An Adaptive Ensemble Surrogate Model Based on Fuzzy C-Means and Its Parallel Infilling Strategy

Infilling strategies have been proposed for decades and are widely used in engineering problems. It is still challenging to achieve an effective trade-off between global exploration and local exploitation. In this paper, a novel decision-making infilling strategy named the Go-inspired hybrid infilling (Go-HI) strategy is proposed. The Go-HI strategy combines multiple individual infilling strategies, such as the mean square error (MSE), expected improvement (EI), and probability of improvement (PoI) strategies. The Go-HI strategy consists of two major parts. In the first part, a tree-like structure consisting of several subtrees is built. In the second part, the decision value for each subtree is calculated using a cross-validation (CV)-based criterion. Key factors that significantly influence the performance of the Go-HI strategy, such as the number of component infilling strategies and the tree depth, are explored. Go-HI strategies with different component strategies and tree depths are investigated and also compared with four baseline adaptive sampling strategies through three numerical functions and one engineering case. Results show that the number of component infilling strategies exerts a larger influence on the global and local performance than the tree depth; the Go-HI strategy with two component strategies performs better than the ones with three; the Go-HI strategy always outperforms the three component infilling strategies and the other four benchmark strategies in global performance and robustness and saves much computational cost.

Surrogate models have been widely applied in the aerodynamic optimization of aircrafts, whereas the traditional individual surrogate models have the defects of low robustness and applicability. In this study, a novel ensemble surrogate model is proposed and applied in the multi-objective optimization of the airfoil. The backpropagation neural network, deep belief network, and kriging surrogate models are selected as the member surrogate models, and the Dirichlet distribution strategy is introduced to adaptively generate the weights of the member surrogate models in constructing the ensemble surrogate model. An improved multi-objective particle swarm optimization (MOPSO) framework is established by employing the α-stable distribution function to enhance the global convergence rate of the algorithm. Based on the improved MOPSO framework in which the ensemble surrogate model is embedded, the multi-objective optimization of the airfoil is conducted. The results indicate that the proposed ensemble-surrogate-model-based optimization obtains better aerodynamic performance of the airfoil under multiple operating conditions, compared to the individual-surrogate-model-based optimization.

Efficient Global Optimization (EGO) has become a standard approach for the global optimization of complex systems with high computational costs. EGO uses a training set of objective function values computed at selected input points to construct a statistical surrogate model, with low evaluation cost, on which the optimization procedure is applied. The training set is sequentially enriched, selecting new points, according to a prescribed infilling strategy, in order to converge to the optimum of the original costly model. Multi-fidelity approaches combining evaluations of the quantity of interest at different fidelity levels have been recently introduced to reduce the computational cost of building a global surrogate model. However, the use of multi-fidelity approaches in the context of EGO is still a research topic. In this work, we propose a new effective infilling strategy for multi-fidelity EGO. Our infilling strategy has the particularity of relying on non-nested training sets, a characteristic that comes with several computational benefits. For the enrichment of the multi-fidelity training set, the strategy selects the next input point together with the fidelity level of the objective function evaluation. This characteristic is in contrast with previous nested approaches, which require estimation all lower fidelity levels and are more demanding to update the surrogate. The resulting EGO procedure achieves a significantly reduced computational cost, avoiding computations at useless fidelity levels whenever possible, but it is also more robust to low correlations between levels and noisy estimations. Analytical problems are used to test and illustrate the efficiency of the method. It is finally applied to the optimization of a fully nonlinear fluid-structure interaction system to demonstrate its feasibility on real large-scale problems, with fidelity levels mixing physical approximations in the constitutive models and discretization refinements.

