offloading decision), and how many resources should be allocated (i.e., resource allocation). This article presents a two-layer optimization method for jointly optimizing the deployment of UAVs and task scheduling, with the aim of minimizing system energy consumption. By analyzing this system, we obtain the following property: the number of UAVs should be as small as possible under the condition that all tasks can be completed. Based on this property, in the upper layer, we propose a differential evolution algorithm with an elimination operator to optimize the deployment of UAVs, in which each individual represents a UAV's location and the entire population represents an entire deployment of UAVs. During the evolution, we first determine the maximum number of UAVs. Subsequently, the elimination operator gradually reduces the number of UAVs until at least one task cannot be executed under delay constraints. This process achieves an adaptive adjustment of the number of UAVs. In the lower layer, based on the given deployment of UAVs, we transform the task scheduling into a 0-1 integer programming problem. Due to the large-scale characteristic of this 0-1 integer programming problem, we propose an efficient greedy algorithm to obtain the near-optimal solution with much less time. The effectiveness of the proposed two-layer optimization method and the established multi-UAV-enabled MEC system is demonstrated on ten instances with up to 1000 mobile users.

The desired rhythmic signals for adaptive walking of humanoid robots should have proper frequencies, phases, and shapes. Matsuoka's central pattern generator (CPG) is able to generate rhythmic signals with reasonable frequencies and phases, and thus has been widely applied to control the movements of legged robots, such as walking of humanoid robots. However, it is difficult for this kind of CPG to generate rhythmic signals with desired shapes, which limits the adaptability of walking of humanoid robots in various environments. To address this issue, a new framework that can generate desired rhythmic signals for Matsuoka's CPG is presented. The proposed framework includes three main parts. First, feature processing is conducted to transform the Matsuoka's CPG outputs into a normalized limit cycle. Second, by combining the normalized limit cycle with robot feedback as the feature inputs and setting the required learning objective, the neural network (NN) learns to generate desired rhythmic signals. Finally, in order to ensure the continuity of the desired rhythmic signals, signal filtering is applied to the outputs of NN, with the aim of smoothing the discontinuous parts. Numerical experiments on the proposed framework suggest that it can not only generate a variety of rhythmic signals with desired shapes but also preserve the frequency and phase properties of Matsuoka's CPG. In addition, the proposed framework is embedded into a control system for adaptive omnidirectional walking of humanoid robot NAO. Extensive simulation and real experiments on this control system demonstrate that the proposed framework is able to generate desired rhythmic signals for adaptive walking of NAO on fixed and changing inclined surfaces. Furthermore, the comparison studies verify that the proposed framework can significantly improve the adaptability of NAO's walking compared with the other methods.

When solving bilevel optimization problems (BOPs) by evolutionary algorithms (EAs), it is necessary to obtain the lower-level optimal solution for each upper-level solution, which gives rise to a large number of lower-level fitness evaluations, especially for large-scale BOPs. It is interesting to note that some upper-level variables may not interact with some lower-level variables. Under this condition, if the value(s) of one/several upper-level variables change(s), we only need to focus on the optimization of the interacting lower-level variables, thus reducing the dimension of the search space and saving the number of lower-level fitness evaluations. This article proposes a new framework (called GO) to identify and utilize the interactions between upper-level and lower-level variables for scalable BOPs. GO includes two phases: 1) the grouping phase and 2) the optimization phase. In the grouping phase, after identifying the interactions between upper-level and lower-level variables, they are divided into three types of subgroups (denoted as types I-III), which contain only upper-level variables, only lower-level variables, and both upper-level and lower-level variables, respectively. In the optimization phase, if type-I and type-II subgroups only include one variable, a multistart sequential quadratic programming is designed; otherwise, a single-level EA is applied. In addition, a criterion is proposed to judge whether a type-II subgroup has multiple optima. If multiple optima exist, by incorporating the information of the upper level, we design new objective function and degree of constraint violation to locate the optimistic solution. As for type-III subgroups, they are optimized by a bilevel EA (BLEA). The effectiveness of GO is demonstrated on a set of scalable test problems by applying it to five representative BLEAs. Moreover, GO is applied to the resource pricing in mobile edge computing.

