The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.

翻译：本文研究了为回归模型参数的最小二乘估计计算精确最优实验设计的问题。我们证明，对于包括A-、I-、G-和MV-最优性准则在内的广泛最优性准则类别，该问题可通过混合整数线性规划（MILP）求解。此方法改进了当前最先进的数学规划公式——混合整数二阶锥规划。MILP公式的核心思想是McCormick松弛，其关键依赖于与最优精确设计相对应的最小二乘估计量协方差矩阵元素的有限区间界。我们提供了构建这些区间界的解析方法和算法方法。我们还展示了MILP方法的独特优势，例如可将多重设计约束纳入优化问题，包括对最小二乘估计量方差和协方差的约束。