Evolutionary Algorithms (EAs) can not handle expensive optimization problems (EOPs) well due to the limited function evaluations in EOPs. To address this challenge, surrogate-assisted evolutionary algorithms (SAEAs) have been widely used and obtained good performance. With the problem dimension increases, SAEAs encounter some challenges in relatively high complexity on the training time and prediction time. To address this, this article proposes a novel expensive optimization algorithm with evolutionary multi-tasking and hybrid restart strategy (HRS-EMT). In the surrogate model construct, two radial basis function (RBF) models with different kernel functions are trained on all evaluated data to provide diversity and then are solved by a multi-tasking optimizer to a better optimization performance. In the surrogate model management, HRS-EMT combines multiple RBF models into an ensemble RBF (ERBF) model, strategically applied in the initial population pre-selection of surrogate model. Based on the prediction of ERBF surrogate model, HRS-EMT can obtain a better initial population in high dimensions. HRS-EMT is validated on twelve benchmark functions and compared with other state-of-the-art SAEAs. Experimental studies have shown the superior or comparable performance to other popular SAEAs in addressing EOPs.

Human mesh recovery from one single image has achieved rapid progress recently, but many methods suffer from the image appearance overfitting since the training data are collected along with accurate 3D annotations in controlled settings of monotonous backgrounds or simple clothes. Some methods regress human mesh vertices from poses to tackle the above problem. However the mesh topologies have not been well exploited, and artifacts are often generated. In this paper, we aim to find an efficient low-cost solution to human mesh reconstruction. To this end, we propose a Progressive Quadric Graph Convolutional Network (PQ-GCN), and design a simple and fast method for 3D human mesh recovery from a single image in the wild. Specifically, we apply quadric-based surface simplification to human meshes and design a progressive graph convolution network, accompanied by mesh feature up-sampling, to deal with the mesh topologies. We carry out a series of studies to validate our method. The results prove that our method achieves superior performance on a challenging in-the-wild dataset, while using 66% fewer parameters than the existing method, Pose2Mesh. Artifacts have also been eliminated and better visual quality has been obtained without any further post-processing and model fitting. Besides, the recovery can be stopped at an earlier stage by adding a decoder head. Consequently, the computational complexity can be reduced greatly.

Multiobjectivization has emerged as a new promising paradigm to solve single-objective optimization problems (SOPs) in evolutionary computation, where an SOP is transformed into a multiobjective optimization problem (MOP) and solved by an evolutionary algorithm to find the optimal solutions of the original SOP. The transformation of an SOP into an MOP can be done by adding helper-objective(s) into the original objective, decomposing the original objective into multiple subobjectives, or aggregating subobjectives of the original objective into multiple scalar objectives. Multiobjectivization bridges the gap between SOPs and MOPs by transforming an SOP into the counterpart MOP, through which multiobjective optimization methods manage to attain superior solutions of the original SOP. Particularly, using multiobjectivization to solve SOPs can reduce the number of local optima, create new search paths from local optima to global optima, attain more incomparability solutions, and/or improve solution diversity. Since the term "multiobjectivization" was coined by Knowles et al. in 2001, this subject has accumulated plenty of works in the last two decades, yet there is a lack of systematic and comprehensive survey of these efforts. This article presents a comprehensive multifacet survey of the state-of-the-art multiobjectivization methods. Particularly, a new taxonomy of the methods is provided in this article and the advantages, limitations, challenges, theoretical analyses, benchmarks, applications, as well as future directions of the multiobjectivization methods are discussed.

Evolutionary multitasking optimization (EMTO) solves multiple optimization tasks meanwhile in the framework of evolutionary algorithm, aiming at improving the solving performance on each task via knowledge transfer among tasks. As one of the representative EMTO algorithms, multifactorial evolutionary algorithm (MFEA) has attracted great attention and has been used to solve many optimization problems. However, most of MFEAs tend to suffer from premature convergence. To deal with this issue, this article designs a novel MFEA by integrating differential evolution, and a hybrid of global and local search strategies, named MFDE-GLS for short. Particularly, the global search strategy is based on an opposition-based learning and a Gaussian perturbation to improve the search ability and maintain population diversity. A local search strategy is introduced by combining 1-dimension search and n-dimension search to accelerate the convergence. Moreover, a new environmental selection mechanism is developed to keep the elite individuals while maintaining the population diversity based on the affinity propagation clustering method. Comprehensive experiments were conducted on both single-objective and multi-objective multi-task benchmark problems to show the effectiveness of the proposed algorithm.

