基于Kriging模型的全局近似与仿真优化方法

发布时间：2018-03-13 13:42

本文选题：Kriging模型　切入点：增量构造方法　出处：《华中科技大学》2015年博士论文　论文类型：学位论文

【摘要】：在科学技术不断发展的今天,为了应对越来越复杂的工程设计优化问题,出现了各种不同的设计策略和研究方法。基于响应面模型的全局近似与仿真优化方法是目前工程设计领域的焦点之一。这类方法主要利用计算机试验设计、响应面构造以及响应面给出的有效信息来实现模型的全局近似和最优解的获珥取,特别适合求解需要“昂贵”估值的“黑箱”函数或计算机仿真模型的近似优化问题。作为一种高精度插值响应面模型,Kriging模型是目前应用较为广‘泛的响应面模型之一,它能够灵活地代替多峰或非线性函数进行“最优参数”估计和近似模型的精度评价。在优化过程中,来自于Kriging模型的函数估值和估计方差等信息能够有效地指导优化搜索朝着全局的方向进行。因此,基于Kriging模型的全局近似和优化方法已被融入到方案设计、结构优化、大数据统计分析以及多学科设计优化过程中,并广泛应用于航空航天、机械工程、车辆工程、地质工程等诸多领域。鉴于此,进一步研究基于Kriging模型的全局近似和优化方法具有一定的理论意义和应用价值。针对确定性的“黑箱”函数或仿真模型,本文基于Kriging模型研究全局近似和优化方法,包括：Kriging模型的增量构造以及全局近似方法：基于正则对偶变换和信任域策略的高效无约束优化方法：并行多采样点的无约束全局优化方法：存在不可行采样点的约束全局优化方法。这些方法拓展和完善了基于响应面的全局近似和优化体系,为具有昂贵估值的优化问题提供了有效的解决方法。本文的研究成果主要体现在如下几个方面： (1)深入分析了标准Kriging模型的结构和参数,提出了Kriging模型的增量构造方法。该方法能够在损失很少精度的前提下,大幅度地提高建模效率。为基于序列采样的全局近似方法提供了理论依据。 (2)依据增量Kriging方法,提出了一种以提高建模效率为目的、在序列优化过程中实现模型全局近似的方法。在确保Kriging模型的稳定性和有效性的前提下对一次增加一个采样点的序列增量构造进行了研究,利用最大化估计方差的方法来寻优下一个采样点,以六西格玛更新准则为判断标准,决定使用计算机试验设计与分析(DACE-Design and Analysis of Computer Experiments)建模或增量Kriging建模。 (3)结合正则对偶/三对偶理论和基于响应面的信任域策略,提出了一种基于Kriging模型的全局优化方法。其中,正则对偶变换能将非凸的Krging模型优化问题转化为凸优化问题,而三对偶原理证明了对偶变换后问题中的所有极值点与原问题所有极值点之间的映射关系,并确保在对偶变量大于0的条件下能够直接获取问题的全局最优点。该方法结合Kriging模型和信任域策略的特点,根据迭代过程中的最优解信息自动调用区间缩减方法来搜索更优的迭代点,有效地改善了复杂无约束问题的收敛效率和精度。 (4)研究了一种基于Kriging模型的多采样点并行序列全局优化算法。该算法利用中点距离最小舍弃方法处理样本中的所有中点,根据目标和方差的近似估计来获取多个新采样点;基于广义EGO (Efficient Global Optimization)方法,利用改进的广义期望改善(GEI-Generalized Expected Improvement)作为填充采样准则(ISC-Infill Sampling Criterion)对新采样点进行并行优化。该算法有效地减少了函数的估值次数,较好地平衡局部搜索与全局搜索,大大提高了优化效率。 (5)提出了一种基于Kriging模型的约束全局优化方法。在初始试验设计不存在可行采样点且目标函数和约束函数都是“黑箱”函数的情况下,该算法通过最大化“满足所有约束的概率”来获取可行采样点；利用目标估计的下限和均方根误差估计的上限建立填充采样准则,通过最小化填充采样准则,确保能够以尽可能少的函数估值次数得到一个可行的全局近似最优解。此外,针对寻优过程中连续出现多个不可行采样点的情况,利用近似约束校正方法将处于可行边界的采样点拉回到实际的可行域内。 (6)基于多学科优化平台FlowComputer,利用开源优化工具包DAKOTA,开发了基于响应面组件的仿真优化模块,实现了经典的EGO及本文中所提出的近似优化算法。最后,通过一个重型汽车燃油经济性的仿真优化问题来验证本文方法的有效性。
[Abstract]:In the continuous development of science and technology today, in order to cope with the increasingly complex optimization problems in engineering design, presents the design strategy and research various methods. The approximate optimization method and simulation is one of the focuses in the field of global engineering design based on response surface model. This kind of method by means of computer experimental design, response surface structure and effective the information given by response surface model to achieve global optimal solution by Joel and approximation, especially suitable for solving the needs of "expensive" valuation of the "black box" function or a computer simulation model of the approximate optimization problem. As a kind of high precision interpolation response surface model, the Kriging model is more widely the response surface model one of the applications, it can replace the multi peak function or nonlinear optimal parameter estimation and approximation accuracy evaluation. In the optimization process. The function of valuation and estimation of variance information from Kriging model can effectively guide the search toward the global direction. Therefore, optimizing the structure of global Kriging model approximation and optimization methods have been integrated into the design, based on the statistical analysis of large data and multidisciplinary design optimization process, and is widely used in aviation aerospace, mechanical engineering, vehicle engineering, geological engineering and other fields. In view of this, further study on global Kriging model approximation and optimization method has certain theoretical significance and application value based.
According to the uncertainty of the "black box" function or simulation model, this paper studies the global Kriging model and approximate optimization method based on Kriging model including incremental construction and global approximation method: canonical dual transformation and trust region strategy, unconstrained optimization method based on unconstrained global optimization method for parallel multi sampling points: there is no feasible sampling the constrained global optimization method. These methods to expand and improve the response surface approximation and global optimization based system provides an effective solution to the optimization problem with expensive valuation. The research results of this paper are mainly embodied in the following aspects:
