  Optimization problems involving mixed variables, i.e., variables of numerical
and categorical nature, can be challenging to solve, especially in the presence
of complex constraints. Moreover, when the objective function is the result of
a complicated simulation or experiment, it may be expensive to evaluate. This
paper proposes a novel surrogate-based global optimization algorithm to solve
linearly constrained mixed-variable problems up to medium-large size (around
100 variables after encoding and 20 constraints) based on constructing a
piecewise affine surrogate of the objective function over feasible samples. We
introduce two types of exploration functions to efficiently search the feasible
domain via mixed-integer linear programming solvers. We also provide a
preference-based version of the algorithm, which can be used when only pairwise
comparisons between samples can be acquired while the underlying objective
function to minimize remains unquantified. The two algorithms are tested on
mixed-variable benchmark problems with and without constraints. The results
show that, within a small number of acquisitions, the proposed algorithms can
often achieve better or comparable results than other existing methods.
