Various real-world applications can be classified as expensive multimodal optimization problems. When surrogate-assisted evolutionary algorithms (SAEAs) are employed to tackle these problems, they not only face a contradiction between the precision of surrogate models and the cost of individual evaluations but also have the difficulty that surrogate models and problem modalities are hard to match. To address this issue, this article studies a dual-surrogate-assisted cooperative particle swarm optimization algorithm to seek multiple optimal solutions. A dual-population cooperative particle swarm optimizer is first developed to simultaneously explore/exploit multiple modalities. Following that, a modal-guided dual-layer cooperative surrogate model, which contains one upper global surrogate model and a group of lower local surrogate models, is constructed with the purpose of reducing the individual evaluation cost. Moreover, a hybrid strategy based on clustering and peak-valley is proposed to detect new modalities. Compared with five existing SAEAs and seven multimodal evolutionary algorithms, the proposed algorithm can simultaneously obtain multiple highly competitive optimal solutions at a low computational cost according to the experimental results of testing both 11 benchmark instances and the building energy conservation problem.

Many real-world applications can be formulated as expensive multimodal optimization problems (EMMOPs). When surrogate-assisted evolutionary algorithms (SAEAs) are employed to tackle these problems, they not only face the problem of selecting surrogate models but also need to tackle the problem of discovering and updating multiple modalities. Different optimization problems and different stages of evolutionary algorithms (EAs) generally require different types of surrogate models. To address this issue, in this article, we present a multisurrogate-assisted multitasking particle swarm optimization algorithm to seek multiple optimal solutions of EMMOPs at a low computational cost. The proposed algorithm first transforms an EMMOP into a multitasking optimization problem by integrating various surrogate models, and designs a multitasking niche particle swarm algorithm to solve it. Following that, a surrogate model management strategy based on the skill factor and clustering is developed to effectively balance the number of real function evaluations and the prediction accuracy of candidate optimal solutions. In addition, an adaptive local search strategy based on the trust region is proposed to enhance the capability of swarm in exploiting potential optimal modalities. We compare the proposed algorithm with five state-of-the-art SAEAs and seven multimodal EAs on 19 benchmark functions and the building energy conservation problem and experimental results show that the proposed algorithm can obtain multiple highly competitive optimal solutions.

Expensive constraint multimodal optimization problems (ECMMOPs) have such characteristics as expensive objectives and constraints, and multiple optimal modalities simultaneously, which pose severe challenges to evolutionary optimization methods. This paper studies an objective-constraint mutual-guided surrogate-assisted particle swarm optimization algorithm for the kind of problem, aiming to discover multiple competing feasible optimal solutions at a lower calculation cost. The algorithm designs firstly a new two-layer cooperative surrogate model framework based on heterogeneous database to effectively adjust the prediction accuracies of objective surrogates and constraint surrogates on different search regions. And, an objective-constraint mutual-guided partial evaluation strategy (O-C-PES) is developed to generate high-quality infilling samples for objective and constraint surrogates respectively, based on which the number of unnecessary real evaluations can be significantly reduced. Moreover, a position feature-guided hybrid update mechanism (PF-HUM) is proposed to find more optimal solutions by searching excellent infeasible and feasible areas at the same time, and a feasible ratio-driven local search (FR-LS) strategy is proposed to improve the algorithm’s exploitation. Compared with four existing surrogate-assisted evolutionary algorithms and one constraint multimodal evolutionary algorithms on 21 benchmark problems and three engineering instances, experiment results show that the proposed algorithm can simultaneously obtain multiple highly-competitive feasible optimal solutions with less computational cost.

In practice, some optimization problems require expensive calculation and exhibit multimodal characteristics simultaneously. These problems are called high-dimensional expensive multimodal optimization problems. When addressing such problems, existing surrogate-assisted evolutionary algorithms (SAEAs) encounter the “curse of dimensionality," which severely affects their capability to search optimal solutions. Therefore, this study proposed a surrogate and autoencoder-assisted multitask particle swarm optimization algorithm. First, an autoencoder-embedded multitask evolutionary framework was established to transform a high-dimensional multimodal optimization problem into multiple low-dimensional subproblems or subtasks. Further, a multi-level surrogate model management mechanism combining mirror learning was proposed. An appropriate local surrogate model can be rapidly generated for each modality of the problem. Moreover, a dual-mode local exploitation strategy was developed to improve the capability of swarm to exploit each subtask. The proposed algorithm was compared with seven existing SAEAs on 33 benchmark functions and the aeroengine aerodynamic design optimization problem. Experimental results revealed that the proposed algorithm can obtain multiple highly competitive optimal solutions, including global optimal solutions.

The refined aerodynamic design optimization of multistage compressors is a typical high-dimensional and expensive optimization problem. This study proposes an integrated surrogate model-assisted evolutionary algorithm combined with a Directly Manipulated Free-Form Deformation (DFFD)-based parametric dimensionality reduction method, establishing a high-precision and efficient global parallel aerodynamic optimization platform for multistage axial compressors. The DFFD method achieves a balance between flexibility and low-dimensional characteristics by directly controlling the surface points of blades, which demonstrates a particular suitability for the aerodynamic design optimization of multistage axial compressors. The integrated surrogate model enhances prediction accuracy by simultaneously identifying optimal solutions and the most uncertain solutions, effectively addressing highly nonlinear design space challenges. A three-stage axial compressor in a heavy-duty gas turbine is selected as the optimization object. The results demonstrate that the optimization task takes less than 48 h and achieves an improvement of 0.6% and 4% in the adiabatic efficiency and surge margin, respectively, while maintaining a nearly unchanged flow rate and pressure ratio at the design point. The proposed approach provides an efficient and reliable solution for complex aerodynamic optimization problems.