多参数非线性优化方法关键技术研究及应用
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摘要
现代CAE技术的发展使得各类优化方法在汽车车身设计领域得到广泛应用。车身设计优化问题的目标函数通常为隐式的黑箱函数，且其CAE仿真过程非常耗时，传统的梯度算法和启发式算法在计算效率上已经远远不能满足工程实践的应用需求。目前，近似模型技术是求解此类优化问题的有效手段。该技术的最大优势在于能够通过有限次数的正问题计算给出目标响应在整个设计空间中的经验性估计。如果能够建立问题的高精度近似模型，则优化的效率将得到大幅度提升。然而，随着设计变量的增多，设计空间的规模迅速扩大，构建精度和效率成为当前近似模型技术的主要瓶颈，因此，建立一种行之有效的多参数近似模型方法很有必要。此外，很多优化问题涉及到离散设计变量，其高维性导致近似模型技术的优势难以得到发挥，为此，需要立足于新的角度，开发适用于离散变量的多参数优化方法。综上所述，本文将围绕多参数优化问题展开研究，具体的研究内容如下：
(1)随着优化问题设计参数的增多，构建近似模型所需样本点的数量将呈近乎指数级增长，对于非线性问题，在可接受的计算成本(训练样本点数)下，难以获取黑箱问题的高精度近似模型，且传统的元模型技术无法有效识别设计参数之间的耦合性以及它们对目标响应的敏感程度，从而难以体现问题的本质。为此，本文提出基于多参数解耦的自适应非线性近似模型方法(Kriging-HDMR)。该方法采用Cut-HDMR多参数解耦模型将高维问题分解成不同阶次耦合项的组合，通过逐一识别两两参数之间的耦合性确定Cut-HDMR中一阶耦合项的组成，同时忽略那些对目标响应影响较弱的高阶耦合项，从而可将样本点数随问题维数增长的数量级由原来的指数级降为多项式级。另一方面，Kriging-HDMR方法利用Cut-HDMR模型将建模的对象由最初的高维问题转化为若干较低维问题，成功地降解了建模的复杂度，因此能够大幅度地提高近似模型的精度。研究结果表明，基于同一组训练样本点，Kriging-HDMR模型的精度显著优于Kriging模型。
(2)目前主流的连续设计变量优化方法是基于近似模型技术和智能布点策略进行的。此类算法的性能在很大程度上依赖近似模型的精度，然而对于多参数问题，传统近似模型的全局精度差，导致优化的效率低且容易陷入局部最优。为此，本文提出了基于连续型设计变量的多参数非线性优化算法(P-HGS)。该算法采用多维投影技术将Kriging-HDMR近似模型与MPS智能布点策略相结合，在每个迭代步，根据Kriging-HDMR模型的结构将新样本点向过中心点的切线和切面内投影，利用生成的投影点更新Kriging-HDMR近似模型中的各分项，并通过添加修正项来确保整个近似模型在新样本点上的插值性，从而可以利用Kriging-HDMR模型的精度优势改善优化的效率和精度。采用不同规模的测试函数对P-HGS方法的性能进行测试，并与MPS方法作比较，测试结果表明，P-HGS方法显著提高了MPS的全局搜索能力、效率和稳健性。
(3)工程优化问题会常常会涉及离散变量，如材料变量。而离散变量的存在会额外地、甚至成倍地增加解向量的维数，这会导致近似模型的精度大幅度下降，难以用于求解优化问题，尤其是多参数优化问题。因此，本文提出了基于离散型设计变量的多参数非线性优化算法(KCHS-UPDA)。算法根据Pareo前沿定义若干特征解，在每个迭代步中，找到每个特征解的位置，进而确定相应的采样集合，从而实现了空间缩减，提高了优化效率。为了进一步提高算法的收敛速度，在每个采样集合内部，采用K-mean聚类方法建立采样概率模型，并随机地生成样本点。该算法不需要建立近似模型，从而有效地避免了近似模型对于离散变量多参数问题的精度缺陷。另一方面，在同一个迭代步中，几个特征解的位置通常是分散的，而采样集合中的样本点在每个特征解周围都有分布，因此，算法的全局搜索性能可以在一定程度上得到提升。通过若干多目标优化问题对算法的可行性进行测试，测试结果表明该算法可以通过较少量的正问题计算找到精度较高的Pareto前沿。
Development of modern CAE technique has boosted wide applications of variousoptimizations to automobile body design. Objective functions of such optimizationproblems are black-box and usually require complicate CAE analysis which is verytime-consuming. For such problems, traditional gradient-based and heuristic methodscan hardly satisfy the application requirement of practical engineering. So far,metamodel technique is an effective approach to solve black-box expensiveoptimizations. The most remarkable advantage of metamodel-based optimization isthat experimental prediction of the objective functions can be drawn by a few functionevaluations, so that the efficiency of optimization can be improved. However, withincreasing number of design variables, the scale of design space will expand rapidly.Accuracy and efficiency become main bottlenecks of the metamodel technique. Thus,constructing an effective metamodel method for multi-variable problems is verynecessary. Moreover, many engineering optimizations involve discrete variables. Thehigh dimensionality of discrete variable limits the advantage of metamodel technique,so a study of discrete variables based multi-variable optimization from anotherperspective is needed. Summarily, this paper conducts research on the basis ofmulti-variable optimization. Details of the research contents are described as thefollows:
(1) With increasing number of design variables, the number of sample pointsused to construct the metamodel grows almost exponentially. For nonlinear problems,metamodels with high accuracy can be hardly obtained under acceptablecomputational cost. Moreover, conventional metamodel techniques lacks capability toeffectively identify coupling relationships among the design variables and theircontributions to the objective response, so essence of the problem cannot be wellreflected. Thus, a multi-variable decoupling based adaptive nonlinear metamodeltechnique (Kriging-HDMR) is proposed in this paper. The method uses multi-variabledecoupling model (Cut-HDMR) to decompose a high-dimensional problem into aseries of coupling terms with different hierarchies. The components of Cut-HDMR aredetermined by identifying the relationship of each pair of variables and neglectinghigh-order coupling terms which have weak contributions to the objective response,so that the computational cost can be reduced from exponential growth to polynomial level. On the other hand, the proposed method uses Cut-HDMR model to transformmetamodeling object from a high-dimensional problem into a few lower-dimensionalsub-problems, successfully reducing the metamodeling complexity, so accuracy of themetamodel can be remarkably improved. Research results indicate that based on asame set of sample points, the accuracy of Kriging-HDMR model is much better thanthat of Kriging model.
(2) So far, popular methods for continuous variable optimizations are based onmetamodel techniques and intelligent sampling strategies. The performance of thesemethods significantly depends on the accuracy of metamodels. However, formulti-variable problems, traditional metamodels usually have low accuracies, leadingto low efficiency and local optimums. Therefore, a projection-based heuristic globalsearch algorithm (P-HGS) is proposed for multi-variable optimization. The algorithmadopts multi-dimensional projection technique to integrate Kriging-HDMR with MPSseamlessly. In each iteration, new samples are projected to the lines and hyper-planescrossing through the cut center. The projected points are used to update the Krigingmodels of the component terms of Kriging-HDMR. In order to ensure theinterpolation of Kriging-HDMR at the new samples, a error modification term is alsoadded. P-HGS takes advantage of Kriging-HDMR to improve the efficiency andaccuracy of the optimizaition. Tested by several nonlinear functions, P-HGSobviously excels MPS in global search, efficiency and robustness.
(3) Engineering optimizations often concern with discrete variables, such as thematerial variable. Existence of discrete variables extraly, even multiply increase thedimensionality of solution vector so that the accuracy of metamodel can hardly beguaranteed, particularly for the multi-variable optimization. Thus, a K-mean Clusterbased Heuristic Sampling with Utopia-Pareto Directing Adaptive Strategy(KCHS-UPDA) is proposed for discrete design variables based multi-objectiveoptimization. The method defines several feature solutions according to the Paretofrontier. In each iteration, the feature solutions are located and correspondingsampling sets are generated, so that space reduction can be easily realized. In order toenhance the convergent rate, in each sampling set, k-mean cluster is employed toconstruct a probabilistic model, according to which new samples are drawnstochastically. The method successfully replaces metamodels so that accuracy defectof metamodel for the discrete variable can be well avoided. Moreover, in a iteration,the locations of the feature solutions are usually disperse and the sampling setsdistribute around each feature solution, so global search ability of the algorithm can be enhanced to some extent. A few benchmark problems are used to test theperformance of the proposed algorithm. Test results indicate that KCHS-UPDA cangenerally converge to the Pareto frontier with a small quantity of number of functionevaluations.
