  Optimizing multiple, non-preferential objectives for mixed-variable,
expensive black-box problems is important in many areas of engineering and
science. The expensive, noisy, black-box nature of these problems makes them
ideal candidates for Bayesian optimization (BO). Mixed-variable and
multi-objective problems, however, are a challenge due to BO's underlying
smooth Gaussian process surrogate model. Current multi-objective BO algorithms
cannot deal with mixed-variable problems. We present MixMOBO, the first
mixed-variable, multi-objective Bayesian optimization framework for such
problems. Using MixMOBO, optimal Pareto-fronts for multi-objective,
mixed-variable design spaces can be found efficiently while ensuring diverse
solutions. The method is sufficiently flexible to incorporate different kernels
and acquisition functions, including those that were developed for
mixed-variable or multi-objective problems by other authors. We also present
HedgeMO, a modified Hedge strategy that uses a portfolio of acquisition
functions for multi-objective problems. We present a new acquisition function,
SMC. Our results show that MixMOBO performs well against other mixed-variable
algorithms on synthetic problems. We apply MixMOBO to the real-world design of
an architected material and show that our optimal design, which was
experimentally fabricated and validated, has a normalized strain energy density
$10^4$ times greater than existing structures.
