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.

翻译：涉及混合变量（即数值型和分类型变量）的优化问题求解可能具有挑战性，尤其是在存在混合变量约束的情况下。此外，当目标函数是复杂仿真或实验的结果时，其评估成本可能很高。本文提出了一种新颖的基于代理模型的全局优化算法，用于求解中等规模（编码后约100个变量）的线性约束混合变量问题。所提方法基于在可行样本上构建目标函数的分段仿射代理模型。我们假设目标函数是黑箱且评估成本高昂，而线性约束是可量化、不可松弛、先验已知且评估成本低廉的。我们引入了两种探索函数，通过混合整数线性规划求解器高效搜索可行域。我们还提供了算法的偏好版本，该版本适用于仅能获取样本间两两比较、而待最小化的底层目标函数本身无法量化的情况。这两种算法在多个无约束和约束混合变量基准问题上进行了评估。结果表明，在所需实验/仿真次数较少的情况下，所提算法通常能取得优于或可比肩其他现有方法的结果。