Computing the exact optimal experimental design has been a longstanding challenge in various scientific fields. This problem, when formulated using a specific information function, becomes a mixed-integer nonlinear programming (MINLP) problem, which is typically NP-hard, thus making the computation of a globally optimal solution extremely difficult. The branch and bound (BnB) method is a widely used approach for solving such MINLPs, but its practical efficiency heavily relies on the ability to solve continuous relaxations effectively within the BnB search tree. In this paper, we propose a novel projected Newton framework, combining with a vertex exchange method for efficiently solving the associated subproblems, designed to enhance the BnB method. This framework offers strong convergence guarantees by utilizing recent advances in solving self-concordant optimization and convex quadratic programming problems. Extensive numerical experiments on A-optimal and D-optimal design problems, two of the most commonly used models, demonstrate the framework's promising numerical performance. Specifically, our framework significantly improves the efficiency of node evaluation within the BnB search tree and enhances the accuracy of solutions compared to state-of-the-art methods. The proposed framework is implemented in an open source Julia package called \texttt{PNOD.jl}, which opens up possibilities for its application in a wide range of real-world scenarios.

翻译：计算精确最优实验设计长期以来一直是各科学领域面临的挑战。当使用特定信息函数表述时，该问题转化为混合整数非线性规划问题，这类问题通常具有NP难特性，使得全局最优解的计算极为困难。分支定界法是求解此类混合整数非线性规划问题的常用方法，但其实际效率在很大程度上依赖于在分支定界搜索树中有效求解连续松弛问题的能力。本文提出一种新颖的投影牛顿框架，结合顶点交换法高效求解相关子问题，旨在增强分支定界法的性能。该框架利用自协调优化与凸二次规划问题求解的最新进展，提供了强有力的收敛性保证。在两种最常用的模型——A最优与D最优设计问题上进行的广泛数值实验表明，该框架具有优异的数值性能。具体而言，与现有先进方法相比，我们的框架显著提升了分支定界搜索树中节点评估的效率，并提高了求解精度。所提出的框架已在开源Julia软件包\texttt{PNOD.jl}中实现，这为其在广泛实际场景中的应用开辟了可能性。