When solving constrained optimization problems by evolutionary algorithms, the core issue is to balance constraints and objective function. This paper is the first attempt to utilize the correlation between constraints and objective function to keep this balance. First of all, the correlation between constraints and objective function is mined and represented by a correlation index. Afterward, a weighted sum updating approach and an archiving and replacement mechanism are proposed to make use of this correlation index to guide the evolution. By the above process, a novel constrained optimization evolutionary algorithm is presented. Experiments on a broad range of benchmark test functions indicate that the proposed method shows better or at least competitive performance against other state-of-the-art methods. Moreover, the proposed method is applied to the gait optimization of humanoid robots.

Finding multiple roots of nonlinear equation systems (NESs) in a single run is one of the most important challenges in numerical computation. We tackle this challenging task by combining the strengths of the repulsion technique, diversity preservation mechanism, and adaptive parameter control. First, the repulsion technique motivates the population to find new roots by repulsing the regions surrounding the previously found roots. However, to find as many roots as possible, algorithm designers need to address a key issue: how to maintain the diversity of the population. To this end, the diversity preservation mechanism is integrated into our approach, which consists of the neighborhood mutation and the crowding selection. In addition, we further improve the performance by incorporating the adaptive parameter control. The purpose is to enhance the search ability and remedy the trial-and-error tuning of the parameters of differential evolution (DE) for different problems. By assembling the above three aspects together, we propose a repulsion-based adaptive DE, called RADE, for finding multiple roots of NESs in a single run. To evaluate the performance of RADE, 30 NESs with diverse features are chosen from the literature as the test suite. Experimental results reveal that RADE is able to find multiple roots simultaneously in a single run on all the test problems. Moreover, RADE is capable of providing better results than the compared methods in terms of both root rate and success rate.

Finding multiple roots of nonlinear equations systems (NESs) in a single run is an important yet difficult task. It requires to keep a balance between explorative and exploitative traits. In this paper, we present a random walk mutation-based differential evolution (DE) with estimation of distribution algorithm (EDA) to address this problem. The major characteristics are: i) the random walk mutation is capable of preserving the population diversity , which guides individuals to move toward different promising regions; ii) probability selection is employed to provide suitable parent individuals for evolution; iii) EDA is used to accelerate the convergence and obtains the roots. To evaluate the performance of our approach, 30 NESs with diverse features are selected as test suite. Experimental results indicate that the proposed approach is able to yield better performance compared with other state-of-the-art methods.

The performance of many nature-inspired optimization algorithms (NIOAs) depends strongly on their implemented coordinate system. However, the commonly used coordinate system is fixed and not well suited for different function landscapes, NIOAs thus might not search efficiently. To overcome this shortcoming, in this paper we propose a framework, named ACoS, to adaptively tune the coordinate systems in NIOAs. In ACoS, an Eigen coordinate system is established by making use of the cumulative population distribution information, which can be obtained based on a covariance matrix adaptation strategy and an additional archiving mechanism. Since the population distribution information can reflect the features of the function landscape to some extent, NIOAs in the Eigen coordinate system have the capability to identify the modality of the function landscape. In addition, the Eigen coordinate system is coupled with the original coordinate system, and they are selected according to a probability vector. The probability vector aims to determine the selection ratio of each coordinate system for each individual, and is adaptively updated based on the collected information from the offspring. ACoS has been applied to two of the most popular paradigms of NIOAs, i.e., particle swarm optimization and differential evolution, for solving 30 test functions with 30D and 50D at the 2014 IEEE Congress on Evolutionary Computation. The experimental studies demonstrate its effectiveness.