Surrogate-based method can dramatically reduce the number of expensive function evaluations in real-world multi- objective optimization problems (MOP). When the number of objectives is small, using surrogate models combined with expected hypervolume improvement (EHVI) infill sampling criterion (ISC) has been proved to be efficient to provide a set of solutions with good diversity and good proximity to the Pareto front (PF) in aerodynamic shape optimization. However, traditional hypervolume-based infilling strategies use only one kind of ISC to generate one or multiple sample points, the advantages of various kinds of ISC cannot be comprehensively utilized and the parallelization is not easy to implement. This paper proposes a combined multi-point infilling strategy based on Kriging models and develops an efficient global multi-objective constrained optimization method (EGMOCO) to solve multi- objective aerodynamic shape optimization with complex constraints. Multiple sample points are generated by using four ISC considering hypervolume at each iteration and then evaluated in parallel. Firstly, the performance of EGMOCO is compared with that of single criterion EHVI strategy on six benchmarks within the same computational budget to prove its effectiveness, and then EGMOCO is implemented in an aerodynamic shape optimization problem with complex constraints. The result shows that EGMOCO has good performance in balancing local exploitation and global exploration with faster convergence rate and high robustness, the whole PF can be fully explored in limited evaluations and the constraint handling is effective especially for real-world problems with complex and nonlinear constraints, the comprehensive aerodynamic performance of the airfoil is greatly improved. It can be confirmed that Kriging-based multi-objective optimization method combined with multi-point infilling strategy performs better than single infilling criterion EHVI, since different sample infilling criteria can complement with each other, both local exploitation and global exploration can be considered and well balanced.

Most practical multi-objective optimization problems are often characterized by two or more expensive and conflicting objectives, which require time-consuming simulations. Commonly used algorithms construct a surrogate model of each objective function from a few high-fidelity solutions. In order to further decrease the computational burden, adaptive infilling strategies for multi-objective problems are developed to guide the next infilling design for expensive evaluation and update the surrogate model as well as the Pareto front in an iterative manner. In this paper, a multi-objective infilling strategy integrating the Kriging model with a two-stage infilling framework is proposed, termed as ATKIS. This method allows exploitation and exploration alternately to pinpoint the infilling solution for improving the Pareto set and avoiding local over-exploitation simultaneously. At the local exploitation stage, Kriging-based prediction and uncertainty estimation are combined with Non-dominant Ranking and Minimum Relative Distance theories for determining a new design solution, which has maximum improvement relative to the current Pareto set. At the global exploration stage, Voronoi tessellation theory is employed to search for the sparsest position in the design space for a new evaluation. The proposed method is compared with five recent infilling strategies to investigate the performance of infilling ability using several numerical benchmarks. The experimental results show that the proposed method outperforms the other three strategies in improving both effectiveness and robustness using the improvement of hypervolume as the evaluating indicator. In addition, a lightweight optimization design of hoist sheaves shows that the proposed method can deal with real engineering problems, while significantly reducing the computational time and the number of expensive simulations of samples.

For the path coverage testing of a Message-Passing Interface (MPI) program, test data generation based on an evolutionary optimization algorithm (EOA) has been widely known. However, during the use of the above technique, it is necessary to evaluate the fitness of each evolutionary individual by executing the program, which is generally computationally expensive. In order to reduce the computational cost, this article proposes a method of integrating an ensemble surrogate model’s estimation into the process of generating test data. The proposed method first produces a number of test inputs using an EOA, and forms a training set together with their real fitness. Then, this article trains an ensemble surrogate model (ESM) based on the training set, which is employed to estimate the fitness of each individual. Finally, a small number of individuals with good estimations are selected to further execute the program, so as to have their real fitness for the subsequent evolution. This article applies the proposed method to seven benchmark MPI programs, which is compared with several state-of-the-art approaches. The experimental results show that the proposed method can generate test data with significantly low computational cost.

