Infill Sampling Criterion Research Articles

Theoretical and experimental investigations on a data-driven trajectory planning scheme for dynamic shape control of piezo-actuated compliant morphing structures

Piezo-actuated compliant morphing structures can transmit motion and force via elastic deformation with lower weight, simplified design, and higher accuracy, a typical example is the variable camber wing. However, the rapid shape change of compliant structures is a challenging control problem because it often results in a high level of vibration due to the structural flexibility necessary to achieve the morphing requirement. This study presents a model-free data-driven trajectory planning approach based on spline curves and surrogate-based optimization (SBO) to obtain smooth “rest-to-rest” morphing trajectories. The macro-fiber composite (MFC) patches are used for piezoelectric actuators to generate elastic deformation. The flexible beam and variable camber wing are used for both linear numerical simulation and experimental tests with various nonlinearities and uncertainties. An optimization criterion is proposed to minimize both transient and residual vibrations during the “rest-to-rest” motion. An extraordinarily terminal displacement error function is added to eliminate the effects of the hysteresis and creep of the actuator. The control inputs, i.e., voltage profiles for the MFCs, are defined using cubic spline curves via only a few control points. Then, the optimization problem is solved by employing a Kriging surrogate model-based algorithm integrated with the expected improvement (EI) infilling-sampling criterion, which constructs an efficient algorithm similar to reinforcement learning. The optimal morphing trajectories can be highly efficiently obtained by using only 4 control points and less than 150 samplings. The experimental results of both the plate model and the variable camber wing model show that the proposed method can not only achieve a smooth dynamic morphing trajectory with minimized vibration but also automatically compensate for hysteresis and creep. The proposed method provides an effective means for the rapid realization of an optimized morphing trajectory for compliant morphing structures.

A noise-resistant infill sampling criterion in surrogate-assisted multi-objective evolutionary algorithms

Most existing surrogate-assisted multi-objective evolutionary algorithms are susceptible to the noise, since the noise affects both the approximation performance of objective surrogate models and the selection accuracy. Therefore, the keys to these algorithms are how to reduce the noise impact without additional function evaluations burden and how to maximize the noise immunity of the algorithms. In this work, a noise-resistant surrogate-assisted multi-objective evolutionary algorithm is proposed to solve the noisy expensive optimization problems. In the proposed algorithm, a novel classification-based noise handling method is used to reduce the noise impact without additional function evaluations burden before optimization. In the optimization process, the denoised data and model selection strategy are used to construct appropriate objective surrogate models to assist in generating promising candidates. Then, a noise-resistant infill sampling criterion considers convergence, diversity, and model uncertainty to select the most potential individual from candidates for re-evaluation. The experimental results on a series of expensive test problems with additive noise have demonstrated the competitiveness of the proposed algorithm against the other comparative algorithms.

Inverse distance weighting and radial basis function based surrogate model for high-dimensional expensive multi-objective optimization

Radial basis function (RBF) models have attracted a lot of attention in assisting evolutionary algorithms for solving computationally expensive optimization problems. However, most RBFs cannot directly provide the uncertainty information of their predictions, making it difficult to adopt principled infill sampling criteria for model management. To overcome this limitation, an inverse distance weighting (IDW) and RBF based surrogate assisted evolutionary algorithm, named IR-SAEA, is proposed to address high-dimensional expensive multi-objective optimization problems. First, an RBF-IDW model is developed, which can provide both the predicted objective values and the uncertainty of the predictions. Moreover, a modified lower confidence bound infill criterion is proposed based on the RBF-IDW for the balance of exploration and exploitation. Extensive experiments have been conducted on widely used benchmark problems with up to 100 dimensions. The empirical results have validated that the proposed algorithm is able to achieve a competitive performance compared with state-of-the-art SAEAs.

Radial projection-based adaptive sampling strategies for surrogate-assisted many-objective optimization

Intelligent manufacturing and industrial control systems frequently encounter expensive many-objective optimization problems (EMaOPs). Surrogate-assisted evolutionary algorithms (SAEAs) build predictive models to substitute the expensive fitness evaluation, enabling them to solve optimization more efficiently. In SAEAs, to enhance the exploration of optimization and the generalization of the surrogate model, the diversity of infill offspring and training database should be well-maintained, which is challenging in high-dimensional spaces or problems with disconnected Pareto front. This paper suggests a radial projection-based surrogate-assisted framework for solving EMaOPs. The radial projection can map the high-dimensional objective space into a 2-dimensional radial space. Based on this, a dynamic quadratic division method is proposed to enhance the diversity of solutions. Furthermore, an adaptive infill sampling criterion is introduced based on the distribution of selected convergent solutions, and a training database updating strategy is designed under the premise of maintaining its diversity and the model training efficiency. The presented framework exhibits a notable level of flexibility and adaptability as it can be effortlessly combined with other multi-objective optimization algorithms. Several experimental results on a set of expensive multi/many-objective test problems have demonstrated the superiority of the framework.