For expensive constrained optimization problems (ECOPs), the computation of objective function and constraints is very time-consuming. This paper proposes a novel global and local surrogate-assisted differential evolution (DE) for solving ECOPs with inequality constraints. The proposed method consists of two main phases: 1) global surrogate-assisted phase and 2) local surrogate-assisted phase. In the global surrogate-assisted phase, DE serves as the search engine to produce multiple trial vectors. Afterward, the generalized regression neural network is used to evaluate these trial vectors. In order to select the best candidate from these trial vectors, two rules are combined. The first is the feasibility rule, which at first guides the population toward the feasible region, and then toward the optimal solution. In addition, the second rule puts more emphasis on the solution with the highest predicted uncertainty, and thus alleviates the inaccuracy of the surrogates. In the local surrogate-assisted phase, the interior point method coupled with radial basis function is utilized to refine each individual in the population. During the evolution, the global surrogate-assisted phase has the capability to promptly locate the promising region and the local surrogate-assisted phase is able to speed up the convergence. Therefore, by combining these two important elements, the number of fitness evaluations can be reduced remarkably. The proposed method has been tested on numerous benchmark test functions from three test suites and two real-world cases. The experimental results demonstrate that the performance of the proposed method is better than that of other state-of-the-art methods.

Constrained multiobjective optimization problems (CMOPs) are frequently encountered in real-world applications, which usually involve constraints in both the decision and objective spaces. However, current artificial CMOPs never consider constraints in the decision space (i.e., decision constraints) and constraints in the objective space (i.e., objective constraints) at the same time. As a result, they have a limited capability to simulate practical scenes. To remedy this issue, a set of CMOPs, named DOC, is constructed in this paper. It is the first attempt to consider both the decision and objective constraints simultaneously in the design of artificial CMOPs. Specifically, in DOC, various decision constraints (e.g., inequality constraints, equality constraints, linear constraints, and nonlinear constraints) are collected from real-world applications, thus making the feasible region in the decision space have different properties (e.g., nonlinear, extremely small, and multimodal). On the other hand, some simple and controllable objective constraints are devised to reduce the feasible region in the objective space and to make the Pareto front have diverse characteristics (e.g., continuous, discrete, mixed, and degenerate). As a whole, DOC poses a great challenge for a constrained multiobjective evolutionary algorithm (CMOEA) to obtain a set of well-distributed and well-converged feasible solutions. In order to enhance current CMOEAs' performance on DOC, a simple and efficient two-phase framework, named ToP, is proposed in this paper. In ToP, the first phase is implemented to find the promising feasible area by transforming a CMOP into a constrained single-objective optimization problem. Then in the second phase, a specific CMOEA is executed to obtain the final solutions. ToP is applied to four state-of-the-art CMOEAs, and the experimental results suggest that it is quite effective.

For solving constrained multiobjective optimization problems (CMOPs), many algorithms have been proposed in the evolutionary computation research community for the past two decades. Generally, the effectiveness of an algorithm for CMOPs is evaluated by artificial test problems. However, after a brief review of current artificial test problems, we have found that they are not well-designed and fail to reflect the characteristics of real-world applications (e.g., small feasibility ratio). Thus, in this paper, we first propose a new constraint construction method to facilitate the systematic design of test problems. Then, on the basis of this method, we design a new test suite consisting of 14 instances, which covers diverse characteristics extracted from real-world CMOPs and can be divided into four types. Considering that the comprehensive performance comparisons among the constraint-handling techniques (CHTs) remain scarce, we choose several representative CHTs and compare their performance on our test suite. The performance comparisons identify the strengths and weaknesses of different CHTs on different types of CMOPs and provide guidelines on how to select/design a CHT in a specific scenario.

When solving constrained optimization problems (COPs) by evolutionary algorithms, the search algorithm plays a crucial role. In general, we expect that the search algorithm has the capability to balance not only diversity and convergence but also constraints and objective function during the evolution. For this purpose, this paper proposes a composite differential evolution (DE) for constrained optimization, which includes three different trial vector generation strategies with distinct advantages. In order to strike a balance between diversity and convergence, one of these three trial vector generation strategies is able to increase diversity, and the other two exhibit the property of convergence. In addition, to accomplish the tradeoff between constraints and objective function, one of the two trial vector generation strategies for convergence is guided by the individual with the least degree of constraint violation in the population, and the other is guided by the individual with the best objective function value in the population. After producing offspring by the proposed composite DE, the feasibility rule and the ε constrained method are combined elaborately for selection in this paper. Moreover, a restart scheme is proposed to help the population jump out of a local optimum in the infeasible region for some extremely complicated COPs. By assembling the above techniques together, a constrained composite DE is proposed. The experiments on two sets of benchmark test functions with various features, i.e., 24 test functions from IEEE CEC2006 and 18 test functions with 10 dimensions and 30 dimensions from IEEE CEC2010, have demonstrated that the proposed method shows better or at least competitive performance against other state-of-the-art methods.

