We propose an algorithm to construct optimal exact designs (EDs). Most of the work in the optimal regression design literature focuses on the approximate design (AD) paradigm due to its desired properties, including the optimality verification conditions derived by Kiefer (1959, 1974). ADs may have unbalanced weights, and practitioners may have difficulty implementing them with a designated run size $n$. Some EDs are constructed using rounding methods to get an integer number of runs at each support point of an AD, but this approach may not yield optimal results. To construct EDs, one may need to perform new combinatorial constructions for each $n$, and there is no unified approach to construct them. Therefore, we develop a systematic way to construct EDs for any given $n$. Our method can transform ADs into EDs while retaining high statistical efficiency in two steps. The first step involves constructing an AD by utilizing the convex nature of many design criteria. The second step employs a simulated annealing algorithm to search for the ED stochastically. Through several applications, we demonstrate the utility of our method for various design problems. Additionally, we show that the design efficiency approaches unity as the number of design points increases.

翻译：我们提出了一种构建最优精确设计（EDs）的算法。在最优回归设计文献中，大多数研究集中于近似设计（AD）范式，这得益于其理想性质，包括由Kiefer（1959，1974）导出的最优性验证条件。AD可能具有不平衡的权重，实践者在按指定运行规模n实施时可能遇到困难。部分ED通过舍入方法获得AD各支撑点上的整数运行次数来构建，但此方法未必能产生最优结果。为构建ED，可能需要针对每个n进行新的组合构造，且目前尚无统一方法来构造它们。因此，我们开发了一种系统方法，可为任意给定n构建ED。我们的方法能在两步内将AD转化为ED，同时保持高统计效率：第一步利用许多设计准则的凸性质构建AD，第二步采用模拟退火算法随机搜索ED。通过多项应用实例，我们展示了该方法解决各类设计问题的实用性。此外，我们证明随着设计点数量的增加，设计效率趋近于1。