(1)随着优化问题设计参数的增多，构建近似模型所需样本点的数量将呈近乎指数级增长，对于非线性问题，在可接受的计算成本(训练样本点数)下，难以获取黑箱问题的高精度近似模型，且传统的元模型技术无法有效识别设计参数之间的耦合性以及它们对目标响应的敏感程度，从而难以体现问题的本质。为此，本文提出基于多参数解耦的自适应非线性近似模型方法(Kriging-HDMR)。该方法采用Cut-HDMR多参数解耦模型将高维问题分解成不同阶次耦合项的组合，通过逐一识别两两参数之间的耦合性确定Cut-HDMR中一阶耦合项的组成，同时忽略那些对目标响应影响较弱的高阶耦合项，从而可将样本点数随问题维数增长的数量级由原来的指数级降为多项式级。另一方面，Kriging-HDMR方法利用Cut-HDMR模型将建模的对象由最初的高维问题转化为若干较低维问题，成功地降解了建模的复杂度，因此能够大幅度地提高近似模型的精度。研究结果表明，基于同一组训练样本点，Kriging-HDMR模型的精度显著优于Kriging模型。
(2)目前主流的连续设计变量优化方法是基于近似模型技术和智能布点策略进行的。此类算法的性能在很大程度上依赖近似模型的精度，然而对于多参数问题，传统近似模型的全局精度差，导致优化的效率低且容易陷入局部最优。为此，本文提出了基于连续型设计变量的多参数非线性优化算法(P-HGS)。该算法采用多维投影技术将Kriging-HDMR近似模型与MPS智能布点策略相结合，在每个迭代步，根据Kriging-HDMR模型的结构将新样本点向过中心点的切线和切面内投影，利用生成的投影点更新Kriging-HDMR近似模型中的各分项，并通过添加修正项来确保整个近似模型在新样本点上的插值性，从而可以利用Kriging-HDMR模型的精度优势改善优化的效率和精度。采用不同规模的测试函数对P-HGS方法的性能进行测试，并与MPS方法作比较，测试结果表明，P-HGS方法显著提高了MPS的全局搜索能力、效率和稳健性。
(3)工程优化问题会常常会涉及离散变量，如材料变量。而离散变量的存在会额外地、甚至成倍地增加解向量的维数，这会导致近似模型的精度大幅度下降，难以用于求解优化问题，尤其是多参数优化问题。因此，本文提出了基于离散型设计变量的多参数非线性优化算法(KCHS-UPDA)。算法根据Pareo前沿定义若干特征解，在每个迭代步中，找到每个特征解的位置，进而确定相应的采样集合，从而实现了空间缩减，提高了优化效率。为了进一步提高算法的收敛速度，在每个采样集合内部，采用K-mean聚类方法建立采样概率模型，并随机地生成样本点。该算法不需要建立近似模型，从而有效地避免了近似模型对于离散变量多参数问题的精度缺陷。另一方面，在同一个迭代步中，几个特征解的位置通常是分散的，而采样集合中的样本点在每个特征解周围都有分布，因此，算法的全局搜索性能可以在一定程度上得到提升。通过若干多目标优化问题对算法的可行性进行测试，测试结果表明该算法可以通过较少量的正问题计算找到精度较高的Pareto前沿。
Development of modern CAE technique has boosted wide applications of variousoptimizations to automobile body design. Objective functions of such optimizationproblems are black-box and usually require complicate CAE analysis which is verytime-consuming. For such problems, traditional gradient-based and heuristic methodscan hardly satisfy the application requirement of practical engineering. So far,metamodel technique is an effective approach to solve black-box expensiveoptimizations. The most remarkable advantage of metamodel-based optimization isthat experimental prediction of the objective functions can be drawn by a few functionevaluations, so that the efficiency of optimization can be improved. However, withincreasing number of design variables, the scale of design space will expand rapidly.Accuracy and efficiency become main bottlenecks of the metamodel technique. Thus,constructing an effective metamodel method for multi-variable problems is verynecessary. Moreover, many engineering optimizations involve discrete variables. Thehigh dimensionality of discrete variable limits the advantage of metamodel technique,so a study of discrete variables based multi-variable optimization from anotherperspective is needed. Summarily, this paper conducts research on the basis ofmulti-variable optimization. Details of the research contents are described as thefollows:
(1) With increasing number of design variables, the number of sample pointsused to construct the metamodel grows almost exponentially. For nonlinear problems,metamodels with high accuracy can be hardly obtained under acceptablecomputational cost. Moreover, conventional metamodel techniques lacks capability toeffectively identify coupling relationships among the design variables and theircontributions to the objective response, so essence of the problem cannot be wellreflected. Thus, a multi-variable decoupling based adaptive nonlinear metamodeltechnique (Kriging-HDMR) is proposed in this paper. The method uses multi-variabledecoupling model (Cut-HDMR) to decompose a high-dimensional problem into aseries