Decomposition-based multiobjective evolutionary algorithms (MOEAs) have received increasing research interests due to their high performance for solving multiobjective optimization problems. However, scalarizing functions (SFs), which play a crucial role in balancing diversity and convergence in these kinds of algorithms, have not been fully investigated. This paper is mainly devoted to presenting two new SFs and analyzing their effect in decomposition-based MOEAs. Additionally, we come up with an efficient framework for decomposition-based MOEAs based on the proposed SFs and some new strategies. Extensive experimental studies have demonstrated the effectiveness of the proposed SFs and algorithm.

This paper presents a differential evolution algorithm with a new encoding mechanism for efficiently solving the optimal layout of the wind farm, with the aim of maximizing the power output. In the modeling of the wind farm, the wake effects among different wind turbines are considered and the Weibull distribution is employed to estimate the wind speed distribution. In the process of evolution, a new encoding mechanism for the locations of wind turbines is designed based on the characteristics of the wind farm layout. This encoding mechanism is the first attempt to treat the location of each wind turbine as an individual. As a result, the whole population represents a layout. Compared with the traditional encoding, the advantages of this encoding mechanism are twofold: 1) the dimension of the search space is reduced to two, and 2) a crucial parameter (i.e., the population size) is eliminated. In addition, differential evolution serves as the search engine and the caching technique is adopted to enhance the computational efficiency. The comparative analysis between the proposed method and seven other state-of-the-art methods is conducted based on two wind scenarios. The experimental results indicate that the proposed method is able to obtain the best overall performance, in terms of the power output and execution time.

Due to the fact that a nonlinear equation system (NES) may contain multiple optimal solutions, solving NESs is one of the most important challenges in numerical computation. When applying evolutionary algorithms to solve NESs, two issues should be considered: 1) how to transform an NES into a kind of optimization problem and 2) how to develop an optimization algorithm to solve the transformed optimization problem. In this paper, we tackle the first issue by transforming an NES into a weighted biobjective optimization problem. By the above transformation, not only do all the optimal solutions of an original NES become the Pareto optimal solutions of the transformed biobjective optimization problem, but also their images are different points on a linear Pareto front in the objective space. In addition, we suggest an adaptive multiobjective differential evolution, the goal of which is to effectively locate the Pareto optimal solutions of the transformed biobjective optimization problem. Once these solutions are found, the optimal solutions of the original NES can also be obtained correspondingly. By combining the weighted biobjective transformation technique with the adaptive multiobjective differential evolution, we propose a generic framework for the simultaneous locating of multiple optimal solutions of NESs. Comprehensive experiments on 38 NESs with various features have demonstrated that our framework provides very competitive overall performance compared with several state-of-the-art methods.

In many real-world engineering applications, a uniform design needs to be conducted in a constrained experimental domain that includes linear/nonlinear and inequality/equality constraints. In general, these constraints make the constrained experimental domain small and irregular in the decision space. Therefore, it is difficult for current methods to produce a predefined number of samples and make the samples distribute uniformly in the constrained experimental domain. This paper presents a two-phase differential evolution for uniform designs in constrained experimental domains. In the first phase, considering the constraint violation as the fitness function, a clustering DE is proposed to guide the population toward the constrained experimental domain from different directions promptly. As a result, a predefined number of samples can be obtained in the constrained experimental domain. In the second phase, maximizing the minimum Euclidean distance among samples is treated as another fitness function. By optimizing this fitness function, the samples produced in the first phase can be scattered uniformly in the constrained experimental domain. The performance of the proposed method has been tested and compared with another state-of-the-art method. Experimental results suggest that our method is significantly better than the compared method in the uniform designs of a new type of automotive crash box and five benchmark test problems. Moreover, the proposed method could be considered as a general and promising framework for other uniform designs in constrained experimental domains.

