Evolutionary multitasking (EMT) endeavors to solve multiple optimization tasks simultaneously via knowledge transfer (KT) among tasks. While several EMT algorithms have obtained promising results, they still encounter uncertainties and challenges in the complex field of multiobjective multitask optimization (MO-MTO). A key limitation of existing approaches is the insufficient attention to manifold structures of the Pareto set (PS), which represent the local regularities of multiobjective optimization problems. Ignoring these structures prevents the algorithm from capturing local features, thereby limiting the efficiency and accuracy of KT. To alleviate this issue, this article proposes a new EMT algorithm for MO-MTO with manifold structure-driven KT, namely EMT-MSKT. EMT-MSKT aims to improve optimization performance by leveraging both global and local structural information. In particular, KT in EMT-MSKT consists of two key strategies: global structure learning (GSL) and local structure learning (LSL). GSL exploits global structural similarity to provide population-level directional information that drives broad exploration across tasks. LSL exploits local structural similarity to support fine-grained exploitation based on manifold features. To implement these two strategies, the global and local search directions are extracted as knowledge, reflecting how solutions evolve within these structures. Moreover, to further enhance transfer efficiency in the LSL process, a local source selection (LSS) strategy is developed to identify relevant knowledge sources based on local manifold similarities. Comprehensive results on four benchmark suites (including a newly constructed suite for complex problems) and three real-world applications demonstrate that EMT-MSKT consistently outperforms other state-of-the-art algorithms.

Evolutionary multitasking (EMT) exploits the correlation among different tasks to help handle them through knowledge transfer (KT) techniques in evolutionary algorithms. In this area, multiobjective multitask optimization (MO-MTO) utilizes EMT to solve multiple multiobjective optimization tasks simultaneously. The key to addressing MO-MTO problems (MO-MTOPs) is to transfer appropriate knowledge among optimization tasks to assist the multiobjective evolutionary process. Both the type and the amount of knowledge can significantly affect the KT process. To achieve better KT behavior, we propose a multiple knowledge types and transfer adaptation (MKTA) framework for handling MO-MTOPs. The MKTA framework incorporates multiple types of knowledge in order to obtain comprehensive KT performance. It also provides transfer adaptation strategies to control: i) the type of knowledge and ii) the amount of knowledge for KT via parameter adaptation approaches, thereby mitigating negative KT. Furthermore, we propose an evolution-path-model-based knowledge type and incorporate the existing unified-search-space-based knowledge type to form the knowledge pool for MKTA. Finally, The MKTA framework is coupled with a ranking-based differential evolution operator to constitute the complete algorithm MTDE-MKTA. In the experimental study, MTDE-MKTA outperformed 10 advanced algorithms on 39 benchmark MO-MTOPs and six groups of real-world application problems.

With the increasing requirement for computational efficiency in evolutionary algorithms (EAs) when tackling many optimization tasks concurrently, many-task optimization (MaTO) has gained much attention in recent years. It involves extracting and transferring the knowledge from successful search experiences across many tasks. As the number of tasks and variable scale grows in MaTO, the need for efficient knowledge transfer in EAs becomes indispensable. In this article, we present a task-averaged natural gradient-assisted natural evolution strategy (TNG-NES) to deal with MaTO efficiently. The task-averaged natural gradient (TNG) captures how a group of task distributions evolves, considering their overall trends of mean and covariance. This can accelerate the optimization process across all tasks by leveraging the evolutionary similarities among multiple task search distributions. Notably, TNG-NES exhibits linear computational complexity concerning the number of tasks for knowledge transfer. Additionally, to adaptively utilize TNG for distribution evolution and mitigate negative transfer, we introduce a transfer adaptive control mechanism for TNG-NES. We conducted extensive experiments on CEC19-MaTO, WCCI22-MaTO, our proposed large-scale MaTO benchmark suite, and real-world applications. The results validate the effectiveness of TNG-NES, outperforming state-of-the-art MaTO algorithms.