In recent years, dynamic multiobjective optimization problems (DMOPs) have drawn increasing interest. Many dynamic multiobjective evolutionary algorithms (DMOEAs) have been put forward to solve DMOPs mainly by incorporating diversity introduction or prediction approaches with conventional multiobjective evolutionary algorithms. Maintaining a good balance of population diversity and convergence is critical to the performance of DMOEAs. To address the above issue, a DMOEA based on decision variable classification (DMOEA-DVC) is proposed in this article. DMOEA-DVC divides the decision variables into two and three different groups in static optimization and changes response stages, respectively. In static optimization, two different crossover operators are used for the two decision variable groups to accelerate the convergence while maintaining good diversity. In change response, DMOEA-DVC reinitializes the three decision variable groups by maintenance, prediction, and diversity introduction strategies, respectively. DMOEA-DVC is compared with the other six state-of-the-art DMOEAs on 33 benchmark DMOPs. The experimental results demonstrate that the overall performance of the DMOEA-DVC is superior or comparable to that of the compared algorithms.

Evolutionary multitasking (EMT) is an emerging research direction in the field of evolutionary computation. EMT solves multiple optimization tasks simultaneously using evolutionary algorithms with the aim to improve the solution for each task via intertask knowledge transfer. The effectiveness of intertask knowledge transfer is the key to the success of EMT. The multifactorial evolutionary algorithm (MFEA) represents one of the most widely used implementation paradigms of EMT. However, it tends to suffer from noneffective or even negative knowledge transfer. To address this issue and improve the performance of MFEA, we incorporate a prior-knowledge-based multiobjectivization via decomposition (MVD) into MFEA to construct strongly related meme helper-tasks. In the proposed method, MVD creates a related multiobjective optimization problem for each component task based on the corresponding problem structure or decision variable grouping to enhance positive intertask knowledge transfer. MVD can reduce the number of local optima and increase population diversity. Comparative experiments on the widely used test problems demonstrate that the constructed meme helper-tasks can utilize the prior knowledge of the target problems to improve the performance of MFEA.

The divide-and-conquer strategy has been widely used in cooperative co-evolutionary algorithms to deal with large-scale global optimization problems, where a target problem is decomposed into a set of lower-dimensional and tractable subproblems to reduce the problem complexity. However, such a strategy usually demands a large number of function evaluations to obtain an accurate variable grouping. To address this issue, a merged differential grouping (MDG) method is proposed in this article based on the subset–subset interaction and binary search. In the proposed method, each variable is first identified as either a separable variable or a nonseparable variable. Afterward, all separable variables are put into the same subset, and the nonseparable variables are divided into multiple subsets using a binary-tree-based iterative merging method. With the proposed algorithm, the computational complexity of interaction detection is reduced to$O(\max \{n,n_{ns}\times \log _{2} k\})$, where$n$,$n_{ns}(\leq n)$, and$k( < n)$indicate the numbers of decision variables, nonseparable variables, and subsets of nonseparable variables, respectively. The experimental results on benchmark problems show that MDG is very competitive with the other state-of-the-art methods in terms of efficiency and accuracy of problem decomposition.

