An approximate optimization strategy using refined hybrid metamodel

An approximate model called metamodel or surrogate model is a mathematical model that numerically approximates response of a system during an engineering simulation process or test. The introduction of a metamodel makes it possible to express response defined in the design problem as a simple mathematical function of design variables. A metamodel can be built with response surface method (RSM), kriging, neural network, radial basis function, and so on. Each method has its advantages and disadvantages. A combined metamodel called hybrid model, ensemble model, or multiple surrogates has been developed to maximize each metamodel's strength. The hybrid model of this research includes RSM and kriging. Besides, a strategy to refine the hybrid metamodel is implemented by reducing design space. In this process, information related to Hessian is utilized for an unconstrained optimization problem, on the contrary feasibility for a constrained optimization problem. This research presents a new hybrid metamodel-based optimization strategy called refined hybrid metamodel. Five mathematical test problems, two-bar design, spring design, and propeller shaft design problems are solved with the suggested method, verifying its usefulness. Most of the optimal results with the proposed method are closer to exact solutions with smaller function evaluations than existing methods.     

Unified uncertainty representation and quantification based on insufficient input data

Uncertainty quantification accuracy of system performance has an important influence on the results of reliability-based design optimization (RBDO). A new uncertain identification and quantification methodology is proposed considering the strong statistical variables, sparse variables, and interval variables simultaneously. Maximum likelihood function and Akaike information criterion (AIC) methods are used to identify the best-fitted distribution types and distribution parameters of sparse variables. The interval variables are represented with evidence theory. Finally, a unified uncertainty quantification framework considering the three types of uncertain design variables is put forward, and then the failure probability of system performance is quantified with belief and plausibility measures. The Kriging metamodel and random sampling method are used to reduce the computational complexity. Three examples are illustrated to verify the effectiveness of the proposed methodology.     

An importance learning method for non-probabilistic reliability analysis and optimization

With the time-consuming computations incurred by nested double-loop strategy and multiple performance functions, the enhancement of computational efficiency for the non-probabilistic reliability estimation and optimization is a challenging problem in the assessment of structural safety. In this study, a novel importance learning method (ILM) is proposed on the basis of active learning technique using Kriging metamodel, which builds the Kriging model accurately and efficiently by considering the influence of the most concerned point. To further accelerate the convergence rate of non-probabilistic reliability analysis, a new stopping criterion is constructed to ensure accuracy of the Kriging model. For solving the non-probabilistic reliability-based design optimization (NRBDO) problems with multiple non-probabilistic constraints, a new active learning function is further developed based upon the ILM for dealing with this problem efficiently. The proposed ILM is verified by two non-probabilistic reliability estimation examples and three NRBDO examples. Comparing with the existing active learning methods, the optimal results calculated by the proposed ILM show high performance in terms of efficiency and accuracy.

     

基于Kriging模型和NSGA-Ⅱ的航空发动机管路卡箍布局优化

针对传统卡箍布局规划存在效率低且难以解决多目标优化问题等难点，提出一种基于Kriging模型和NSGA-Ⅱ算法的航空发动机管路卡箍多目标布局优化方法。提出并建立反映卡箍位置与管路振动性能关系的Kriging代理模型，同时结合拉丁超立方实验设计方法以及粒子群优化技术提高建模精度。以管路一阶固有频率和二阶固有频率为优化目标，应用NSGA-Ⅱ对管路卡箍位置进行布局规划以避免共振。在优化计算过程中，应用所建近似模型代替CAE分析程序对适应值函数进行评价，大大减小了计算量。发动机管路卡箍布局算例表明:所提方法可一次性获得多个满足要求的非支配解方案，不仅提升了管路系统的可靠性，而且计算效率亦得到了很大提高。最后仿真结果证明了所提方法的有效性。     

Extended Gaussian Kriging for computer experiments in engineering design

Metamodeling or surrogate modeling is becoming increasingly popular for product design optimization in manufacture industries. In this paper, an extended Gaussian Kriging method is proposed to improve the prediction performance of widely used ordinary Kriging in engineering design. Unlike the forgoing approaches, the proposed method places a variance-varying Gaussian prior on the unknown regression coefficients in the mean model of Kriging and makes prediction at untried design points based on the principle of Bayesian maximum a posterior. The achieved regression mean model is adaptive, therefore capable of capturing more effectively the overall trend of computer responses and leading to a more accurate metamodel. Particularly, the regression coefficients in the mean model are estimated by a fast numerical algorithm, making extended Gaussian Kriging implemented roughly as efficient as ordinary Kriging. Experiment results on several examples are presented, showing remarkable improvement in prediction using extended Gaussian Kriging over ordinary Kriging and several other metamodeling methods.     