Solving constrained multiobjective optimization problems is one of the most challenging areas in the evolutionary computation research community. To solve a constrained multiobjective optimization problem, an algorithm should tackle the objective functions and the constraints simultaneously. As a result, many constraint-handling techniques have been proposed. However, most of the existing constraint-handling techniques are developed to solve test instances (e.g., CTPs) with low dimension and large feasible region. On the other hand, experimental comparisons on different constraint-handling techniques remain scarce. In view of these two issues, in this paper we first construct 18 test instances, each of which exhibits different properties. Afterward, we choose three representative constraint-handling techniques and combine them with nondominated sorting genetic algorithm II to study the performance difference on various conditions. By the experimental studies, we point out the advantages and disadvantages of different constraint-handling techniques.

Differential Evolution (DE) is a popular paradigm of evolutionary algorithms, which has been successfully applied to solve different kinds of optimization problems. To design an effective DE, it is necessary to consider different requirements of the exploration and exploitation at different evolutionary stages. Motivated by this consideration, a new DE with a two-stage optimization mechanism, called TSDE, has been proposed in this paper. In TSDE, based on the number of fitness evaluations, the whole evolutionary process is divided into two stages, namely the former stage and the latter stage. TSDE focuses on improving the search ability in the former stage and emphasizes the convergence in the latter stage. Hence, different trial vector generation strategies have been utilized at different stages. TSDE has been tested on 25 benchmark test functions from IEEE CEC2005 and 30 benchmark test functions from IEEE CEC2014. The experimental results suggest that TSDE performs better than four other state-of-the-art DE variants.

When solving constrained optimization problems by evolutionary algorithms, an important issue is how to balance constraints and objective function. This paper presents a new method to address the above issue. In our method, after generating an offspring for each parent in the population by making use of differential evolution (DE), the well-known feasibility rule is used to compare the offspring and its parent. Since the feasibility rule prefers constraints to objective function, the objective function information has been exploited as follows: if the offspring cannot survive into the next generation and if the objective function value of the offspring is better than that of the parent, then the offspring is stored into a predefined archive. Subsequently, the individuals in the archive are used to replace some individuals in the population according to a replacement mechanism. Moreover, a mutation strategy is proposed to help the population jump out of a local optimum in the infeasible region. Note that, in the replacement mechanism and the mutation strategy, the comparison of individuals is based on objective function. In addition, the information of objective function has also been utilized to generate offspring in DE. By the above processes, this paper achieves an effective balance between constraints and objective function in constrained evolutionary optimization. The performance of our method has been tested on two sets of benchmark test functions, namely, 24 test functions at IEEE CEC2006 and 18 test functions with 10-D and 30-D at IEEE CEC2010. The experimental results have demonstrated that our method shows better or at least competitive performance against other state-of-the-art methods. Furthermore, the advantage of our method increases with the increase of the number of decision variables.

This paper investigates a practical variant of the vehicle routing problem (VRP), called VRP with simultaneous delivery and pickup and time windows (VRPSDPTW), in the logistics industry. VRPSDPTW is an important logistics problem in closed-loop supply chain network optimization. VRPSDPTW exhibits multiobjective properties in real-world applications. In this paper, a general multiobjective VRPSDPTW (MO-VRPSDPTW) with five objectives is first defined, and then a set of MO-VRPSDPTW instances based on data from the real-world are introduced. These instances represent more realistic multiobjective nature and more challenging MO-VRPSDPTW cases. Finally, two algorithms, multiobjective local search (MOLS) and multiobjective memetic algorithm (MOMA), are designed, implemented and compared for solving MO-VRPSDPTW. The simulation results on the proposed real-world instances and traditional instances show that MOLS outperforms MOMA in most of instances. However, the superiority of MOLS over MOMA in real-world instances is not so obvious as in traditional instances.

Many real-world problems often have several, usually conflicting objectives. Traditional multi-objective optimization problems (MOPs) usually search for the Pareto-optimal solutions for this predicament. A special class of MOPs, the convex hull maximization problems which prefer solutions on the convex hull, has posed a new challenge for existing approaches for solving traditional MOPs, as a solution on the Pareto front is not necessarily a good solution for convex hull maximization. In this work, the difference between traditional MOPs and the convex hull maximization problems is discussed and a new Evolutionary Convex Hull Maximization Algorithm (ECHMA) is proposed to solve the convex hull maximization problems. Specifically, a Convex Hull-based sorting with Convex Hull of Individual Minima (CH-CHIM-sorting) is introduced, as well as a novel selection scheme, Extreme Area Extract-based selection (EAE-selection). Experimental results show that ECHMA significantly outperforms the existing approaches for convex hull maximization and evolutionary multi-objective optimization approaches in achieving a better approximation to the convex hull more stably and with a more uniformly distributed set of solutions.