Adaptive surrogate model based multi-objective transfer trajectory optimization between different libration points

As an important part of the lifting platform of pallet forklift trucks, how to reduce the deformation of the pallet rack under working conditions while reducing the mass to ensure the safety performance of forklift trucks is the most concerning issue in the design of forklift truck structure. The pallet rack structure is complex, and optimizing simulation using traditional high-precision simulation models takes much time and effort. Therefore, this paper takes the lifting platform of an unmanned AVG forklift truck as the research object and establishes a parametric model of the pallet rack using the 3D modelling software SolidWorks and the finite element analysis software ANSYS to carry out static analysis of it. Optimization design variables are selected, a single surrogate model and ensemble surrogate model are chosen for various surrogate model techniques, a small number of sample points are used to construct a low-precision model, and adaptive infilling technology is used to improve the model accuracy, and the structure is optimized, and the results show that, while the pallet rack structure meets the requirements of light weight and strength, the mass is reduced by 1.2%, and the morphology is reduced by 17.2%. Moreover, a global sensitivity analysis of each design parameter was carried out under the guidance of the surrogate model, highlighting the most influential design variable as the height of the channel steel and establishing the design variables that should be taken into account in the structural design. This paper compares the performance of the mainstream single-surrogate model and ensemble-surrogate model as well as the adaptive infilling strategy in engineering design and proves that the surrogate model optimization method has a certain guiding significance for the structural optimization design of pallet racking.

To achieve time–energy–impact multi-objective optimization in the trajectory control of underwater manipulators, this paper proposes a Fast Non-Dominated Sorting Tuna Swarm Optimization algorithm (FNS-TSO). The algorithm integrates a fast non-dominated sorting mechanism into the Tuna Swarm Optimization algorithm, improves initialization through Optimal Latin Hypercubic Sampling (OLHS) to enhance population distribution uniformity, and incorporates a nonlinear dynamic weight to refine the spiral foraging strategy, thereby strengthening algorithmic robustness. To verify FNS-TSO’s effectiveness, we conducted comparative evaluations using standard test functions against three established algorithms: Multi-Objective Particle Swarm Optimization (MOPSO), Multi-Objective Jellyfish Search Optimization (MOJSO), and the Non-Dominated-Sorting Genetic Algorithm (NSGA-II). Results demonstrate superior overall performance, particularly regarding convergence speed and solution diversity, with solution set distributions showing enhanced uniformity. In practical implementation, we applied FNS-TSO to the multi-objective optimization of an underwater manipulator using quintic spline curves for trajectory planning. Simulation outcomes reveal respective reductions of 11.03% in total operation time, 19.02% in energy consumption, and 24.69% in mechanical impacts, with the optimized manipulator achieving stable point-to-point motion transitions.

Multi-objective Fuzzy Optimization Design of Helical Gear Drive

Gaussian-Process based optimization methods have become very popular in recent years for the global optimization of complex systems with high computational costs. These methods rely on the sequential construction of a statistical surrogate model, using a training set of computed objective function values, which is refined according to a prescribed infilling strategy. However, this sequential optimization procedure can stop prematurely if the objective function cannot be computed at a proposed point. Such a situation can occur when the search space encompasses design points corresponding to an unphysical configuration, an ill-posed problem, or a non-computable problem due to the limitation of numerical solvers. To avoid such a premature stop in the optimization procedure, we propose to use a classification model to learn non-computable areas and to adapt the infilling strategy accordingly. Specifically, the proposed method splits the training set into two subsets composed of computable and non-computable points. A surrogate model for the objective function is built using the training set of computable points, only, whereas a probabilistic classification model is built using the union of the computable and non-computable training sets. The classifier is then incorporated in the surrogate-based optimization procedure to avoid proposing new points in the non-computable domain while improving the classification uncertainty if needed. The method has the advantage to automatically adapt both the surrogate of the objective function and the classifier during the iterative optimization process. Therefore, non-computable areas do not need to be a priori known. The proposed method is applied to several analytical problems presenting different types of difficulty, and to the optimization of a fully nonlinear fluid-structure interaction system. The latter problem concerns the drag minimization of a flexible hydrofoil with cavitation constraints. The efficiency of the proposed method compared favorably to a reference evolutionary algorithm, except for situations where the feasible domain is a small portion of the design space.

