We typically construct optimal designs based on a single objective function. To better capture the breadth of an experiment's goals, we could instead construct a multiple objective optimal design based on multiple objective functions. While algorithms have been developed to find multi-objective optimal designs (e.g. efficiency-constrained and maximin optimal designs), it is far less clear how to verify the optimality of a solution obtained from an algorithm. In this paper, we provide theoretical results characterizing optimality for efficiency-constrained and maximin optimal designs on a discrete design space. We demonstrate how to use our results in conjunction with linear programming algorithms to verify optimality.

翻译：通常我们基于单一目标函数构建最优设计。为更好地反映实验目标的广泛性，可基于多个目标函数构建多目标最优设计。尽管已有算法（如效率约束优化设计和最大最小优化设计）可求解多目标最优设计，但如何验证算法所得解的优化性仍远未明确。本文给出了离散设计空间上效率约束优化设计与最大最小优化设计最优性的理论刻画，并展示了如何结合线性规划算法运用这些结果验证最优性。