of coupling terms with different hierarchies. The components of Cut-HDMR aredetermined by identifying the relationship of each pair of variables and neglectinghigh-order coupling terms which have weak contributions to the objective response,so that the computational cost can be reduced from exponential growth to polynomial level. On the other hand, the proposed method uses Cut-HDMR model to transformmetamodeling object from a high-dimensional problem into a few lower-dimensionalsub-problems, successfully reducing the metamodeling complexity, so accuracy of themetamodel can be remarkably improved. Research results indicate that based on asame set of sample points, the accuracy of Kriging-HDMR model is much better thanthat of Kriging model.
(2) So far, popular methods for continuous variable optimizations are based onmetamodel techniques and intelligent sampling strategies. The performance of thesemethods significantly depends on the accuracy of metamodels. However, formulti-variable problems, traditional metamodels usually have low accuracies, leadingto low efficiency and local optimums. Therefore, a projection-based heuristic globalsearch algorithm (P-HGS) is proposed for multi-variable optimization. The algorithmadopts multi-dimensional projection technique to integrate Kriging-HDMR with MPSseamlessly. In each iteration, new samples are projected to the lines and hyper-planescrossing through the cut center. The projected points are used to update the Krigingmodels of the component terms of Kriging-HDMR. In order to ensure theinterpolation of Kriging-HDMR at the new samples, a error modification term is alsoadded. P-HGS takes advantage of Kriging-HDMR to improve the efficiency andaccuracy of the optimizaition. Tested by several nonlinear functions, P-HGSobviously excels MPS in global search, efficiency and robustness.
(3) Engineering optimizations often concern with discrete variables, such as thematerial variable. Existence of discrete variables extraly, even multiply increase thedimensionality of solution vector so that the accuracy of metamodel can hardly beguaranteed, particularly for the multi-variable optimization. Thus, a K-mean Clusterbased Heuristic Sampling with Utopia-Pareto Directing Adaptive Strategy(KCHS-UPDA) is proposed for discrete design variables based multi-objectiveoptimization. The method defines several feature solutions according to the Paretofrontier. In each iteration, the feature solutions are located and correspondingsampling sets are generated, so that space reduction can be easily realized. In order toenhance the convergent rate, in each sampling set, k-mean cluster is employed toconstruct a probabilistic model, according to which new samples are drawnstochastically. The method successfully replaces metamodels so that accuracy defectof metamodel for the discrete variable can be well avoided. Moreover, in a iteration,the locations of the feature solutions are usually disperse and the sampling setsdistribute around each feature solution, so global search ability of the algorithm can be enhanced to some extent. A few benchmark problems are used to test theperformance of the proposed algorithm. Test results indicate that KCHS-UPDA cangenerally converge to the Pareto frontier with a small quantity of number of functionevaluations.
引文
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