Surrogate Ensemble-Assisted Hyper-Heuristic Algorithm for Expensive Optimization Problems

This paper proposes a novel surrogate ensemble-assisted hyper-heuristic algorithm (SEA-HHA) to solve expensive optimization problems (EOPs). A representative HHA consists of two parts: the low-level and the high-level components. In the low-level component, we regard the surrogate-assisted technique as a type of search strategy and design the four search strategy archives: exploration strategy archive, exploitation strategy archive, surrogate-assisted estimation archive, and mutation strategy archive as low-level heuristics (LLHs), each archive contains one or more search strategies. Once the surrogate-assisted estimation archive is activated to generate the offspring individual, SEA-HHA first selects the dataset for model construction from three principles: All Data, Recent Data, and Neighbor, which correspond to the global and the local surrogate model, respectively. Then, the dataset is randomly divided into training and validation data, and the most accurate model built by polynomial regression (PR), support vector regression (SVR), and Gaussian process regression (GPR) cooperates with the infill sampling criterion is employed for solution estimation. In the high-level component, we design a random selection function based on the pre-defined probabilities to manipulate a set of LLHs. In numerical experiments, we compare SEA-HHA with six optimization techniques on 5-D, 10-D, and 30-D CEC2013 benchmark functions and three engineering optimization problems with only 1000 fitness evaluation times (FEs). The experimental and statistical results show that our proposed SEA-HHA has broad prospects for dealing with EOPs.

Cooperative coevolutionary surrogate ensemble-assisted differential evolution with efficient dual differential grouping for large-scale expensive optimization problems

This paper proposes a novel algorithm named surrogate ensemble assisted differential evolution with efficient dual differential grouping (SEADECC-EDDG) to deal with large-scale expensive optimization problems (LSEOPs) based on the CC framework. In the decomposition phase, our proposed EDDG inherits the framework of efficient recursive differential grouping (ERDG) and embeds the multiplicative interaction identification technique of Dual DG (DDG), which can detect the additive and multiplicative interactions simultaneously without extra fitness evaluation consumption. Inspired by RDG2 and RDG3, we design the adaptive determination threshold and further decompose relatively large-scale sub-components to alleviate the curse of dimensionality. In the optimization phase, the SEADE is adopted as the basic optimizer, where the global and the local surrogate model are constructed by generalized regression neural network (GRNN) with all historical samples and Gaussian process regression (GPR) with recent samples. Expected improvement (EI) infill sampling criterion cooperated with random search is employed to search elite solutions in the surrogate model. To evaluate the performance of our proposal, we implement comprehensive experiments on CEC2013 benchmark functions compared with state-of-the-art decomposition techniques. Experimental and statistical results show that our proposed EDDG is competitive with these advanced decomposition techniques, and the introduction of SEADE can accelerate the convergence of optimization significantly.

A Surrogate-Assisted Evolutionary Framework With Regions of Interests-Based Data Selection for Expensive Constrained Optimization

Optimization problems whose evaluations of the objective and constraints involve costly numerical simulations or physical experiments are referred to as expensive constrained optimization (ECO) problems. Such problems can be solved by evolutionary algorithms (EAs) in conjunction with computationally cheap surrogates that separately approximate the expensive objective and constraint functions. During the process of the ECO, the interested regions of surrogate models for the objective and constraints usually have a small overlap only. Specifically, the surrogate model for the objective function should focus on the prediction accuracy in the promising region, while the models for constraint functions should concentrate on the accuracy at the boundary of the feasible region. However, most existing methods neglect such differences and train those different models using the same training data, barely resulting in satisfactory performance. Therefore, we propose a general framework for solving expensive optimization problems with inequality constraints. In the proposed framework, the objective and constraints are separately trained with two different sets of training data to enhance the prediction accuracy and reliability in the interested regions. A novel infill sampling criterion is tailored to decide whether potentially better or more uncertain solutions should be sampled. Moreover, a new strategy, termed search intensity adjustment, is designed for adjusting the number of search generations on new surrogate models. We attempt to embed three competitive constrained EAs into our framework to verify its generality. The experimental results obtained on numerous benchmark functions from CEC2006, CEC2010, and CEC2017 have demonstrated the superiority of our approach over three state-of-the-art surrogate-assisted EAs.

