  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 in 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.
