Hybrid meta-model based search method for expensive problems

The complexity and opaque characteristics of the practical expensive problems hinder the further applications of the single meta-model based optimization algorithms. In this work, a hybrid meta-model based search method (HMBSM) is presented. In this method, an important region is firstly constructed using a part of the expensive points which are evaluated by the expensive problems. Then, three meta-models with different fitting techniques are used together both in the important region and the remaining region. The whole design space will also be searched simultaneously to further avoid the local optima. Through intensive test by six benchmark math functions with the variables ranging from 10 to 24 and compared with the efficient global optimization (EGO), hybrid meta-model based design space management (HMDSM) method and multiple meta-model based design space differentiation (MDSD) method, the proposed HMBSM method shows excellent accuracy, efficiency and robustness. Then, the proposed method is applied in a vehicle lightweight design involving finite element analysis with 30 design variables, reducing 11.4 kg of weight.     

Ensemble of surrogate based global optimization methods using hierarchical design space reduction

For computationally expensive black-box problems, surrogate models are widely employed to reduce the needed computation time and efforts during the search of the global optimum. However, the construction of an effective surrogate model over a large design space remains a challenge in many cases. In this work, a new global optimization method using an ensemble of surrogates and hierarchical design space reduction is proposed to deal with the optimization problems with computation-intensive, black-box objective functions. During the search, an ensemble of three representative surrogate techniques with optimized weight factors is used for selecting promising sample points, narrowing down space exploration and identifying the global optimum. The design space is classified into: Original Global Space (OGS), Promising Joint Space (PJS), Important Local Space (ILS), using the newly proposed hierarchical design space reduction (HSR). Tested using eighteen representative benchmark and two engineering design optimization problems, the newly proposed global optimization method shows improved capability in identifying promising search area and reducing design space, and superior search efficiency and robustness in identifying the global optimum.     

Multi-stage design space reduction and metamodeling optimization method based on self-organizing maps and fuzzy clustering

Design space exploration and metamodeling techniques have gained rapid dominance in complex engineering design problems. It is observed that the modeling efficiency and accuracy are directly associated with the design space. In order to reduce the complexity of the design space and improve modeling accuracy, a multi-stage design space reduction and metamodeling optimization methodology based on self-organizing maps and fuzzy clustering is proposed in this paper. By using the proposed three-stage optimization approach, the design space is systematically reduced to a relatively small promising region. Self-organizing maps are introduced to act as the preliminary reduction approach through analyzing the underlying mapping relations between design variables and system responses within the original samples. GK (Gustafson & Kessel) clustering algorithm is employed to determine the proper number of clusters by utilizing clustering validity indices, and sample points are clustered using fuzzy c-means (FCM) clustering method with the known number of clusters, so that the search can focus on the most promising area and be better supported by the constructed kriging metamodel. Empirical studies on benchmark problems with multi-humps and two practical nonlinear engineering design problems show the accurate results can be obtained within the reduced design space, which improve the overall efficiency significantly.     

Space exploration and global optimization for computationally intensive design problems: a rough set based approach

Modern engineering design problems often involve computation-intensive analysis and simulation processes. Design optimization based on such processes is desired to be efficient, informative and transparent. This work proposes a rough set based approach that can identify multiple sub-regions in a design space, within which all of the design points are expected to have a performance value equal to or less than a given level. The rough set method is applied iteratively on a growing sample set. A novel termination criterion is also developed to ensure a modest number of total expensive function evaluations to identify these sub-regions and search for the global optimum. The significance of the proposed method is twofold. First, it provides an intuitive method to establish the mapping from the performance space to the design space, i.e. given a performance level, its corresponding design region(s) can be identified. Such a mapping could be potentially used to explore and visualize the entire design space. Second, it can be naturally extended to a global optimization method. It also bears potential for more broad application to problems such as metamodeling-based design and robust design optimization. The proposed method was tested with a number of test problems and compared with a few well-known global optimization algorithms.     