In the field of evolutionary computation, there has been a growing interest in applying evolutionary algorithms to solve multimodal optimization problems (MMOPs). Due to the fact that an MMOP involves multiple optimal solutions, many niching methods have been suggested and incorporated into evolutionary algorithms for locating such optimal solutions in a single run. In this paper, we propose a novel transformation technique based on multiobjective optimization for MMOPs, called MOMMOP. MOMMOP transforms an MMOP into a multiobjective optimization problem with two conflicting objectives. After the above transformation, all the optimal solutions of an MMOP become the Pareto optimal solutions of the transformed problem. Thus, multiobjective evolutionary algorithms can be readily applied to find a set of representative Pareto optimal solutions of the transformed problem, and as a result, multiple optimal solutions of the original MMOP could also be simultaneously located in a single run. In principle, MOMMOP is an implicit niching method. In this paper, we also discuss two issues in MOMMOP and introduce two new comparison criteria. MOMMOP has been used to solve 20 multimodal benchmark test functions, after combining with nondominated sorting and differential evolution. Systematic experiments have indicated that MOMMOP outperforms a number of methods for multimodal optimization, including four recent methods at the 2013 IEEE Congress on Evolutionary Computation, four state-of-the-art single-objective optimization based methods, and two well-known multiobjective optimization based approaches.

Nonlinear equation systems may have multiple optimal solutions. The main task of solving nonlinear equation systems is to simultaneously locate these optimal solutions in a single run. When solving nonlinear equation systems by evolutionary algorithms, usually a nonlinear equation system should be transformed into a kind of optimization problem. At present, various transformation techniques have been proposed. This paper presents a simple and generic transformation technique based on multiobjective optimization for nonlinear equation systems. Unlike the previous work, our transformation technique transforms a nonlinear equation system into a biobjective optimization problem that can be decomposed into two parts. The advantages of our transformation technique are twofold: 1) all the optimal solutions of a nonlinear equation system are the Pareto optimal solutions of the transformed problem, which are mapped into diverse points in the objective space, and 2) multiobjective evolutionary algorithms can be directly applied to handle the transformed problem. In order to verify the effectiveness of our transformation technique, it has been integrated with nondominated sorting genetic algorithm II to solve nonlinear equation systems. The experimental results have demonstrated that, overall, our transformation technique outperforms another state-of-the-art multiobjective optimization based transformation technique and four single-objective optimization based approaches on a set of test instances. The influence of the types of Pareto front on the performance of our transformation technique has been investigated empirically. Moreover, the limitation of our transformation technique has also been identified and discussed in this paper.

During the past decade, solving constrained optimization problems with evolutionary algorithms has received considerable attention among researchers and practitioners. Cai and Wang's method (abbreviated as CW method) is a recent constrained optimization evolutionary algorithm proposed by the authors. However, its main shortcoming is that a trial-and-error process has to be used to choose suitable parameters. To overcome the above shortcoming, this paper proposes an improved version of the CW method, called CMODE, which combines multiobjective optimization with differential evolution to deal with constrained optimization problems. Like its predecessor CW, the comparison of individuals in CMODE is also based on multiobjective optimization. In CMODE, however, differential evolution serves as the search engine. In addition, a novel infeasible solution replacement mechanism based on multiobjective optimization is proposed, with the purpose of guiding the population toward promising solutions and the feasible region simultaneously. The performance of CMODE is evaluated on 24 benchmark test functions. It is shown empirically that CMODE is capable of producing highly competitive results compared with some other state-of-the-art approaches in the community of constrained evolutionary optimization.