Multitasking optimization can achieve better performance than traditional single-tasking optimization by leveraging knowledge transfer between tasks. However, the current multitasking optimization algorithms suffer from some deficiencies. Particularly, on high similar problems, the existing algorithms might fail to take full advantage of knowledge transfer to accelerate the convergence of the search, or easily get trapped in the local optima. Whereas, on low similar problems, they tend to suffer from negative transfer, resulting in performance degradation. To solve these issues, this article proposes an evolutionary multitasking optimization algorithm for multiobjective/many-objective optimization with two-stage adaptive knowledge transfer based on population distribution. The resultant algorithm named EMT-PD can improve the convergence performance of the target optimization tasks based on the knowledge extracted from the probability model that reflects the search trend of the whole population. At the first stage of knowledge transfer, an adaptive weight is used to adjust the search step size of each individual, which can reduce the impact of negative transfer. At the second stage of knowledge transfer, the search range of each individual is further adjusted dynamically, which can improve the population diversity and be beneficial for jumping out of the local optima. Experimental results on multitasking multiobjective optimization test suites show that EMT-PD is superior to other state-of-the-art evolutionary multitasking/single-tasking algorithms. To further investigate the effectiveness of EMT-PD on many-objective optimization problems, a multitasking many-objective optimization test suite is also designed in this article. The experimental results on the new test suite also demonstrate the competitiveness of EMT-PD.

In recent decades, multi-objective multifactorial evolutionary algorithm (MOMFEA) has become a very promising research direction. How to achieve effective knowledge transfer between similar tasks is the key issue to affect the performance of the algorithm. In this paper, an adaptive MOMFEA (AMOMFEA) is proposed by exploiting the mixture Gaussian distribution of the population distributions of related tasks to help solve the target task. Wasserstein distance is used to measure the inter-task relevance in that the weight coefficient in the mixture distribution is proportional to the inter-task relevance. Experimental results on benchmark problems validate the effectiveness and efficiency of the proposed method in comparison with MOMFEA and NSGA-II.

Mobile edge computing (MEC) can supplement cloud computing by helping to overcome the limitations of long physical transmission distances and accelerating the responsiveness of edge computing servers. In 5G (fifth generation) cellular networks, adopting MEC can guarantee ultralow latency. To enhance the MEC quality, optimization of the user service profile migration according to the user mobility is essential. However, this optimization establishes an NP-hard problem. Moreover, high-speed 5G base stations with MEC servers often experience high energy consumption. As conventional service migration algorithms such as those based on profile tracking and game theory tend to fall in local optima and neglect energy consumption constraints, we propose a memetic algorithm based on community detection local search (MA-CDLS) to continuously optimize the service migration in 5G MEC scenarios. During busy periods or in crowded areas, MA-CDLS adopts a single-objective optimization of user-perceived latency to achieve high-performance 5G services. During light-load periods or in uncrowded areas, MA-CDLS uses two measures, namely the user-perceived latency and energy consumption, to realize energy-efficient 5G services. MA-CDLS effectively reduces the search space and speeds up the elite selection in the meme operator. Experiments in simulated scenarios show that MA-CDLS achieves a lower user-perceived latency and energy consumption, than the traditional profile tracking and game theory methods, especially during congestion.

Balancing population diversity and convergence is critical for evolutionary algorithms to solve many-objective optimization problems (MaOPs). In this paper, a two-round environmental selection strategy is proposed to pursue good tradeoff between population diversity and convergence for many-objective evolutionary algorithms (MaOEAs). Particularly, in the first round, the solutions with small neighborhood density are picked out to form a candidate pool, where the neighborhood density of a solution is calculated based on a novel adaptive position transformation strategy. In the second round, the best solution in terms of convergence is selected from the candidate pool and inserted into the next generation. The procedure is repeated until a new population is generated. The two-round selection strategy is embedded into an MaOEA framework and the resulting algorithm, namely, 2REA, is compared with eight state-of-the-art MaOEAs on various benchmark MaOPs. The experimental results show that 2REA is very competitive with the compared MaOEAs and the two-round selection strategy works well on balancing population diversity and convergence.

