Global and Preference-based Optimization with Mixed Variables using Piecewise Affine Surrogates

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 simulation or experiment, it may be expensive to evaluate. In this paper, we propose a novel surrogate-based global optimization algorithm, called PWAS, 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 (MILP) solvers. We also provide a preference-based version of the algorithm, called PWASp, which can be used when only pairwise comparisons between samples can be acquired while the objective function remains unquantified. PWAS and PWASp are tested on mixed-variable benchmark problems with and without constraints. The results show that, within a small number of acquisitions, PWAS and PWASp can often achieve better or comparable results than other existing methods.