Robust aerodynamic shape design based on an adaptive stochastic optimization framework

Optimization techniques combined with uncertainty quantification are computationally expensive for robust aerodynamic optimization due to expensive CFD costs. Surrogate model technology can be used to improve the efficiency of robust optimization. In this paper, non-intrusive polynomial chaos method and Kriging model are used to construct a surrogate model that associate stochastic aerodynamic statistics with airfoil shapes. Then, global search algorithm is used to optimize the model to obtain optimal airfoil fast. However, optimization results always depend on the approximation accuracy of the surrogate model. Actually, it is difficult to achieve a high accuracy of the model in the whole design space. Therefore, we introduce the idea of adaptive strategy to robust aerodynamic optimization and propose an adaptive stochastic optimization framework. The surrogate model is updated adaptively by increasing training airfoils according to historical optimization results to guarantee the accuracy near the optimal design point, which can greatly reduce the number of training airfoils. The proposed method is applied to a robust aerodynamic shape optimization for drag minimization considering uncertainty of Mach number in transonic region. It can be concluded that the proposed method can obtain better optimal results more efficiently than the traditional robust optimization method and global surrogate model method.     

Efficient adaptive response surface method using intelligent space exploration strategy

This article presents a novel intelligent space exploration strategy (ISES), which is then integrated with the adaptive response surface method (ARSM) for higher global optimization efficiency. ISES consists of two novel elements for space reduction and sequential sampling: i) Significant design space (SDS) identification algorithm, which is developed to identify the promising design space and balance local exploitation and global exploration during the search, and ii) An iterative maximin sequential Latin hypercube design (LHD) sampling scheme and tailored termination criteria. Moreover, an adaptive penalty method is developed for handling expensive constraints. The new global optimization strategy, notated as ARSM-ISES, is then tested with numerical benchmark problems on optimization efficiency, global convergence, robustness, and algorithm execution overhead. Comparative results show that ARSM-ISES not only outperforms the original ARSM and IARSM, in general it also converges to better optima with fewer function evaluations and less algorithm execution time as compared to state-of-the-art metamodel-based design optimization algorithms including MPS, EGO, and MSEGO. For high dimensional (HD) problems, ARSM-ISES shows promises as it performs better on chosen test problems than TR-MPS, which is especially designed for solving HD problems. ARSM-ISES is then applied to the optimal design of a lifting surface of hypersonic flight vehicles. Finally, main features and limitations of the proposed algorithm are discussed.     

基于改进细菌觅食算法的飞控系统多模态参数优化

针对飞控系统参数优化过程中存在的解空间非凸性问题,或由于多约束条件下导致的全局最优不可达问题,提出一种基于改进细菌觅食算法的多模态参数优化方法.采用基于格型准则的采样方法以尽可能广泛地搜索解空间,并利用K均值聚类的小生境技术使得多个细菌种群能够分别搜索各自的区域,以尽可能多地获得解空间中不同位置的可行解.同时研究一种自适应深度搜索策略,确保算法在整个寻优过程中的鲁棒性.所提出算法可以在完成对系统优化的基础上,探寻飞控系统中各参数本身的可行域及其在解空间中所处的位置,也能够在一定程度上揭示解空间本身的特性.仿真结果验证了所提出算法可以有效地简化系统调参的过程,更为快速地获得一个满足设计性能期望的飞控系统.     

Improving surrogate-assisted variable fidelity multi-objective optimization using a clustering algorithm

Surrogate-assisted evolutionary optimization has proved to be effective in reducing optimization time, as surrogates, or meta-models can approximate expensive fitness functions in the optimization run. While this is a successful strategy to improve optimization efficiency, challenges arise when constructing surrogate models in higher dimensional function space, where the trade space between multiple conflicting objectives is increasingly complex. This complexity makes it difficult to ensure the accuracy of the surrogates. In this article, a new surrogate management strategy is presented to address this problem. A k-means clustering algorithm is employed to partition model data into local surrogate models. The variable fidelity optimization scheme proposed in the author's previous work is revised to incorporate this clustering algorithm for surrogate model construction. The applicability of the proposed algorithm is illustrated on six standard test problems. The presented algorithm is also examined in a three-objective stiffened panel optimization design problem to show its superiority in surrogate-assisted multi-objective optimization in higher dimensional objective function space. Performance metrics show that the proposed surrogate handling strategy clearly outperforms the single surrogate strategy as the surrogate size increases.     