Many-objective optimization problems (MaOPs) pose a big challenge to the traditional Pareto-based multiobjective evolutionary algorithms (MOEAs). As the number of objectives increases, the number of mutually nondominated solutions explodes and MOEAs become invalid due to the loss of Pareto-based selection pressure. Indicator-based many-objective evolutionary algorithms (MaOEAs) have been proposed to address this issue by enhancing the environmental selection. Indicator-based MaOEAs are easy to implement and of good versatility, however, they are unlikely to maintain the population diversity and coverage very well. In this article, a new indicator-based MaOEA with boundary protection, namely, MaOEA-IBP, is presented to relieve this weakness. In MaOEA-IBP, a worst elimination mechanism based on the${I}_{{\epsilon }^{+}}$indicator and boundary protection strategy is devised to enhance the balance of population convergence, diversity, and coverage. Specifically, a pair of solutions with the smallest${I}_{{\epsilon }^{+}}$value are first identified from the population. If one solution dominates the other, the dominated solution is eliminated. Otherwise, one solution is eliminated by the boundary protection strategy. MaOEA-IBP is compared with four indicator-based algorithms (i.e.,${I}_{{{ {SDE}}}^{+}}$, SRA, MaOEAIGD, and ARMOEA) and other five state-of-the-art MaOEAs (i.e., KnEA, MaOEA-CSS, 1by1EA, RVEA, and EFR-RR) on various benchmark MaOPs. The experimental results demonstrate that MaOEA-IBP can achieve competitive performance with the compared algorithms.

Multi-tasking optimization (MTO) has attracted increasing attention in the domain of evolutionary computation. Different from single-tasking optimization, MTO can solve multiple optimization tasks simultaneously to improve the performance of solving each optimization task by inter-task knowledge transfer. Multifactorial evolutionary algorithm (MFEA) is one of the most widely used MTO algorithm based on assortative mating and vertical cultural transmission. This work extends MFEA by integrating bone route and large neighborhood local search to solve multi-objective vehicle routing problem with time window (VRPTW). The VRPTW is modeled as two related tasks, i.e., one is a multi-objective version of VRPTW (the main task), and the other is a single-objective version of VRPTW (the auxiliary task). The resultant new algorithm namely multi-objective multi-factorial memetic algorithm (MOMFMA) solve the two tasks simultaneously where the information between the tasks is exchanged in the evolutionary process. In addition to the implicit information transfer of MFEA, the bone route is introduced to enable explicit information transfer between tasks. Particularly, bone routes are constructed as semi-finished product solutions and used in large neighborhood local search. The bone route and the large neighborhood local search work together to speed up the convergence of the algorithm. MOMFMA is tested on Solomon's 56 datasets and the experimental results demonstrate that the efficiency of MOMFMA.

Multiobjective evolutionary algorithms based on decomposition (MOEA/D) have attracted tremendous attention and achieved great success in the fields of optimization and decision-making. MOEA/Ds work by decomposing the target multiobjective optimization problem (MOP) into multiple single-objective subproblems based on a set of weight vectors. The subproblems are solved cooperatively in an evolutionary algorithm framework. Since weight vectors define the search directions and, to a certain extent, the distribution of the final solution set, the configuration of weight vectors is pivotal to the success of MOEA/Ds. The most straightforward method is to use predefined and uniformly distributed weight vectors. However, it usually leads to the deteriorated performance of MOEA/Ds on solving MOPs with irregular Pareto fronts. To deal with this issue, many weight vector adjustment methods have been proposed by periodically adjusting the weight vectors in a random, predefined, or adaptive way. This article focuses on weight vector adjustment on a simplex and presents a comprehensive survey of these weight vector adjustment methods covering the weight vector adaptation strategies, theoretical analyses, benchmark test problems, and applications. The current limitations, new challenges, and future directions of weight vector adjustment are also discussed.

Recently, the multifactorial evolutionary algorithm (MFEA) has achieved remarkable success in multi-task optimization (MTO) and received extensive attention from academia and industry. The key idea of MFEA is to use the inter-task knowledge transfer to produce the mutual promotion effect of all tasks. However, MFEA still has some limitations in accelerating convergence and enhancing global search ability, especially when the optima of different optimization tasks are far away. To relieve this issue, this paper integrates a new cross-task knowledge transfer, which is based on a search direction instead of an individual. The proposed knowledge transfer strategy generates offspring by the sum of an elite individual of one task and a difference vector from another task. As a basic vector, the elite individual is used to speed up the population convergence. Adding the elite individual with a difference vector from another task can enhance the search diversity. The experimental studies have shown the effectiveness and efficiency of the proposed cross-task knowledge transfer strategy, compared with the classical MFEA on a set of benchmark problems with different degrees of similarities.