Surrogate-based optimization of microstructural features of structural materials

This paper proposes a methodology for surrogate-based microstructural optimization of structural metals that integrates a limited number of 3D image-based numerical simulations with microstructural quantification, coarsening and optimisation processes. The support vector machine that was used had an infill sampling criterion to reduce the number of numerical trials, and the proposed methodology was found to be effective for wrought 2024 aluminium alloy with irregularly shaped particles. Appropriate objective functions were defined to assess particle damage. The number of design parameters, which quantitatively express the size, shape, and spatial distribution of particles, was initially 41, but they were reduced to four during a two-step coarsening process. The surrogate model provided highly accurate predictions, and the size, shape, and spatial distribution values of the optimal and weakest particles were successfully identified. It was shown that the optimal particle was small, spherical, sparsely dispersed, and perpendicular to the loading direction. However, it was also found that the smallest and most independent particle with a spherical shape was not necessarily strong, which implies the effects of particle clustering. It was also concluded that the dependency of in-situ particle strength on size was of crucial importance for weaker particles. The shape and spatial distribution of stronger particles were, however, more crucial for suppressing their internal stress than was their size. The results show that the proposed methodology offers a cost-efficient solution for microstructural designs involving 3D high-fidelity simulations that cannot be obtained with the existing approaches for developing materials.

Shape optimization of closed-box girder considering dynamic and aerodynamic effects on flutter: a CFD-enabled and Kriging surrogate-based strategy

Shape optimization of single box girder allows the bridge to achieve the most favourable aerodynamic performance. Currently, aerodynamic optimization of bridge deck is usually achieved by traversing all potential schemes using wind tunnel tests or computational fluid dynamics (CFD), which are time-consuming, laborious, and restricted to few numbers of geometric parameter. This study proposed an aerodynamic optimization strategy using CFD technique, uniform design, Kriging surrogate model and hybrid infill sampling criteria to achieve the most favourable girder shape with the largest flutter wind speed. The effect of the variation of section shape on the dynamic characteristics of a real suspension bridge is considered. The Kriging surrogate model is then applied to approximately describe the aerodynamic parameters including static force coefficients and flutter derivatives as well as dynamic characteristics of the bridge. The cross-validation and samples infill are conducted to validate and improve the accuracy of the model. The optimal ranges, influence degree of each factor and their interactions on flutter performance are examined by statistical analysis. The contributions of aerodynamic parameters and dynamic characteristics to flutter performance are compared and discussed. The girder shape with the best flutter performance is obtained in updated surrogate model.

A collaborative model calibration framework under uncertainty considering parameter distribution

Model calibration is critical to update unknown parameters and model differences between simulation and experiment. While some of the current methods can quantify response uncertainty and estimate the likelihood function of unknown parameters, calibration remains challenging when the parameters are uncertainties of unknown distribution. Considering model uncertainty inversion, we propose a collaborative calibration framework that quantifies and calibrates the distribution of unknown parameters. We used the Nested Stochastic Kriging (NSK) model to estimate the global trend and uncertainty of response data. The unknown parameters are then updated using the optimization-based calibration (OBC) method, where an infill sampling criterion is proposed. Finally, the variance of the parameter is calibrated. The calibrated model can reflect the uncertainty of the response data and can be further used to estimate the distribution of a new design. The proposed approach is tested using two numerical instances, and the bolt simulation model under tension-bending condition is calibrated using 30 experimental data points. The results demonstrate that the proposed method can achieve the best performance in terms of accuracy and efficiency.

A kriging-based sequential optimization algorithm with hybrid infill sampling strategy

The appropriate infill sampling criteria to generate more promising updated points is crucial for kriging-based global optimization. For this purpose, a kriging-based sequential optimization algorithm with hybrid infill sampling strategy (KSO-HIS) is proposed. In each iteration, three efficient sampling criteria (i.e., predicted objective minimization criterion, improved expected improvement and new curvature maximization criterion) based on kriging are respectively optimized to produce three optimal solutions by TR (Trust Region) method. Then, a new screening strategy is adopted to determine final expensive evaluation points from the three optimal solutions. The proposed method is compared with three other optimization methods. The test results of eight benchmark functions and a simulation case verify that KSO-HIS can deliver better sampling and convergence performance.