Based on our previous work, this paper presents a dynamic hybrid framework, called DyHF, for solving constrained optimization problems. This framework consists of two major steps: global search model and local search model. In the global and local search models, differential evolution serves as the search engine, and Pareto dominance used in multiobjective optimization is employed to compare the individuals in the population. Unlike other existing methods, the above two steps are executed dynamically according to the feasibility proportion of the current population in this paper, with the purpose of reasonably distributing the computational resource for the global and local search during the evolution. The performance of DyHF is tested on 22 benchmark test functions. The experimental results clearly show that the overall performance of DyHF is highly competitive with that of a number of state-of-the-art approaches from the literature.

This paper proposes a (μ + λ)-differential evolution and an improved adaptive trade-off model for solving constrained optimization problems. The proposed (μ + λ)-differential evolution adopts three mutation strategies (i.e., rand/1 strategy, current-to-best/1 strategy, and rand/2 strategy) and binomial crossover to generate the offspring population. Moreover, the current-to-best/1 strategy has been improved in this paper to further enhance the global exploration ability by exploiting the feasibility proportion of the last population. Additionally, the improved adaptive trade-off model includes three main situations: the infeasible situation, the semi-feasible situation, and the feasible situation. In each situation, a constraint-handling mechanism is designed based on the characteristics of the current population. By combining the (μ + λ)-differential evolution with the improved adaptive trade-off model, a generic method named (μ + λ)-constrained differential evolution ((μ + λ)-CDE) is developed. The (μ + λ)-CDE is utilized to solve 24 well-known benchmark test functions provided for the special session on constrained real-parameter optimization of the 2006 IEEE Congress on Evolutionary Computation (CEC2006). Experimental results suggest that the (μ + λ)-CDE is very promising for constrained optimization, since it can reach the best known solutions for 23 test functions and is able to successfully solve 21 test functions in all runs. Moreover, in this paper, a self-adaptive version of (μ + λ)-CDE is proposed which is the most competitive algorithm so far among the CEC2006 entries.

Trial vector generation strategies and control parameters have a significant influence on the performance of differential evolution (DE). This paper studies whether the performance of DE can be improved by combining several effective trial vector generation strategies with some suitable control parameter settings. A novel method, called composite DE (CoDE), has been proposed in this paper. This method uses three trial vector generation strategies and three control parameter settings. It randomly combines them to generate trial vectors. CoDE has been tested on all the CEC2005 contest test instances. Experimental results show that CoDE is very competitive.

Based on our previous work, a hybrid differential evolution is proposed in this paper and combined with an adaptive trade-off model to solve constrained optimization problems. The proposed hybrid differential evolution adopts three mutation strategies to generate mutant vectors. Unlike the existing methods, direction information has been utilized in this paper. The proposed method is tested on thirteen well-known benchmark test functions. Our method outperforms or performs similarly to two state-of-the-art approaches in terms of effectiveness and efficiency.

In this paper, an adaptive tradeoff model (ATM) is proposed for constrained evolutionary optimization. In this model, three main issues are considered: (1) the evaluation of infeasible solutions when the population contains only infeasible individuals; (2) balancing feasible and infeasible solutions when the population consists of a combination of feasible and infeasible individuals; and (3) the selection of feasible solutions when the population is composed of feasible individuals only. These issues are addressed in this paper by designing different tradeoff schemes during different stages of a search process to obtain an appropriate tradeoff between objective function and constraint violations. In addition, a simple evolutionary strategy (ES) is used as the search engine. By integrating ATM with ES, a generic constrained optimization evolutionary algorithm (ATMES) is derived. The new method is tested on 13 well-known benchmark test functions, and the empirical results suggest that it outperforms or performs similarly to other state-of-the-art techniques referred to in this paper in terms of the quality of the resulting solutions.