基于粒子群优化算法的自适应IIR滤波器设计

针对自适应无限冲激响应(infinite impulse response,IIR)数字滤波器的设计实质上是一个多参数优化问题,提出了一种用粒子群优化算法(particle swarm optimization,PSO)设计IIR数字滤波器的方法.将滤波器的设计问题转化为滤波器参数的优化问题,利用粒子群优化算法对整个参数空间进行高效并行搜索以获得参数的最优化,基于多个典型系统的随机数值仿真以及与最小二乘方法的比较研究,验证了该方法的有效性、全局性和对初值的鲁棒性.     

A multi-point sampling method based on kriging for global optimization

In general, sampling strategy plays a very important role in metamodel based design optimization, especially when computationally expensive simulations are involved in the optimization process. The research on new optimization methods with less sampling points and higher convergence speed receives great attention in recent years. In this paper, a multi-point sampling method based on kriging (MPSK) is proposed for improving the efficiency of global optimization. The sampling strategy of this method is based on a probabilistic distribution function converted from the expected improvement (EI) function. It can intelligently draw appropriate new samples in an area with certain probability according to corresponding EI values. Besides, three strategies are also proposed to speed up the sequential sampling process and the corresponding convergence criterions are put forward to stop the searching process reasonably. In order to validate the efficiency of this method, it is tested by several numerical benchmark problems and applied in two engineering design optimization problems. Moreover, an overall comparison between the proposed method and several other typical global optimization methods has been made. Results show that the higher global optimization efficiency of this method makes it particularly suitable for design optimization problems involving computationally expensive simulations.     

基于改进果蝇优化的压电精密定位平台迟滞Bouc-Wen模型辨识

提出一种基于改进果蝇优化算法（Improved Fruit fly Optimization Algorithm,IFOA）的压电精密定位平台迟滞特性的Bouc-Wen模型参数辨识方法。通过引入交叉因子和自适应搜索步长,IFOA可有效实现全局搜索与局部优化的动态平衡,并通过一种新的搜索策略提高整体搜索效率和优化精度。将IFOA应用于压电精密定位平台迟滞Bouc-Wen模型的参数辨识。实验辨识结果验证了该方法的有效性和潜力。     

Self-adaptive constrained artificial bee colony for constrained numerical optimization

The artificial bee colony is a simple and effective global optimization algorithm. It has been successfully applied to solve a wide range of real-world optimization problem, and later, it was extended to constrained design problems as well. This paper describes a self-adaptive constrained artificial bee colony algorithm for constrained optimization problem based on feasible rule method and multiobjective optimization method. The employed bee colony severs as the global search engine for each population based on feasible rule. Then, the onlooker bee colony can explore the new search space based on the multiobjective optimization. In order to enhance the convergence rate of the proposed algorithm, a self-adaptive modification rate is proposed to make the algorithm can change many parameters. To verify the performance of our approach, 24 well-known constrained problems from 2006 IEEE congress on Evolution Computation (CEC2006) are employed. Experimental results indicate that the proposed algorithm performs better than, or at least comparable to, state-of-the-art approaches in terms of the quality of the resulting solutions from literature.     

一种基于PSO的RBF-SVM模型优化新方法

针对使用径向基核函数的支持向量机,采用粒子群优化方法实现模型优化.基于训练集中样本之间的最近平均距离和最远平均距离,给出参数σ的取值空间,从而减小了超参数搜索的范围,并采用对数刻度进一步提高粒子群优化方法的参数搜索效率.与遗传算法和网格法的对比实验表明,所提出的方法收敛速度更快,得出的超参数更优.     