(1) in-depth analysis of the structure and parameters of the standard Kriging model. The incremental construction method of Kriging model is proposed. This method can greatly improve the modeling efficiency under the premise of little loss accuracy, which provides a theoretical basis for the global approximation method based on sequential sampling.
(2) based on the incremental Kriging method, put forward a kind of in order to improve the efficiency of modeling methods, implementation model of global approximation in the sequence optimization process. In the premise of ensuring the stability of the Kriging model and the effectiveness of the time for a sampling point sequence of incremental construction were studied, using the maximum estimate the variance method to optimize the next sampling point, with six sigma criteria for judgment standard, decided to use the design and analysis of computer experiments (DACE-Design and Analysis of Computer Experiments) modeling or incremental Kriging modeling.
(3 / three) combined with the canonical dual duality theory and the response surface based trust region strategy, this paper presents a global optimization method based on Kriging model. The canonical dual transformation can be non convex Krging model optimization problem into a convex optimization problem, and proves that the three principle of duality mapping between all extreme points of dual transformation after the problems in the original problem and all extreme points, and to ensure that global direct access to the problem of more than 0 conditions in the dual variables of the most advantages. This method combines the characteristics of Kriging model and trust region strategy, according to the optimal solution of the iterative process of information automatic call interval reduction method to search the iteration point better and effectively improve the complexity of the unconstrained problem of convergence efficiency and accuracy.
(4) on a Kriging model of multi sample parallel global optimization algorithm based on sequence. The algorithm uses minimum distance method to handle all abandon the midpoint in the sample according to the approximate midpoint, estimation of target and variance to obtain multiple sampling points; based on the generalized EGO (Efficient Global Optimization) method, using generalized expectation improvement the improved (GEI-Generalized Expected Improvement) as the filling sampling criteria (ISC-Infill Sampling Criterion) of the new sampling points for parallel optimization. This algorithm can reduce the number of times the valuation function, better balance between global search and local search, improves the efficiency of optimization.
(5) proposed a global optimization method based on Kriging model. In the initial design there is no feasible sampling points and the objective function and constraint function are the "black box" function, the algorithm to obtain the feasible sampling points by maximizing the "probability" of all constraints by using the lower bound estimation of the target; and the root mean square error estimation of the upper bound of a packed sampling criterion, by minimizing the sampling criteria to ensure the filling, function evaluation times as little as possible to obtain a feasible global optimal solution. In addition, for a number of feasible sampling points of consecutive optimization process, using the correction method of approximate constraint will be feasible in the the boundary of the feasible domain sampling point back to the actual.
(6) multidisciplinary design optimization platform based on FlowComputer, using open-source DAKOTA toolkit to develop the simulation optimization, response surface optimization module based on component, realizes the approximate optimization algorithm proposed by EGO and the classic in. Finally, through a heavy vehicle fuel economy simulation optimization problem to verify the validity of this method.

【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：U462.34;TB21

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