Recently, multi-tasking optimization (MTO) has become a rising research topic in the field of evolutionary computation that has attracted increasing attention of academia. Comparing with single-objective optimization (SOO) and multi-objective optimization (MOO), MTO can solve different optimization tasks simultaneously by utilizing inter-task similarities and complementarities. Based on crossover operator, the classical multifactorial evolutionary algorithm (MFEA) transfers inter-task knowledge. To broaden the search region and accelerate the convergence, this paper integrates differential evolution (DE) and opposition-based learning (OBL) into MFEA and hence proposes MFEA/DE-OBL. The motivation of integrating DE and OBL is that they have different search neighborhoods and strong complementarity with simulated binary crossover (SBX) used in MFEA. Furthermore, integrating DE and OBL can help MFEA jump out of local optima. The effectiveness and efficiency of integrating DE and OBL into MFEA are experimentally studied on a set of benchmark problems with different degrees of similarities. Experimental results demonstrate that the proposed MFEA/DE-OBL dramatically improves the performance compared with the MFEA.

This paper presents a multi-objective memetic algorithm based on correlation priority to solve route planning of electric vehicles in pickup-and-delivery problems. Four objectives namely route length, waiting time, charging times, and the number of vehicles are optimized using multi-objective memetic algorithm, which is a combination of multi-objective genetic algorithm, greedy strategy, and a correlation priority based local search. The correlation between two customer nodes is used to fine-tune the route to accelerate the convergence of the algorithm. The algorithm is tested on three sets of data with different scales and the experimental results demonstrate the efficiency of the proposed algorithm.

The first cooperative co-evolutionary algorithm (CCEA) was proposed by Potter and De Jong in 1994 and since then many CCEAs have been proposed and successfully applied to solving various complex optimization problems. In applying CCEAs, the complex optimization problem is decomposed into multiple subproblems, and each subproblem is solved with a separate subpopulation, evolved by an individual evolutionary algorithm (EA). Through cooperative co-evolution of multiple EA subpopulations, a complete problem solution is acquired by assembling the representative members from each subpopulation. The underlying divide-and-conquer and collaboration mechanisms enable CCEAs to tackle complex optimization problems efficiently, and hence CCEAs have been attracting wide attention in the EA community. This paper presents a comprehensive survey of these CCEAs, covering problem decomposition, collaborator selection, individual fitness evaluation, subproblem resource allocation, implementations, benchmark test problems, control parameters, theoretical analyses, and applications. The unsolved challenges and potential directions for their solutions are discussed.

Tchebycheff decomposition represents one of the most widely used decomposition approaches that can convert a multiobjective optimization problem into a set of scalar optimization subproblems. Nevertheless, the geometric properties of the subproblem objective functions in Tchebycheff decomposition have not been explicitly studied. This paper proposes a Tchebycheff decomposition with lp-norm constraint on direction vectors in which the subproblem objective functions are endowed with clear geometric property. Especially, the Tchebycheff decomposition with l2-norm constraint on direction vectors is taken as an example to illustrate its advantage. A new unary R2indicator is also introduced to approximate the hyper-volume metric and justify the efficiency of the proposed Tchebycheff decomposition. A resultant Tchebycheff decomposition-based multiobjective evolutionary algorithm (MOEA) with l2-norm constraint and a new population update strategy is proposed to solve multiobjective optimization problems. The experimental results on both benchmark and real-world multiobjective optimization problems show that the proposed algorithm is capable of obtaining high quality solutions compared with other state-of-the-art MOEAs.