An intelligent sampling approach for metamodel-based multi-objective optimization with guidance of the adaptive weighted-sum method

In order to reduce the computational cost of multi-objective optimization (MOO) with expensive black-box simulation models, an intelligent sampling approach (ISA) is proposed with the guidance of the adaptive weighted-sum method (AWS) to construct a metamodel for MOO gradually. The initial metamodel is built by using radial basis function (RBF) with Latin Hypercube Sampling (LHS) to distribute samples over the design space. An adaptive weighted-sum method is then employed to obtain the Pareto Frontier (POF) efficiently based on the metamodel constructed. The design variables related to extreme points on the frontier and an extra point interpolated between the maximal-minimal-distance point along the frontier and the nearest boundary point are selected as the concerned points to update the metamodel, which could improve the metamodel accuracy gradually. This iterative updating strategy is performed until the optimization problem is converged. A series of representative mathematical examples are systematically investigated to demonstrate the effectiveness of the proposed method, and finally it is employed for the design of a bus body frame.     

Improving solution characteristics of particle swarm optimization using digital pheromones

In this paper, a new approach to particle swarm optimization (PSO) using digital pheromones to coordinate swarms within an n-dimensional design space is presented. In a basic PSO, an initial randomly generated population swarm propagates toward the global optimum over a series of iterations. The direction of the swarm movement in the design space is based on an individual particle’s best position in its history trail (pBest), and the best particle in the entire swarm (gBest). This information is used to generate a velocity vector indicating a search direction toward a promising location in the design space. The premise of the research presented in this paper is based on the fact that the search direction for each swarm member is dictated by only two candidates—pBest and gBest, which are not efficient to locate the global optimum, particularly in multi-modal optimization problems. In addition, poor move sets specified by pBest in the initial stages of optimization can trap the swarm in a local minimum or cause slow convergence. This paper presents the use of digital pheromones for aiding communication within the swarm to improve the search efficiency and reliability, resulting in improved solution quality, accuracy, and efficiency. With empirical proximity analysis, the pheromone strength in a region of the design space is determined. The swarm then reacts accordingly based on the probability that this region may contain an optimum. The additional information from pheromones causes the particles within the swarm to explore the design space thoroughly and locate the solution more efficiently and accurately than a basic PSO. This paper presents the development of this method and results from several multi-modal test cases.     

A prior-knowledge input LSSVR metamodeling method with tuning based on cellular particle swarm optimization for engineering design

Engineering design is usually a daunting optimization task which often involving time-consuming, even computation-prohibitive process. To reduce the computational expense, metamodels are commonly used to replace the actual expensive simulations or experiments. In this paper, a new and efficient metamodeling method named prior-knowledge input least square support vector regression (PKI-LSSVR) is developed, in which samples from different levels of fidelity are incorporated to gain an accurate approximation with limited times of the high-fidelity (HF) expensive simulations. The low-fidelity (LF) output serves as a prior-knowledge of the real response function, and then is used as the input variables of least square support vector regression (LSSVR). When the corresponding HF response is gained, a function that maps the LF outputs to HF outputs is constructed via LSSVR. The predictive accuracy of LSSVR models is highly dependent on their learning parameters. Therefore, a novel optimization method, cellular particle swarm optimization (CPSO), is exploited to seek the optimal hyper-parameters for PKI-LSSVR in order to improve its generalization capability. To get a better optimization performance, a new neighborhood function is developed for CPSO where the global and local search is efficiently balanced by adaptively varied neighbor radius. Several numerical experiments and one engineering case verify the efficiency of the proposed PKI-LSSVR method. Sample quality merits including sample sizes and noise, and metamodel performance evaluation measures incorporating accuracy, robustness, and efficiency are considered.     

