Within the framework of complex system design, it is often necessary to solve mixed variable optimization problems, in which the objective and constraint functions can depend simultaneously on continuous and discrete variables. Additionally, complex system design problems occasionally present a variable-size design space. This results in an optimization problem for which the search space varies dynamically (with respect to both number and type of variables) along the optimization process as a function of the values of specific discrete decision variables. Similarly, the number and type of constraints can vary as well. In this paper, two alternative Bayesian optimization-based approaches are proposed in order to solve this type of optimization problems. The first one consists of a budget allocation strategy allowing to focus the computational budget on the most promising design sub-spaces. The second approach, instead, is based on the definition of a kernel function allowing to compute the covariance between samples characterized by partially different sets of variables. The results obtained on analytical and engineering related test-cases show a faster and more consistent convergence of both proposed methods with respect to the standard approaches.

在复杂系统设计的框架内，通常需要解决混合变量优化问题，其中目标函数和约束函数可能同时依赖于连续变量和离散变量。此外，复杂的系统设计问题有时会提供可变大小的设计空间。这导致了一个优化问题，对于该问题，搜索空间将根据特定离散决策变量的值在优化过程中动态变化（相对于变量的数量和类型）。同样，约束的数量和类型也可以变化。为了解决这类优化问题，本文提出了两种基于贝叶斯优化的方法。第一个包括预算分配策略，该策略允许将计算预算集中在最有前途的设计子空间上。相反，第二种方法基于核函数的定义，该核函数允许计算以部分不同的变量集为特征的样本之间的协方差。在分析和工程相关的测试用例上获得的结果表明，相对于标准方法，两种方法的收敛速度更快，一致性更高。