Many-objective optimization problems pose challenges to the Pareto-based multi-objective optimization algorithms. Recent studies have suggested that decomposition is a promising method to improve the performance of multi-objective evolutionary algorithms on many-objective optimization problem. Various methods based on decomposition have been developed to solve many-objective problems in recent years. However, the existing experimental comparative studies are usually limited to only a few methods based on decomposition. This paper offers a systematic comparison of seven representative decomposition-based approaches tested on two groups of widely used problems. The experimental results have demonstrated that none of the compared algorithms has a clear advantage over the others, although different algorithms are competitive on different test problems. Therefore, a careful selection of algorithms is necessary in handling a many-objective problem in hand.

This paper formulates one-to-many-to-one pickup and delivery problems with dynamic customer requests and traffic information. A multi-objective memetic algorithm namely prioLSH-MOMA is proposed to solve the problems. The new algorithm is characterized with a priority and locality-sensitive hashing based local search. prioLSH-MOMA is designed to find an optimal route of a dynamic pickup and delivery problem in terms of route length and workload. Particularly, a re-planning strategy is introduced to handle the dynamic information. Priority and locality-sensitive hashing based local search is applied to fine-tune the candidate routes during the evolution process. prioLSH-MOMA is evaluated with two dynamic pickup and delivery problems simulated on real-world maps and the results demonstrate the efficiency of the proposed algorithm.

State-of-the-art multiobjective evolutionary algorithms (MOEAs) treat all the decision variables as a whole to optimize performance. Inspired by the cooperative coevolution and linkage learning methods in the field of single objective optimization, it is interesting to decompose a difficult high-dimensional problem into a set of simpler and low-dimensional subproblems that are easier to solve. However, with no prior knowledge about the objective function, it is not clear how to decompose the objective function. Moreover, it is difficult to use such a decomposition method to solve multiobjective optimization problems (MOPs) because their objective functions are commonly conflicting with one another. That is to say, changing decision variables will generate incomparable solutions. This paper introduces interdependence variable analysis and control variable analysis to deal with the above two difficulties. Thereby, an MOEA based on decision variable analyses (DVAs) is proposed in this paper. Control variable analysis is used to recognize the conflicts among objective functions. More specifically, which variables affect the diversity of generated solutions and which variables play an important role in the convergence of population. Based on learned variable linkages, interdependence variable analysis decomposes decision variables into a set of low-dimensional subcomponents. The empirical studies show that DVA can improve the solution quality on most difficult MOPs. The code and supplementary material of the proposed algorithm are available athttp://web.xidian.edu.cn/fliu/paper.html.

Recently, MOEA/D (multi-objective evolutionary algorithm based on decomposition) has achieved great success in the field of evolutionary multi-objective optimization and has attracted a lot of attention. It decomposes a multi-objective optimization problem (MOP) into a set of scalar subproblems using uniformly distributed aggregation weight vectors and provides an excellent general algorithmic framework of evolutionary multi-objective optimization. Generally, the uniformity of weight vectors in MOEA/D can ensure the diversity of the Pareto optimal solutions, however, it cannot work as well when the target MOP has a complex Pareto front (PF; i.e., discontinuous PF or PF with sharp peak or low tail). To remedy this, we propose an improved MOEA/D with adaptive weight vector adjustment (MOEA/D-AWA). According to the analysis of the geometric relationship between the weight vectors and the optimal solutions under the Chebyshev decomposition scheme, a new weight vector initialization method and an adaptive weight vector adjustment strategy are introduced in MOEA/D-AWA. The weights are adjusted periodically so that the weights of subproblems can be redistributed adaptively to obtain better uniformity of solutions. Meanwhile, computing efforts devoted to subproblems with duplicate optimal solution can be saved. Moreover, an external elite population is introduced to help adding new subproblems into real sparse regions rather than pseudo sparse regions of the complex PF, that is, discontinuous regions of the PF. MOEA/D-AWA has been compared with four state of the art MOEAs, namely the original MOEA/D, Adaptive-MOEA/D, [Formula: see text]-MOEA/D, and NSGA-II on 10 widely used test problems, two newly constructed complex problems, and two many-objective problems. Experimental results indicate that MOEA/D-AWA outperforms the benchmark algorithms in terms of the IGD metric, particularly when the PF of the MOP is complex.