An adaptive sampling strategy for Kriging metamodel based on Delaunay triangulation and TOPSIS

Metamodels have been widely used in engineering design and optimization. Sampling method plays an important role in the constructing of metamodels. This paper proposes an adaptive sampling strategy for Kriging metamodel based on Delaunay triangulation and TOPSIS (KMDT). In the proposed KMDT, Delaunay triangulation is employed to partition the design space according to current sample points. The area of each partitioned triangle is used to indicate the degree of dispersion of sample points, and the prediction error of Kriging metamodel at each triangle’s centroid is used to represent the local error of each triangle region. By calculating the weight of the area and prediction error for each triangle region using the entropy method and TOPSIS, the degree of dispersion of sample points and local errors of metamodel are taken into consideration to make a trade-off between global exploration and local exploitation during the sequential sampling process. As a demonstration, the proposed approach is compared to other three sampling methods using several numerical cases and the modeling of the aerodynamic coefficient for a three-dimensional aircraft. The result reveals that the proposed approach provides more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.     

Evolutionary Algorithms for Minimax Problems in Robust Design

Many robust design problems can be described by minimax optimization problems. Classical techniques for solving these problems have typically been limited to a discrete form of the problem. More recently, evolutionary algorithms, particularly coevolutionary optimization techniques, have been applied to minimax problems. A new method of solving minimax optimization problems using evolutionary algorithms is proposed. The performance of this algorithm is shown to compare favorably with the existing methods on test problems. The performance of the algorithm is demonstrated on a robust pole placement problem and a ship engineering plant design problem.     

Metamodel-based collaborative optimization framework

This paper focuses on the metamodel-based collaborative optimization (CO). The objective is to improve the computational efficiency of CO in order to handle multidisciplinary design optimization problems utilising high fidelity models. To address these issues, two levels of metamodel building techniques are proposed: metamodels in the disciplinary optimization are based on multi-fidelity modelling (the interaction of low and high fidelity models) and for the system level optimization a combination of a global metamodel based on the moving least squares method and trust region strategy is introduced. The proposed method is demonstrated on a continuous fiber-reinforced composite beam test problem. Results show that methods introduced in this paper provide an effective way of improving computational efficiency of CO based on high fidelity simulation models.     

Multi-objective optimization of double suction centrifugal pump using Kriging metamodels

This paper proposes a new multi-objective optimization method for a family of double suction centrifugal pumps with various blade shapes, using a Simulation-Kriging model-Experiment (SKE) approach. The Kriging metamodel is established to approximate the characteristic performance functions of a pump, namely, the efficiency and required net positive suction head (NPSHr). Hence, the two objectives are to maximize the efficiency and simultaneously to minimize NPSHr. The Non-dominated Sorting Genetic Algorithm II (NSGA II) and Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) have been applied to the multi-objective optimization problem, respectively. The Pareto solution set is obtained by a more effective and efficient manner of the two multi-objective optimization algorithms. A tradeoff optimal design point is selected from the Pareto solution set by means of a robust design based on Monte Carlo simulations, and the optimal solution is further compared with the value of the physical prototype test. The results show that the solution of the proposed multi-objective optimization method is in line with the experiment test.     

Multi-objective efficient global optimization of expensive simulation-based problem in presence of simulation failures

The multi-objective efficient global optimization (MOEGO), an extension of the single-objective efficient global optimization algorithm with the intention to handle multiple objectives, is one of the most frequently studied surrogate model-based optimization algorithms. However, the evaluation of the infill point obtained in each MOEGO update iteration using simulation tool may fail. Such evaluation failures are critical to the sequential MOEGO method as it leads to a premature halt of the optimization process due to the impossibility of updating the Kriging models approximating objectives. In this paper, a novel strategy to prevent the premature halt of the sequential MOEGO method is proposed. The key point is to introduce an additional Kriging model to predict the success possibility of the simulation at an unvisited point. Multi-objective expected improvement-based criteria incorporating the success possibility of the simulation are proposed. Experiments are performed on a set of six analytic problems, five low-fidelity airfoil shape optimization problems, and a high-fidelity axial flow compressor tandem cascade optimization problem. Results suggest that the proposed MOEGO-Kriging method is the only method that consistently performs well on analytic and practical problems. The methods using the least-square support vector machine (LSSVM) or weighted LSSVM as the predictor of success possibility perform competitively or worse compared with MOEGO-Kriging. The penalty-based method, assigning high objective values to the failed evaluations in minimization problem, yields the worst performance.

