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 mixed-variable 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 size (around 100 variables after encoding). The proposed approach is based on constructing a piecewise affine surrogate of the objective function over feasible samples. We assume the objective function is black-box and expensive-to-evaluate, while the linear constraints are quantifiable, unrelaxable, a priori known, and are cheap to evaluate. 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 designed for situations where only pairwise comparisons between samples can be acquired, while the underlying objective function to minimize remains unquantified. The two algorithms are evaluated on several unconstrained and constrained mixed-variable benchmark problems. The results show that, within a small number of required experiments/simulations, the proposed algorithms can often achieve better or comparable results than other existing methods.