An Ensemble Surrogate-Based Framework for Expensive Multiobjective Evolutionary Optimization

Surrogate-assisted evolutionary algorithms (SAEAs) have become very popular for tackling computationally expensive multiobjective optimization problems (EMOPs), as the surrogate models in SAEAs can approximate EMOPs well, thereby reducing the time cost of the optimization process. However, with the increased number of decision variables in EMOPs, the prediction accuracy of surrogate models will deteriorate, which inevitably worsens the performance of SAEAs. To deal with this issue, this article suggests an ensemble surrogate-based framework for tackling EMOPs. In this framework, a global surrogate model is trained under the entire search space to explore the global area, while a number of surrogate submodels are trained under different search subspaces to exploit the subarea, so as to enhance the prediction accuracy and reliability. Moreover, a new infill sampling criterion is designed based on a set of reference vectors to select promising samples for training the models. To validate the generality and effectiveness of our framework, three state-of-the-art evolutionary algorithms [nondominated sorting genetic algorithm III (NSGA-III), multiobjective evolutionary algorithm based on decomposition with differential evolution (MOEA/D-DE) and reference vector-guided evolutionary algorithm (RVEA)] are embedded, which significantly improve their performance for solving most of the test EMOPs adopted in this article. When compared to some competitive SAEAs for solving EMOPs with up to 30 decision variables, the experimental results also validate the advantages of our approach in most cases.

Interval uncertainty propagation by a parallel Bayesian global optimization method

This paper is concerned with approximating the scalar response of a complex computational model subjected to multiple input interval variables. Such task is formulated as finding both the global minimum and maximum of a computationally expensive black-box function over a prescribed hyper-rectangle. On this basis, a novel non-intrusive method, called ‘triple-engine parallel Bayesian global optimization’, is proposed. The method begins by assuming a Gaussian process prior (which can also be interpreted as a surrogate model) over the response function. The main contribution lies in developing a novel infill sampling criterion, i.e., triple-engine pseudo expected improvement strategy, to identify multiple promising points for minimization and/or maximization based on the past observations at each iteration. By doing so, these identified points can be evaluated on the real response function in parallel. Besides, another potential benefit is that both the lower and upper bounds of the model response can be obtained with a single run of the developed method. Four numerical examples with varying complexity are investigated to demonstrate the proposed method against some existing techniques, and results indicate that significant computational savings can be achieved by making full use of prior knowledge and parallel computing.

Comparison of parallel infill sampling criteria based on Kriging surrogate model

One of the key issues that affect the optimization effect of the efficient global optimization (EGO) algorithm is to determine the infill sampling criterion. Therefore, this paper compares the common efficient parallel infill sampling criterion. In addition, the pseudo-expected improvement (EI) criterion is introduced to minimizing the predicted (MP) criterion and the probability of improvement (PI) criterion, which helps to improve the problem of MP criterion that is easy to fall into local optimum. An adaptive distance function is proposed, which is used to avoid the concentration problem of update points and also improves the global search ability of the infill sampling criterion. Seven test problems were used to evaluate these criteria to verify the effectiveness of these methods. The results show that the pseudo method is also applicable to PI and MP criteria. The DMP and PEI criteria are the most efficient and robust. The actual engineering optimization problems can more directly show the effects of these methods. So these criteria are applied to the inverse design of RAE2822 airfoil. The results show the criterion including the MP has higher optimization efficiency.

A Kriging-Assisted Two-Archive Evolutionary Algorithm for Expensive Many-Objective Optimization

Only a small number of function evaluations can be afforded in many real-world multiobjective optimization problems (MOPs) where the function evaluations are economically/computationally expensive. Such problems pose great challenges to most existing multiobjective evolutionary algorithms (EAs), which require a large number of function evaluations for optimization. Surrogate-assisted EAs (SAEAs) have been employed to solve expensive MOPs. Specifically, a certain number of expensive function evaluations are used to build computationally cheap surrogate models for assisting the optimization process without conducting expensive function evaluations. The infill sampling criteria in most existing SAEAs take all requirements on convergence, diversity, and model uncertainty into account, which is, however, not the most efficient in exploiting the limited computational budget. Thus, this article proposes a Kriging-assisted two-archive EA for expensive many-objective optimization. The proposed algorithm uses one influential point-insensitive model to approximate each objective function. Moreover, an adaptive infill criterion that identifies the most important requirement on convergence, diversity, or uncertainty is proposed to determine an appropriate sampling strategy for reevaluations using the expensive objective functions. The experimental results on a set of expensive multi/many-objective test problems have demonstrated its superiority over five state-of-the-art SAEAs.

