Due to the increasing demand for high performance and cost reduction within the framework of complex system design, numerical optimization of computationally costly problems is an increasingly popular topic in most engineering fields. In this paper, several variants of the Efficient Global Optimization algorithm for costly constrained problems depending simultaneously on continuous decision variables as well as on quantitative and/or qualitative discrete design parameters are proposed. The adaptation that is considered is based on a redefinition of the Gaussian Process kernel as a product between the standard continuous kernel and a second kernel representing the covariance between the discrete variable values. Several parameterizations of this discrete kernel, with their respective strengths and weaknesses, are discussed in this paper. The novel algorithms are tested on a number of analytical test-cases and an aerospace related design problem, and it is shown that they require fewer function evaluations in order to converge towards the neighborhoods of the problem optima when compared to more commonly used optimization algorithms.

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Within the framework of complex system analyses, such as aircraft and launch vehicles, the presence of computationally intensive models (e.g., finite element models and multidisciplinary analyses) coupled with the dependence on discrete and unordered technological design choices results in challenging modeling problems. In this paper, the use of Gaussian process surrogate modeling of mixed continuous/discrete functions and the associated challenges are extensively discussed. A unifying formalism is proposed in order to facilitate the description and comparison between the existing covariance kernels allowing to adapt Gaussian processes to the presence of discrete unordered variables. Furthermore, the modeling performances of these various kernels are tested and compared on a set of analytical and aerospace engineering design related benchmarks with different characteristics and parameterizations. Eventually, general tendencies and recommendations for such types of modeling problem using Gaussian process are highlighted.