Solving the Optimal Experiment Design Problem with Mixed-Integer Convex Methods
Deborah Hendrych, Mathieu Besançon, Sebastian Pokutta
TL;DR
This work addresses solving the Mixed-Integer Convex Optimal Experiment Design Problem by preserving problem structure and leveraging the Boscia.jl framework to solve node relaxations with Frank-Wolfe methods. It introduces a unified nonlinear formulation over a truncated simplex, proves convergence under Lipschitz smoothness or generalized self-concordance, and demonstrates superior performance on large-scale OEDP instances versus traditional MINLP approaches. The study focuses on D- and A-optimal criteria (and GTI generalizations), presents multiple solver paradigms (Boscia, OA, Direct Conic, SOCP, Co-BnB), and provides extensive empirical results showing faster solution times and higher optimality rates for Boscia on challenging problems. This approach enables scalable, structure-preserving optimization for experimental design with practical implications for statistical estimation and engineering applications.
Abstract
We tackle the Optimal Experiment Design Problem, which consists of choosing experiments to run or observations to select from a finite set to estimate the parameters of a system. The objective is to maximize some measure of information gained about the system from the observations, leading to a convex integer optimization problem. We leverage Boscia.jl, a recent algorithmic framework, which is based on a nonlinear branch-and-bound algorithm with node relaxations solved to approximate optimality using Frank-Wolfe algorithms. One particular advantage of the method is its efficient utilization of the polytope formed by the original constraints which is preserved by the method, unlike alternative methods relying on epigraph-based formulations. We assess the method against both generic and specialized convex mixed-integer approaches. Computational results highlight the performance of the proposed method, especially on large and challenging instances.