Competitive multiobjective multitask optimization (CMO-MTO) problems involve multiple tasks with comparable objectives but heterogeneous decision variables. The final Pareto front in CMO-MTO consists of multiple subsets corresponding to different tasks. Since the Pareto front subset of one task may be dominated by that of another, competition arises among tasks. Additionally, there may be exploitable similarities among tasks that evolutionary multitasking methods can leverage. For a comprehensive study of CMO-MTO, we construct 12 benchmark CMO-MTO problems with varied competitive relationships and inter-task similarities. To effectively solve CMO-MTO problems, we propose a reference vector contribution-based multitask evolutionary algorithm (RVC-MTEA). RVC-MTEA facilitates both global and local knowledge transfer based on vector contributions and integrates global archives to gather non-dominated solutions across multiple competitive tasks. Comparative results with four popular single-task and six state-of-the-art multitask evolutionary algorithms demonstrate the efficacy of RVC-MTEA. Finally, we apply RVC-MTEA to several real-world applications, showcasing the potential of CMO-MTO in practical decision-making scenarios.

Achieving a tradeoff between convergence, feasibility, and diversity is critical for solving constrained multiobjective optimization problems (CMOPs). Existing constrained multiobjective evolutionary algorithms (CMOEAs) primarily focus on constraint-handling techniques to balance constraint satisfaction and objective optimization. However, individual diversity is generally considered to be low. Owing to the insufficient enhancement of diversity, CMOEAs are unable to disperse well in the objective space to enhance the search for the constrained Pareto front (CPF) when handling CMOPs with complex constraints. To address this limitation, this study develops a diversity-enhanced tri-stage framework with three different evolutionary stages. First, sufficient convergence is enabled to move the population across the infeasible regions. Afterward, an angle-domination strategy is designed, aiming to spread the population evenly in the objective space while maintaining the achieved convergence. Third, we propose a minimum neighborhood-based domination strategy to ensure that the population searches the CPF by pursuing an even distribution in the objective space. Moreover, a weight vector preselection strategy is proposed to reduce computational overhead by avoiding ineffective searches in regions that do not include the CPF. Extensive experiments with 48 benchmark instances and 25 real-world instances validate the effectiveness of our approach over nine state-of-the-art methods.

Multiobjective multitask optimization (MO-MTO) aims to exploit the similarities among different multiobjective optimization tasks through knowledge transfer (KT), facilitating their simultaneous resolution. The effective design of KT techniques embedded in multiobjective evolutionary optimizers is crucial for enhancing the performance of multiobjective multitask evolutionary algorithms (MO-MTEAs). However, a significant limitation of existing KT techniques in MO-MTEAs is their equal treatment of particles/individuals for transferred knowledge reception, which can negatively impact the balance of diversity and convergence in population evolution. To remedy this limitation, this article proposes a new MO-MTEA, named MTEA-DCK, which incorporates diversity-oriented KT (DKT) and convergence-oriented KT (CKT) techniques tailored for different particles in the population. MTEA-DCK utilizes a strength-Pareto-based competitive mechanism to divide particles into winners and losers: 1) for winners, DKT is conducted via an intertask domain alignment approach to enhance population diversity and 2) for losers, CKT is executed within the unified search space to improve convergence. Additionally, to ensure robust performance on complex task combinations, we introduce two automatic parameter control strategies specifically designed for these KT techniques. MTEA-DCK was performed on 39 benchmark MO-MTO problems and demonstrated superior performance compared to eight state-of-the-art MO-MTEAs and six multiobjective evolutionary algorithms. Finally, we present three real-world MO-MTO application cases, where our approach also yielded better results than other algorithms.

EMaTO endeavors to solve more than three optimization tasks simultaneously by leveraging similarities among tasks. While existing algorithms have shown promising results, they face significant challenges in low-similarity scenarios. First, existing transfer techniques, which rely on population location and distribution, become ineffective. Second, the difficulty of selecting appropriate knowledge increases significantly. To address these challenges, we introduce a new concept: distribution direction knowledge, i.e., the evolutionary direction (ED) of elite solutions. It enables the target task to learn the search experience of source tasks with similar evolutionary trends. To utilize this knowledge effectively, an evolutionary many-task optimization (EMaTO) algorithm with distribution direction-assist two-stage knowledge transfer (DTSKT) is proposed. First, an ED-based MSS strategy is proposed to obtain appropriate knowledge in different circumstances. Second, we design a two-stage knowledge transfer (TSKT) strategy to search promising regions, consisting of exploration-oriented and EiKT. In addition, to directly obtain distribution direction knowledge, the EDA is applied as the basic optimizer, explicitly revealing the ED of populations by employing probability distributions. Afterward, to validate the ability of DTSKT to handle tasks with different similarities, we utilize a test problem generator to create a more challenging many-task benchmark suite, named STOP. The results on the WCCI20 and STOP benchmark suites, along with a real-world application, demonstrate that DTSKT generally outperforms seven state-of-the-art algorithms.

