  A mathematical framework for modelling constrained mixed-variable
optimization problems is presented in a blackbox optimization context. The
framework introduces a new notation and allows solution strategies. The
notation framework allows meta and categorical variables to be explicitly and
efficiently modelled, which facilitates the solution of such problems. The new
term meta variables is used to describe variables that influence which
variables are acting or nonacting: meta variables may affect the number of
variables and constraints. The flexibility of the solution strategies supports
the main blackbox mixed-variable optimization approaches: direct search methods
and surrogate-based methods (Bayesian optimization). The notation system and
solution strategies are illustrated through an example of a hyperparameter
optimization problem from the machine learning community.