Solving constrained optimization problems (COPs) via evolutionary algorithms (EAs) has attracted much attention recently. A new constrained optimization evolutionary algorithm by using good point set (COEAGP) is presented in this paper. In the process of population evolution, multi-objective optimization techniques and good point set in number theory are integrated into our algorithm. The approach transforms COP into a bi-objective optimization problem firstly. Then the crossover operator is designed by using the principle of good point set The purpose of the new crossover is to enrich the exploration and exploitation abilities of the approach proposed. The new crossover operator can produce a small but representative set of points as the potential offspring. After that the BGA mutation operator is applied to potential offspring for enhancing the diversity of the potential offspring population. Furthermore, the update operator incorporates Pareto dominance and the tournament selection operator to choose the best individuals in the current offspring for the next generation. The new approach is tested on 8 well-known benchmark functions, and the empirical evidence suggests that it is robust and efficient when handling linear/nonlinear equality/inequality constraints and that COEAGP outperforms or performs similarly to the other techniques referred in this paper in terms of the quality of the resulting solutions.

Good Nodes Set(GNS) is a concept in number theory. To overcome the deficiency of orthogonal design to handle constrained optimization problems(COPs), this paper presents a method that incorporate GNS principle to enhance the crossover operator of the evolution strategy (ES) can make the resulting evolutionary algorithm more robust and statically sound. In order to gain the rapid and stable rate of converging to the feasible region, traditional crossover operator is split into two steps. GNS initialization methods is applied to ensure the initial population span evenly in relatively large search space and reliably locate the good points for further exploration in subsequent iterations. The proposed method achieves the same sound results just as the orthogonal method does, but its precision is not confined by the dimension of the space. The simplex selected and diversity mechanism similar to Carlos's SMES is used to enrich the exploration and exploitation abilities of the approach proposed. Experiment results on a set of benchmark problems show the efficiency of our methods.

This paper presents a novel evolutionary algorithm (EA) for constrained optimization problems, i.e., the hybrid constrained optimization EA (HCOEA). This algorithm effectively combines multiobjective optimization with global and local search models. In performing the global search, a niching genetic algorithm based on tournament selection is proposed. Also, HCOEA has adopted a parallel local search operator that implements a clustering partition of the population and multiparent crossover to generate the offspring population. Then, nondominated individuals in the offspring population are used to replace the dominated individuals in the parent population. Meanwhile, the best infeasible individual replacement scheme is devised for the purpose of rapidly guiding the population toward the feasible region of the search space. During the evolutionary process, the global search model effectively promotes high population diversity, and the local search model remarkably accelerates the convergence speed. HCOEA is tested on 13 well-known benchmark functions, and the experimental results suggest that it is more robust and efficient than other state-of-the-art algorithms from the literature in terms of the selected performance metrics, such as the best, median, mean, and worst objective function values and the standard deviations.

A considerable number of constrained optimization evolutionary algorithms (COEAs) have been proposed due to increasing interest in solving constrained optimization problems (COPs) by evolutionary algorithms (EAs). In this paper, we first review existing COEAs. Then, a novel EA for constrained optimization is presented. In the process of population evolution, our algorithm is based on multiobjective optimization techniques, i.e., an individual in the parent population may be replaced if it is dominated by a nondominated individual in the offspring population. In addition, three models of a population-based algorithm-generator and an infeasible solution archiving and replacement mechanism are introduced. Furthermore, the simplex crossover is used as a recombination operator to enrich the exploration and exploitation abilities of the approach proposed. The new approach is tested on 13 well-known benchmark functions, and the empirical evidence suggests that it is robust, efficient, and generic when handling linear/nonlinear equality/inequality constraints. Compared with some other state-of-the-art algorithms, our algorithm remarkably outperforms them in terms of the best, mean, and worst objective function values and the standard deviations. It is noteworthy that our algorithm does not require the transformation of equality constraints into inequality constraints

This paper presents a novel evolutionary algorithm for constrained optimization. During the evolutionary process, our algorithm is based on multiobjective optimization techniques, i.e., an individual in the parent population may be replaced if it is dominated by a nondominated individual chosen from the offspring population. In addition, a model of population-based algorithm-generator and an infeasible solutions archiving and replacement mechanism are introduced. Furthermore, the simplex crossover is used as a recombination operator to enrich the exploration and exploitation abilities of the approach proposed. The new approach is tested on thirteen well-known benchmark functions, and the empirical evidences suggest that it is robust, efficient and generic when handling linear/nonlinear equality/inequality constraints. Compared with some other state-of-the-art algorithms, our algorithm remarkably outperforms them in terms of the best, median, mean, and worst objective function values and the standard deviations.