含有定性因子的Kriging模型的稳健设计优化与算法研究

Metamodel-based robust design optimization is a hot research topic in the field of design and analysis of computer experiments. According to the literature survey,however, most current design optimization methods are adaptive to deterministic simulation experiments which only take into account of the quantitative factors.In this project, we concentrate on those stochastic simulation experiments with both quantitative and qualitative factors, with construction of a reliable and effective metamodel between input factors and the output response as the primary research problem. By jointly exploring the hierarchical sparse Bayesian prior for the linear regression and the sparsity projection constraint with respect to the reproducing kernel Hilbert space for the kernel regression,a novel error-tolerant robust design optimization method is proposed based on the developed stochasticsparse Kriging metamodel. First of all, fast sparse Kriging metamodeling is studied based on the hierarchical sparse Bayesian learning, which can automatically screen important quantitative and qualitative factors. Subsequently, taking into account of the implementation error caused by the uncontrollable factors, a regularized reweighted least squares method is proposed for efficient construction of the stochastic Kriging, based on the sparsity projection constraint of the reproducing kernel Hilbert space. Finally, a fast algorithm for stochastic sparse Kriging-based robust design optimization is developed by exploiting the robust counterpart theory. The research in this project will have agreat value in the metamodel-based design of complex products/processes.

基于元模型的稳健设计优化是计算机试验领域的热门课题。然而，目前大多数方法只适用于含有定量因子的确定性仿真试验。本项目以定性定量因子均存在的随机性仿真试验为研究对象，以建立输入因子和输出响应之间高效可靠的计算机试验元模型为首要科学问题，通过联合挖掘线性回归的多层稀疏贝叶斯先验与核回归的再生核Hilbert空间稀疏投影约束，提出基于随机稀疏Kriging元建模的误差可容忍稳健设计优化新方法。首先，基于多层稀疏贝叶斯学习，研究定性定量因子自动筛选的稀疏Kriging快速元建模方法；其次，针对不可控设计因子引起的随机性实现误差，进一步研究基于再生核Hilbert空间稀疏投影约束的加权最小二乘随机Kriging元模型；最后，利用鲁棒对等理论，研究基于随机稀疏Kriging的仿真与元建模误差可容忍稳健设计优化快速算法。本项目在基于元模型的复杂产品/过程设计中将具有十分重要的应用价值。

基于元模型的稳健设计优化是计算机试验领域的热门课题。然而，目前大多数方法只适用于含有定量因子的确定性仿真试验。项目组围绕含有定性因子的Kriging模型的稳健设计优化与算法研究这一课题，按照稀疏约束的因子可筛选Kriging元建模、误差可容忍的序贯分支因子筛选、误差可容忍的随机Kriging元建模及其仿真优化、含有定性因子的混合多响应Kriging元建模及稳健设计优化四个方面展开研究。在稀疏约束的因子可筛选Kriging元建模方面，项目提出广义鲁棒损失惩罚的Ordinary Kriging元建模方法、以及基于SCAD罚函数和稀疏贝叶斯学习的Universal Kriging元建模方法。在误差可容忍的序贯分支因子筛选方面，项目先后提出基于位置和散度稳健统计量的误差可容忍序贯分支因子筛选方法、以及基于参数化自助抽样检验策略的偏正态分布响应序贯分支因子筛选方法。在误差可容忍的的随机Kriging元建模及其仿真优化方面，项目提出基于位置和散度稳健统计量的随机Kriging元建模方法及其仿真优化实例验证。在含有定性因子的混合多响应Kriging元建模及稳健设计优化方面，项目结合模糊集理论，进一步深入探讨了含有定性定量输入因子和定性定量输出响应的元建模及稳健参数设计问题，提出一种针对混合多响应问题的隐变量Kriging元建模及稳健设计优化方法。项目研究成果对于日益复杂的产品/过程质量设计以及工程设计优化等诸多实际应用和应用基础问题提供了重要的技术支撑，具有重要的经济价值和社会价值。