     

结合Kriging和物理规划的多目标代理优化算法

遗传算法处理高耗时且具有黑箱性的工程优化问题效率不足。为了提高工程优化效率，结合Kriging代理优化和物理规划，提出了基于Kriging和物理规划的多目标代理优化算法。在处理多目标问题时，使用物理规划将多目标问题转换成单目标问题，再使用Kriging代理优化对单目标问题进行求解。通过两个多目标数值算例和一个工程实例对提出的算法进行验证。结果表明，提出的算法能够求出符合偏好设置的Pareto最优解，且算法的效率更高。     

Sequential improvement for robust optimization using an uncertainty measure for radial basis functions

The performance of the sequential metamodel based optimization procedure depends strongly on the chosen building blocks for the algorithm, such as the used metamodeling method and sequential improvement criterion. In this study, the effect of these choices on the efficiency of the robust optimization procedure is investigated. A novel sequential improvement criterion for robust optimization is proposed, as well as an improved implementation of radial basis function interpolation suitable for sequential optimization. The leave-one-out cross-validation measure is used to estimate the uncertainty of the radial basis function metamodel. The metamodeling methods and sequential improvement criteria are compared, based on a test with Gaussian random fields as well as on the optimization of a strip bending process with five design variables and two noise variables. For this process, better results are obtained in the runs with the novel sequential improvement criterion as well as with the novel radial basis function implementation, compared to the runs with conventional sequential improvement criteria and kriging interpolation.     

基于灰支持向量回归机预测适应值的交互式集合进化计算

个体适应值的高精度预测和高效的进化策略对于提高进化优化算法性能至关重要.针对现有大规模种群交互式进化计算个体适应值估计误差较大以及传统进化策略搜索效率较低的问题,提出一种基于灰支持向量回归机的个体适应值预测方法和大规模种群集合进化策略.建立基于灰支持向量回归机的适应值预测模型,给出4种集合进化个体比较测度,同时提出新的集合进化个体自适应交叉和变异概率.基于上述策略,采用NSGA-II范式设计一种交互式集合进化优化算法.将该算法应用于RGB颜色One-max优化问题,以表明所提出个体适应值预测方法和集合进化策略的有效性.     

基于PRS元模型的粒子群优化算法及其应用*

针对传统粒子群算法优化黑箱模型过程中存在巨大计算开销的问题,提出一种基于PRS元模型的改进粒子群优化算法—PPSO算法。在该算法迭代过程中,构建PRS元模型,利用其最优值点辅助粒子种群的更新,此外仅选择元模型预估集中优值集的粒子进行目标函数的计算仿真。将PPSO算法与基本粒子群算法、混沌粒子群算法进行数值测试对比,并应用于模糊控制器的优化设计,仿真结果表明该算法可减少真实估值次数,提高优化搜索能力。     

Heuristic particle filter: applying abstraction techniques to the design of visual tracking algorithms

Abstract: Many real‐world visual tracking applications have a high dimensionality, i.e. the system state is defined by a large number of variables. This kind of problem can be modelled as a dynamic optimization problem, which involves dynamic variables whose values change in time. Most applied research on optimization methods have focused on static optimization problems but these static methods often lack explicit adaptive methodologies. Heuristics are specific methods for solving problems in the absence of an algorithm for formal proof. Metaheuristics are approximate optimization methods which have been applied to more general problems with significant success. However, particle filters are Monte Carlo algorithms which solve the sequential estimation problem by approximating the theoretical distributions in the state space by simulated random measures called particles. However, particle filters lack efficient search strategies. In this paper, we propose a general framework to hybridize heuristics/metaheuristics with particle filters properly. The aim of this framework is to devise effective hybrid visual tracking algorithms naturally, guided by the use of abstraction techniques. Resulting algorithms exploit the benefits of both complementary approaches. As a particular example, a memetic algorithm particle filter is derived from the proposed hybridization framework. Finally, we show the performance of the memetic algorithm particle filter when it is applied to a multiple object tracking problem.     