基于SQP 局部搜索的混沌粒子群优化算法

提出一种基于序贯二次规划(SQP)法的混沌粒子群优化方法(CPSO-SQP).将混沌PSO作为全局搜索器,并用SQP加速局部搜索,使得粒子能够在快速局部寻优的基础上对整个空间进行搜索,既保证了算法的收敛性,又大大增加了获得全局最优的几率.仿真结果表明,算法精度高、成功率大、全局收敛速度快,明显优于现有算法.将所提出的算法用于高密度聚乙烯(HDPE)装置串级反应过程的乙烯单耗优化,根据工业反应机理以及现场操作经验分析可知,所提出的算法是可行的.     

Robust optimization of constrained mechanical system with joint clearance and random parameters using multi-objective particle swarm optimization

Joint clearance and uncertainty are inevitable in mechanical systems due to design tolerance, abrasion, manufacture error, assembly error and imperfections. In this study, kinematic analysis and robust optimization of constrained mechanical systems with joint clearance and random parameters were performed. Joint clearance was modeled by Lankarani-Nikravesh contact force model, and probability space was applied for characterizing uncertain parameters. A kinematic analysis method based on Baumgarte approach and confidence region method was presented to predict kinematic error of the mechanical system. Slider-crank mechanism, an illustrative example was presented to show the influence of clearance and uncertainty on the kinematic accuracy. Then, a novel multi-objective robust optimization methodology was presented for kinematic accuracy robust optimization design of the constrained mechanical system. In this approach, a multi-objective robust optimization model derived from 95% confidence region is constructed to reduce the effects of clearance and parameter uncertainty on 95% confidence region of kinematic error. The robust optimization model is a double-loop process. A multi-objective robust optimization strategy, combing Kriging surrogate model, multi-objective particle swarm optimization, confidence region and Monte Carlo methods, was proposed to search the design variables for minimizing the optimization objectives derived from confidence region while balancing computational accuracy and efficiency of the optimization process. The optimal results of the slider-crank mechanism demonstrated the validity and feasibility of the proposed robust optimization method.     

SSTT: Efficient local search for GSI global routing

In this paper, a novel global routing algorithm is presented for congestion opti-mization based on efficient local search, named SSTT (search space traversing technology). This method manages to traverse the whole search space. A hybrid optimization strategy is adopted,consisting of three optimization sub-strategies: stochastic optimization, deterministic optimiza-tion and local enumeration optimization, to dynamically reconstruct the problem structure. Thus,“transition” can be made from a local minimum point to reach other parts of the search space,traverse the whole search space, and obtain the global (approximate) optimal routing solution.Since any arbitrary initial routing solution can be used as the start point of the search, the initial-ization in SSTT algorithm is greatly simplified. SSTT algorithm has been tested on both MCNC benchmark circuits and industrial circuits, and the experimental results were compared with those of typical existing algorithms. The experimental results show that SSTT algorithm can obtain the global (approximate) optimal routing solution easily and quickly. Moreover, it can meet the needs of practical applications. The SSTT global routing algorithm gives a general-purpose routing solution.     

Design optimization of balloon-expandable coronary stent

To prevent the dogboning effect of stent implantation (i.e. the ends of a stent opening first during expansion), an adaptive optimization method based on the kriging surrogate model is proposed to reduce the absolute value of the dogboning rate. Integrating design of experiment (DOE) methods with the kriging surrogate model can approximate the functional relationship between the dogboning rate and the geometrical design parameters of the stent, replacing the expensive reanalysis of the stent dogboning rate during the optimization process. In this adaptive process, an infilling sampling criterion termed expected improvement (EI) is used to balance local and global search and tends to find the global optimal design. Finite element method is used to analyze stent expansion. As an example, a typical diamond-shaped coronary stent is investigated, where four key geometries are selected to be the design variables. Numerical results demonstrate that the proposed adaptive optimization method can effectively decrease the absolute value of the dogboning rate of stent dilation.