An efficient multiple meta-model-based global optimization method for computationally intensive problems

The presence of black-box functions in engineering design, which are usually computation-intensive, demands efficient global optimization methods. This work proposes a new global optimization method for black-box functions. The global optimization method is based on a novel mode-pursuing sampling (MPS) method which systematically generates more sample points in the neighborhood of the function mode while statistically covers the entire search space. Quadratic regression is performed to detect the region containing the global optimum. The sampling and detection process iterates until the global optimum is obtained. Through intensive testing, this method is found to be effective, efficient, robust, and applicable to both continuous and discontinuous functions. It supports simultaneous computation and applies to both unconstrained and constrained optimization problems. Because it does not call any existing global optimization tool, it can be used as a standalone global optimization method for inexpensive problems as well. Limitation of the method is also identified and discussed.

In the surrogate model-based optimization of turbine airfoils, often only the prediction values for objective and constraints are employed, without considering uncertainties in the prediction. This is also the case for multi-fidelity optimization strategies based on e.g. the Gappy-POD approach, in which results from analyses of different fidelities are incorporated. However, the consideration of uncertainties in global optimization has the advantage that a balanced coverage of the design space between unexplored regions and regions close to the current optimum takes place. This means that on the one hand regions are covered in which so far only a few sample points are present and thus a high degree of uncertainty exists (global exploration), and on the other hand regions with promising objective and constraint values are investigated (local exploitation). The genuine new contribution in this work is the quantification of the uncertainty of the multi-fidelity Gappy-POD method and an adapted optimization strategy based on it. The uncertainty quantification is based on the error of linear fitting of low-fidelity values to the POD basis and subsequent forward propagation to the high-fidelity values. The uncertainty quantification is validated for random airfoil designs in a design of experiment. Based on this, a global optimization strategy for constrained problems is presented, which is based on the well-known Efficient Global Optimization (EGO) strategy and the Feasible Expected Improvement criterion. This means that Kriging models are created for both the objective and the constraint values depending on the design variables that consider both the predictions and the uncertainties. This approach offers the advantage that existing and widely used programs or libraries can be used for multi-fidelity optimization that support the (single-fidelity) EGO algorithm. Finally, the method is demonstrated for an industrial test case. A comparison between a single-fidelity optimization and a multi-fidelity optimization is made, each with the EGO strategy. A coupling of 2D/3D simulations is used for multi-fidelity analyses. The proposed method achieves faster feasible members in the optimization, resulting in faster turn-around compared to the single-fidelity strategy.

<div class="section abstract"><div class="htmlview paragraph">Plenum system of automotive vehicles draws atmospheric air using HVAC blower motor through air inlet grille at the bottom of the windshield and uses lower plenum for rainwater drainage after which air flow enters into awning and sugar scoop for evaporator air flow requirement. Awning and sugar scoop component is used in plenum air flow system to ensure the air flow with the required velocity that would meet the performance metric of total pressure drop of plenum air flow system. The optimal design of plenum component is essential since blower motor energy needs to be drawn from the expensive battery pack and any wastage of energy would lead to reduction in range of the Electric Vehicle. The conventional design process utilizes incremental design iteration, engineering intuition and requires significant effort from engineers to meet performance targets within the specified design space. This also may or may not ensure an optimal design.</div><div class="htmlview paragraph">In this work, a Machine learning tool with extreme gradient boost algorithm is used for development of a robust meta model that is used for quick prediction of awning and sugar scoop performance parameters such as total pressure drop and velocity of air prediction. On this response surface, application of a meta heuristic [<span class="xref">1</span>] based global optimization method is demonstrated which uses simplex homological global optimization (SHGO) and differential evolutionary (DE) method for optimum awning and sugar scoop design. Simplex homological global optimization (SHGO) is the concept of homology group growth which is used to solve a global and derivative free optimization problem of arbitrarily high dimensions. Similarly Differential evolution (DE) is a population-based metaheuristic search algorithm [<span class="xref">2</span>] that optimizes a problem which can quickly explore very large design spaces. Using these global optimization algorithms, it is observed that the optimized design of the awning and sugar scoop does result in significant reduction of total pressure drop that helps improve the required performance substantially compared to baseline design parameters within the given design space.</div></div>

Material parameters identification by inverse analysis using finite element computations leads to the resolution of complex and time-consuming optimization problems. One way to deal with these complex problems is to use meta-models to limit the number of objective function computations. In this paper, the Efficient Global Optimization (EGO) algorithm is used. The EGO algorithm is applied to specific objective functions, which are representative of material parameters identification issues. Isotropic and anisotropic correlation functions are tested. For anisotropic correlation functions, it leads to a significant reduction of the computation time. Besides, they appear to be a good way to deal with the weak sensitivity of the parameters. In order to decrease the computation time, a parallel strategy is defined. It relies on a virtual enrichment of the meta-model, in order to compute q new objective functions in a parallel environment. Different methods of choosing the qnew objective functions are presented and compared. Speed-up tests show that Kriging Believer (KB) and minimum Constant Liar (CLmin) enrichments are suitable methods for this parallel EGO (EGO-p) algorithm. However, it must be noted that the most interesting speed-ups are observed for a small number of objective functions computed in parallel. Finally, the algorithm is successfully tested on a real parameters identification problem.

