PNOD: An Efficient Projected Newton Framework for Exact Optimal Experimental Designs

Abstract: 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.