Metamodels for mixed variables based on moving least squares

Surrogate-based optimization has become a major field in engineering design, due to its capacity to handle complex systems involving expensive simulations. However, the majority of general-purpose surrogates (also called metamodels) are restricted to continuous variables, although versatile problems involve additional types of variables (discrete, integer, and even categorical to model technological options). Therefore, the main contribution of this paper consists in the development of metamodels specifically dedicated to handle mixed variables, in particular continuous and unordered categorical variables, and their comparison with state-of-the-art approaches. This task is performed in three steps: (i) considering an appropriate parametrization (integer mapping, regular simplex, dummy, effect codings) for the mixed variable design vector; (ii) defining metrics to compare pairs of design vectors; (iii) carrying out an ordinary or moving least square regression scheme based on the parametrization and metric previously defined. The proposed metamodels have been tested on six analytical benchmark test cases, and applied to the structural finite element analysis model of a rigid frame characterized by continuous and categorical variables. In particular, it is demonstrated that using a standard regular simplex representation for the nominal categorical variables usually outperforms a direct conversion of the nominal parameters to integer values, while offering an efficient and systematic way to encompass all types of variables in a common framework. It is also shown that the choice of a given variable representation has a higher impact on the results than the selected scheme (ordinary or moving least squares), or than the metric used for calculating distances between samples.