Parallel radial basis function methods for the global optimization of expensive functions

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.

In this paper, we first establish some sufficient and some necessary global optimality conditions for quadratic integer programming problems. Then we present a new local optimization method for quadratic integer programming problems according to its necessary global optimality conditions. A new global optimization method is proposed by combining its sufficient global optimality conditions, local optimization method and an auxiliary function. The numerical examples are also presented to show that the proposed optimization methods for quadratic integer programming problems are very efficient and stable.

To improve the efficiency and quality of solving expensive engineering design optimization problems, recently metamodel-based design and optimization technologies have been widely employed. In this paper, a novel adaptive metamodel-based optimization method called sequential radial basis function using support vector machine is proposed to solve the practical engineering optimization problems involving computationally expensive objective and constraints. The proposed metamodel-based optimization method uses the radial basis function metamodel to approximate the expensive simulations in the interesting sampling region, which is a subregion of the design space. To identify the interesting sampling region, a novel feature space fuzzy -means clustering method is proposed to obtain the cluster center of superior cheap points classified by support vector machine. The procedure of sequential radial basis function using support vector machine is presented first, and the interesting sampling region and feature space fuzzy -means clustering methods are discussed in detail. Several numerical benchmark problems are used to compare the proposed method with other well-known metamodel-based design and optimization methods including efficient global optimization, multiple surrogate efficient global optimization methodology, mode-pursuing sampling, constraint importance mode pursuing sampling, and constrained optimization by radial basis function interpolation. The comparison results show that the proposed method generally outperforms the competitive methods in terms of efficiency, convergence, and robustness. Finally, sequential radial basis function using support vector machine is applied in an Earth observation satellite multidisciplinary design optimization problem. The results illustrate that compared with genetic algorithm and sequential quadratic programming, the proposed method can find a better solution with fewer function evaluations, which demonstrates the practicality and effectiveness of the proposed method in solving real-world engineering design optimization problems.

*† ‡ § An efficient and high-fidelity design approach is proposed by combining global and local optimization methods for wing planform and surface design. For enhanced design results, aerodynamic shape optimization process is carried out via 2-stage with different optimization strategy. In the first stage, global optimization techniques are applied to planform design with a few geometric design variables. In the second stage, local optimization techniques are used for wing surface design with a lot of design variables to maintain a sufficient design space with high DOF (Degree of Freedom) geometric change. For global optimization, meta-modeling techniques such as RS (Response Surface) and Kriging methods are used in conjunction with Genetic Algorithm (GA). For local optimization, a discrete adjoint variable method is used. By the successive combination of global and local optimization techniques, drag minimization is performed for a multi-body aircraft configuration while maintaining the baseline lift and the wing weight at the same time. Through the design process, performances of the test models are remarkably improved in comparison with the single stage design approach. The capability of proposed design framework including wing planform design variables can be evaluated by the drag decomposition method which can provide improvement of induced drag and wave drag, respectively.

Global optimization in optical design is particularly challenging for aspheric and freeform surfaces due to their complex, non-symmetric nature and the presence of numerous local minima in high-dimensional design spaces. Traditional optimization methods often struggle to efficiently escape these local minima, leading to suboptimal solutions. To address this, we propose the Simple Saddle Point Detection (SSPD) Algorithm, which enhances optimization by systematically identifying transition points that connect different design regions. By leveraging these pathways, the algorithm enables a more structured exploration of the design space, improving the convergence toward high-performance solutions. This study applies the SSPD approach to optimize complex optical systems, including catadioptric and multiple (folded) imaging mirror systems, where conventional methods face significant limitations. The results demonstrate that this approach is highly effective in refining aspheric and freeform optical designs, facilitating more efficient and reliable global optimization. Finally, we present the global search results as a closed network, highlighting the capability of SSPD to navigate complex design landscapes and achieve superior optical performance.