Evolutionary multitasking has attracted much attention over the past years due to its inter-task knowledge transfer capability. In this area, multiobjective multitask optimization (MO-MTO), aims to handle multiple multiobjective optimization tasks faster and better simultaneously via population synergies among tasks. Existing multiobjective multitask evolutionary algorithms (MO-MTEAs) for MO-MTO mostly transfer positions, i.e., decision variables, which may invoke negative knowledge transfer on tasks with low optimal domain similarities. However, such low similarities are common in practice. To address this issue, this paper proposes a new MO-MTEA, named MTEA/D-TSD, which transfers search directions, rather than positions, among decomposed subtasks for MO-MTO. In addition to position-neighborhood in the decomposition method, MTEA/D-TSD constructs and adaptively updates the search-direction-neighborhood for each subtask. It transfers successful search directions among neighbor subtasks to accelerate population evolution. Moreover, to further improve the efficiency of knowledge transfer, a transfer rate self-adaptation strategy is designed for MTEA/D-TSD. Experimental results on MO-MTO benchmark problems and a real-world application of sensor coverage problems demonstrated the superior performance of MTEA/D-TSD against state-of-the-art MO-MTEAs.

Evolutionary multitask optimization employs similarities among tasks via evolutionary algorithms (EAs) with knowledge transfer techniques to address multiple optimization tasks simultaneously. Although existing knowledge transfer techniques achieved success on population-based EAs, they are inappropriate for evolution strategies (ESs) that employ probability distribution sampling. These techniques will face two difficulties when applied to ESs: 1) distribution adaptation errors and 2) convergence difficulties. This paper proposes a knowledge-guided external sampling (KGxS) method to provide effective knowledge transfer in multitask ESs (MTESs) for solving multitask optimization problems (MTOPs). KGxS guides the distribution evolution in the target task by transferring solutions from source tasks as external samples. Since these external samples are close to the target distribution, they can handle the difficulty of distribution adaptation errors. In addition, the convergence difficulty caused by negative knowledge transfer is also handled through a mitigation strategy, which adaptively controls the number of external samples. Besides, the external samples carry two kinds of knowledge: 1) domain knowledge which employs the similarity of the optimal domains among tasks, and 2) shape knowledge that utilizes the function shapes similarity among tasks. Furthermore, a general boundary constraint handling technique is proposed for ESs to adapt to unconstrained and constrained optimization environments. Empirical results show that KGxS can significantly enhance the positive transfer effect on different types of ES on MTOPs. Moreover, the proposed method obtained superior performance over 20 state-of-the-art algorithms on 38 benchmark problems and three types of real-world applications, including multitask, many-task, and constrained multitask optimization.

Ensemble of constraint handling techniques (CHTs) is an effective way for solving constrained optimization problems (COPs). However, the use of multiple CHTs might cause more resource overhead. How to make reasonable resource allocation and cooperation among different CHTs is the issue that requires much more attention to improve the effectiveness of the ensemble of CHTs. Based on this consideration, we tried to handle the issue with adaptive resource allocation and population co-evolution for competition and cooperation among different CHTs. First, the competitive and cooperative evolutionary framework (CCEF) was proposed to construct multiple populations for CHTs. It assigns separate populations to each CHT and performs resource allocation based on success history records calculated by the performance of each CHT population. In addition, it promotes population co-evolution through parent recombination and offspring diffusion. Second, four CHTs with different types were chosen to compose a technique pool for the implementation of CCEF. Third, to obtain stronger search ability, we designed a heuristic and historical-based differential evolutionary operator (EO) embedded into CCEF. Experimental results on five COP benchmark suites (116 COPs in total) and 33 real-world COPs demonstrated that the proposed CCEF outperforms other advanced algorithms. The results revealed that CCEF can absorb the advantages of different CHTs and appropriately allocate resources to the more suitable CHTs.