This study introduces an information granular-based fuzzy radial basis function neural networks (FRBFNN) based on multiobjective optimization and weighted least square (WLS). An improved multiobjective space search algorithm (IMSSA) is proposed to optimize the FRBFNN. In the design of FRBFNN, the premise part of the rules is constructed with the aid of Fuzzy C-Means (FCM) clustering while the consequent part of the fuzzy rules is developed by using four types of polynomials, namely constant, linear, quadratic, and modified quadratic. Information granulation realized with C-Means clustering helps determine the initial values of the apex parameters of the membership function of the fuzzy neural network. To enhance the flexibility of neural network, we use the WLS learning to estimate the coefficients of the polynomials. In comparison with ordinary least square commonly used in the design of fuzzy radial basis function neural networks, WLS could come with a different type of the local model in each rule when dealing with the FRBFNN. Since the performance of the FRBFNN model is directly affected by some parameters such as e.g., the fuzzification coefficient used in the FCM, the number of rules and the orders of the polynomials present in the consequent parts of the rules, we carry out both structural as well as parametric optimization of the network. The proposed IMSSA that aims at the simultaneous minimization of complexity and the maximization of accuracy is exploited here to optimize the parameters of the model. Experimental results illustrate that the proposed neural network leads to better performance in comparison with some existing neurofuzzy models encountered in the literature.

In this paper, we propose a grayscale image segmentation method based on a multiobjective optimization approach that optimizes two complementary criteria (region and edge based). The region-based fitness used is the improved spatial fuzzy c-means clustering measure that is shown performing better than the standard fuzzy c-means (FCM) measure. The edge-based fitness used is based on the contour statistics and the number of connected components in the image segmentation result. The optimization algorithm used is the multiobjective particle swarm optimization (MOPSO), which is well suited to handle continuous variables problems, the case of FCM clustering. In our case, each particle of the swarm codes the centers of clusters. The result of the multiobjective optimization technique is a set of Pareto-optimal solutions, where each solution represents a segmentation result. Instead of selecting one solution from the Pareto front, we propose a method that combines all solutions to get a better segmentation. The combination method takes place in two steps. The first step is the detection of high-confidence points by exploiting the similarity between the results and the membership degrees. The second step is the classification of the remaining points by using the high-confidence extracted points. The proposed method was evaluated on three types of images: synthetic images, simulated MRI brain images and real-world MRI brain images. This method was compared to the most widely used FCM-based algorithms of the literature. The results demonstrate the effectiveness of the proposed technique.

A multiobjective and fixed elements based modification of the evolutionary structural optimization method

Since the fuzzy local information C-means (FLICM) segmentation algorithm cannot take into account the impact of different features on clustering segmentation results, a local fuzzy clustering segmentation algorithm based on a feature selection Gaussian mixture model was proposed. First, the constraints of the membership degree on the spatial distance were added to the local information function. Second, the feature saliency was introduced into the objective function. By using the Lagrange multiplier method, the optimal expression of the objective function was solved. Neighborhood weighting information was added to the iteration expression of the classification membership degree to obtain a local feature selection based on feature selection. Each of the improved FLICM algorithm, the fuzzy C-means with spatial constraints (FCM_S) algorithm, and the original FLICM algorithm were then used to cluster and segment the interference images of Gaussian noise, salt-and-pepper noise, multiplicative noise, and mixed noise. The performances of the peak signal-to-noise ratio and error rate of the segmentation results were compared with each other. At the same time, the iteration time and number of iterations used to converge the objective function of the algorithm were compared. In summary, the improved algorithm significantly improved the ability of image noise suppression under strong noise interference, improved the efficiency of operation, facilitated remote sensing image capture under strong noise interference, and promoted the development of a robust anti-noise fuzzy clustering algorithm.