Efficient Global Optimization (EGO) is a popular method that searches sequentially for the global optimum of a simulated system. EGO treats the simulation model as a black-box, and balances local and global searches. In deterministic simulation, EGO uses ordinary Kriging (OK), which is a special case of universal Kriging (UK). In our EGO variant we use intrinsic Kriging (IK), which eliminates the need to estimate the parameters that quantify the trend in UK. In random simulation, EGO uses stochastic Kriging (SK), but we use stochastic IK (SIK). Moreover, in random simulation, EGO needs to select the number of replications per simulated input combination, accounting for the heteroscedastic variances of the simulation outputs. A popular selection method uses optimal computer budget allocation (OCBA), which allocates the available total number of replications over simulated combinations. We derive a new allocation algorithm. We perform several numerical experiments with deterministic simulations and random simulations. These experiments suggest that (1) in deterministic simulations, EGO with IK outperforms classic EGO; (2) in random simulations, EGO with SIK and our allocation rule does not differ significantly from EGO with SK combined with the OCBA allocation rule.

To improve the design quality and optimization efficiency of complicated engineering systems such as flight vehicle, metamodel based optimization is applied widely. Trust region is imported into the metamodel optimization and the updating strategy for sampling space using trust region is proposed and then the optimization strategy using dynamic radial basis function metamodel based on trust region is proposed. The metamodel is constructed with radial basis function and initial sampling points selected by Maximin latin hypercube design method. Global optimization algorithm is employed to optimize current metamodel to find the potential global optimum of the true optimization problem. According to the current known information, the trust region sampling space is updated. During optimization process, the new sampling points in the trust region sampling space are added, and the metamodel is updated, until the potential global optimum of the true optimization problem is satisfied the convergence conditions. The optimization strategy is validated by using five benchmark numerical test problems and an I-beam design problem. As the optimization results shown, the capability of TR-DRBF in both aspects of optimization efficiency and global convergency is good compared with the study fruit inland and overseas at present. Especially for high dimension problems, the performance of TR-DRBF is appealing.

Metamodel-based approaches are very often used for nonlinear optimization. The Successive Response Surface Method (SRSM) (Roux et al. in Int J Numer Methods Eng 42:517–534, 1998; Stander and Craig in Eng Comput 19:431–450, 2002) is one example for the application of such an approach. SRSM computes a smoothing approximation for each system response in each iteration based on a set of sample points generated in a defined domain around the current optimum. This domain is shifted and contracted in each iteration according to the resulting current best point. By default, a linear polynomial is used as the approximation function. A new Linear Interpolation Approach (LInA) has been developed to reduce the number of nonlinear analyses required during the optimization. In contrast to SRSM, LInA uses interpolating linear polynomials rather than regressions. Linear approximation is used to exclude artificial local minima between sample points. Due to the fact that no oversampling is required anymore, interpolation results in a lower number of nonlinear analyses without losing accuracy. A second reduction of the required analyses is achieved by avoiding a complete resampling in each iteration. It is assumed that the vectors defined from the current best point to the remaining sample points should be as orthogonal as possible. Therefore, LInA starts with initial sample points from a Koshal design of experiments (DOE) locally around the initial design point. After having computed the new best point based on the linear approximation, only a few existing sample points are deleted in each iteration and replaced by new ones. Complete QR decomposition is used to generate the new sample points in an orthogonal complementary subspace. First studies with two nonlinear structural optimization problems have shown that the LInA approach leads to a noteworthy reduction of the number of required nonlinear analyses in comparison to SRSM. Furthermore, LInA is compared to two kriging approaches using a subdomain similar as LInA does. LInA shows results comparable to kriging with respect to the number of analysis runs.

PurposeThe purpose of this paper is to investigate the mechanical properties’ behaviors of woven composite cut-out structures with specific parameters. Because of the complexity of micro-scale and meso-scale structure, it is difficult to accurately predict the mechanical material behavior of woven composites. Numerical simulations are increasingly necessary for the design and optimization of test procedures for composite structures made by the woven composite. The results of the proposed method are well satisfied with the results obtained from the experiment and other studies. Moreover, parametric studies on different plates based on the stacking sequences are investigated.Design/methodology/approachA multi-scale modeling approach is suggested. Back-propagation neural networks (BPNN), radial basis function (RBF) and least square support vector regression are integrated with efficient global optimization (EGO) to reduce the weight of assigned structure. Optimization results are verified by finite element analysis.FindingsCompared with other similar studies, the advantage of the suggested strategy uses homogenized properties behaviors with more complex analysis of woven composite structures. According to investigation results, it can be found that 450/−450 ply-orientation is the best buckling load value for all the cut-out shape requirements. According to the optimal results, the BPNN-EGO is the best candidate for the EGO to optimize the woven composite structures.Originality/valueA multi-scale approach is used to investigate the mechanical properties of a complex woven composite material architecture. Buckling of different cut-out shapes with the same area is surveyed. According to investigation, 45°/−45° ply-orientation is the best for all cut-out shapes. Different surrogate models are integrated in EGO for optimization. The BPNN surrogate model is the best choice for EGO to optimization difficult problems of woven composite materials.

A reconsideration on the resolution of phase stability analysis using stochastic global optimization methods: Proposal of a reliable set of benchmark problems