Most real optimization problems are defined over a mixed search space where the variables are both discrete and continuous. In engineering applications, the objective function is typically calculated with a numerically costly black-box simulation. General mixed and costly optimization problems are therefore of a great practical interest, yet their resolution remains in a large part an open scientific question. In this article, costly mixed problems are approached through Gaussian processes where the discrete variables are relaxed into continuous latent variables. The continuous space is more easily harvested by classical Bayesian optimization techniques than a mixed space would. Discrete variables are recovered either subsequently to the continuous optimization, or simultaneously with an additional continuous-discrete compatibility constraint that is handled with augmented Lagrangians. Several possible implementations of such Bayesian mixed optimizers are compared. In particular, the reformulation of the problem with continuous latent variables is put in competition with searches working directly in the mixed space. Among the algorithms involving latent variables and an augmented Lagrangian, a particular attention is devoted to the Lagrange multipliers for which a local and a global estimation techniques are studied. The comparisons are based on the repeated optimization of three analytical functions and a beam design problem.

中文翻译：

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混合变量贝叶斯优化方法的比较

大多数真正的优化问题是在混合搜索空间中定义的，其中变量既是离散的又是连续的。在工程应用中，目标函数通常使用数值昂贵的黑盒模拟来计算。因此，一般的混合和昂贵的优化问题具有很大的实际意义，但它们的解决在很大程度上仍然是一个开放的科学问题。在本文中，通过高斯过程处理代价高昂的混合问题，其中离散变量被放松为连续潜变量。连续空间比混合空间更容易通过经典贝叶斯优化技术获得。在连续优化之后恢复离散变量，或者与一个额外的连续离散兼容性约束同时处理，该约束由增强拉格朗日函数处理。比较了这种贝叶斯混合优化器的几种可能实现。特别是，具有连续潜在变量的问题的重新表述与直接在混合空间中进行的搜索竞争。在涉及潜在变量和增广拉格朗日算子的算法中，特别关注拉格朗日乘数，其中研究了局部和全局估计技术。比较基于三个分析函数的重复优化和一个梁设计问题。具有连续潜在变量的问题的重新表述与直接在混合空间中进行的搜索竞争。在涉及潜在变量和增广拉格朗日算子的算法中，特别关注拉格朗日乘数，其中研究了局部和全局估计技术。比较基于三个分析函数的重复优化和一个梁设计问题。具有连续潜在变量的问题的重新表述与直接在混合空间中进行的搜索竞争。在涉及潜在变量和增广拉格朗日算子的算法中，特别关注拉格朗日乘数，其中研究了局部和全局估计技术。比较基于三个分析函数的重复优化和一个梁设计问题。