A framework for expensive many-objective optimization with Pareto-based bi-indicator infill sampling criterion

A framework for expensive many-objective optimization with Pareto-based bi-indicator infill sampling criterion

Design by adaptive infill sampling with multi-objective optimization for exploitation and exploration

A surrogate-based design optimization with an adaptive sampling technique based on the innovative infill sampling criteria (ISC) is developed, where key issues are mathematically formulated on where in the design space and how many sample points are infilled to guarantee desirable accuracy at minimal computational cost. The ISC developed in the study involve multi-objective optimization (MOO) to determine infilling sample points separately for exploitation and exploration of the design space, which represent the global and the local accuracy of the surrogate model, respectively. The infill samples are found by the MOO on the Pareto front in terms of variance of estimation uncertainty and a predicted function value. To dynamically control the location and the count of the infilling points per iteration for the sample infilling, two criteria of the balancing and the dynamic switching approach are developed.The balancing approach selects infill sample points equally from the two far ends of the Pareto front as well as on the middle of it. The dynamic switching approach uses cut-off variance of uncertainty estimation to dynamically switch the ISC exclusively from the exploration to the exploitation, or vice versa adaptively to the accuracy of the model. Solution optimality and computation efficiency of the present method for the EGO, are compared for two analytic functions with those of the EGO with a conventional, multi-point Expected Improvement (q-EI) ISC and a Latin Hypercube Sampling (LHS) method. The gradient-based optimization without using the surrogate model was also carried out independently for the comparison purpose on the solution accuracy and efficiency. The proposed method shows the greatest efficiency, requiring the smallest number sample points in the training set and becomes even compatible with the gradient-based optimization method. For the practical design problem, high-life multi-element airfoil is chosen to maximize a lift coefficient with non-increasing drag constraints. The proposed method showed about 18% increase of lift force.

Global Optimization Algorithm Based on Kriging Using Multi-Point Infill Sampling Criterion and Its Application in Transportation System

Public traffic has a great influence, especially with the background of COVID-19. Solving simulation-based optimization (SO) problem is efficient to study how to improve the performance of public traffic. Global optimization based on Kriging (KGO) is an efficient method for SO; to this end, this paper proposes a Kriging-based global optimization using multi-point infill sampling criterion. This method uses an infill sampling criterion which obtains multiple new design points to update the Kriging model through solving the constructed multi-objective optimization problem in each iteration. Then, the typical low-dimensional and high-dimensional nonlinear functions, and a SO based on 445 bus line in Beijing city, are employed to test the performance of our algorithm. Moreover, compared with the KGO based on the famous single-point expected improvement (EI) criterion and the particle swarm algorithm (PSO), our method can obtain better solutions in the same amount or less time. Therefore, the proposed algorithm expresses better optimization performance, and may be more suitable for solving the tricky and expensive simulation problems in real-world traffic problems.

Comparison of parallel infill sampling criteria for optimization problems

The efficient global optimization (EGO) algorithm based on the Kriging surrogate model is popular. One of the main problems is to determine the infill sampling criteria. An adaptive distance function is proposed and applied to the EI, PI, and MP criteria. The criteria base on a fix distance is also investigated. Seven test problems were used to evaluate these criteria. The results show that the MMP criterion has poor global search ability. PEI, KB and DMP criteria perform better and have better convergence speed and stability.

A Comparative Study of Infill Sampling Criteria for Computationally Expensive Constrained Optimization Problems

Engineering optimization problems often involve computationally expensive black-box simulations of underlying physical phenomena. This paper compares the performance of four constrained optimization algorithms relying on a Gaussian process model and an infill sampling criterion under the framework of Bayesian optimization. The four infill sampling criteria include expected feasible improvement (EFI), constrained expected improvement (CEI), stepwise uncertainty reduction (SUR), and augmented Lagrangian (AL). Numerical tests were rigorously performed on a benchmark set consisting of nine constrained optimization problems with features commonly found in engineering, as well as a constrained structural engineering design optimization problem. Based upon several measures including statistical analysis, our results suggest that, overall, the EFI and CEI algorithms are significantly more efficient and robust than the other two methods, in the sense of providing the most improvement within a very limited number of objective and constraint function evaluations, and also in the number of trials for which a feasible solution could be located.