An optimization algorithm employing multiple metamodels and optimizers

Modern engineering design optimization often relies on computer simulations to evaluate candidate designs, a setup which results in expensive black-box optimization problems. Such problems introduce unique challenges, which has motivated the application of metamodel-assisted computational intelligence algorithms to solve them. Such algorithms combine a computational intelligence optimizer which employs a population of candidate solutions, with a metamodel which is a computationally cheaper approximation of the expensive computer simulation. However, although a variety of metamodels and optimizers have been proposed, the optimal types to employ are problem dependant. Therefore, a priori prescribing the type of metamodel and optimizer to be used may degrade its effectiveness. Leveraging on this issue, this study proposes a new computational intelligence algorithm which autonomously adapts the type of the metamodel and optimizer during the search by selecting the most suitable types out of a family of candidates at each stage. Performance analysis using a set of test functions demonstrates the effectiveness of the proposed algorithm, and highlights the merit of the proposed adaptation approach.     

A framework for the application of metaheuristics to?tasks-to-processors assignation problems

The optimization of the execution time of a parallel algorithm can be achieved through the use of an analytical cost model function representing the running time. Typically the cost function includes a set of parameters that model the behavior of the system and the algorithm. In order to reach an optimal execution, some of these parameters must be fitted according to the input problem and to the target architecture. An optimization problem can be stated where the modeled execution time for the algorithm is used to estimate the parameters. Due to the large number of variable parameters in the model, analytical minimization techniques are discarded. Exhaustive search techniques can be used to solve the optimization problem, but when the number of parameters or the size of the computational system increases, the method is impracticable due to time restrictions. The use of approximation methods to guide the search is also an alternative. However, the dependence on the algorithm modeled and the bad quality of the solutions as a result of the presence of many local optima values in the objective functions are also drawbacks to these techniques. The problem becomes particularly difficult in complex systems hosting a large number of heterogeneous processors solving non-trivial scientific applications. The use of metaheuristics allows for the development of valid approaches to solve general problems with a large number of parameters. A well-known advantage of metaheuristic methods is the ability to obtain high-quality solutions at low running times while maintaining generality. We propose combining the parameterized analytical cost model function and metaheuristic minimization methods, which contributes to a novel real alternative to minimize the parallel execution time in complex systems. The success of the proposed approach is shown with two different algorithmic schemes on parallel heterogeneous systems. Furthermore, the development of a general framework allows us to easily develop and experiment with different metaheuristics to adjust them to particular problems.     

Sequential kriging optimization using multiple-fidelity evaluations

When cost per evaluation on a system of interest is high, surrogate systems can provide cheaper but lower-fidelity information. In the proposed extension of the sequential kriging optimization method, surrogate systems are exploited to reduce the total evaluation cost. The method utilizes data on all systems to build a kriging metamodel that provides a global prediction of the objective function and a measure of prediction uncertainty. The location and fidelity level of the next evaluation are selected by maximizing an augmented expected improvement function, which is connected with the evaluation costs. The proposed method was applied to test functions from the literature and a metal-forming process design problem via finite element simulations. The method manifests sensible search patterns, robust performance, and appreciable reduction in total evaluation cost as compared to the original method.     

基于IWOA算法求解并联冷机负荷分配问题

中央空调系统并联冷水机组系统能耗非常大,如果操作不当,能耗会大大增加。针对OCL问题提出了一种求解连续非线性优化问题的改进鲸群优化算法,首先,为使后续迭代寻优的搜索空间更精确,运用混沌映射初始化种群,使初始解均匀遍布解空间。其次引入变异指数对收敛因子进行改进,平衡了局部勘探和全局勘探的关系。之后引入正弦和余弦使算法收敛到全局最优解,防止了算法过早收敛,提高了算法的收敛精度。最后,通过两个典型的案例来评估IWOA算法的性能,并将其与应用于OCL问题的其他优化算法进行了比较。结果表明,IWOA算法是解决OCL问题的有效方法。此外,算法性能的比较显示,IWOA算法在收敛速度和电能消耗方面相比于其他应用于OCL问题的优化方法提供了更好的解决方案。     

On improving normal boundary intersection method for generation of Pareto frontier

Gradient-based methods, including Normal Boundary Intersection (NBI), for solving multi-objective optimization problems require solving at least one optimization problem for each solution point. These methods can be computationally expensive with an increase in the number of variables and/or constraints of the optimization problem. This paper provides a modification to the original NBI algorithm so that continuous Pareto frontiers are obtained “in one go,” i.e., by solving only a single optimization problem. Discontinuous Pareto frontiers require solving a significantly fewer number of optimization problems than the original NBI algorithm. In the proposed method, the optimization problem is solved using a quasi-Newton method whose history of iterates is used to obtain points on the Pareto frontier. The proposed and the original NBI methods have been applied to a collection of 16 test problems, including a welded beam design and a heat exchanger design problem. The results show that the proposed approach significantly reduces the number of function calls when compared to the original NBI algorithm.