Fast Computation of Highly G-optimal Exact Designs via Particle Swarm Optimization

TL;DR

This paper introduces an extension of particle swarm optimization (PSO) for efficiently computing highly G-optimal exact designs, outperforming existing algorithms in quality and comparable in computational cost for response surface models.

Contribution

The paper develops a novel PSO-based algorithm for G-optimal design computation and demonstrates its superiority over traditional methods across multiple experimental scenarios.

Findings

PSO finds improved G-optimal designs for up to 5 factors.

PSO achieves comparable computational cost to state-of-the-art algorithms.

PSO outperforms existing methods in design quality for response surface models.

Abstract

Computing proposed exact $G$-optimal designs for response surface models is a difficult computation that has received incremental improvements via algorithm development in the last two-decades. These optimal designs have not been considered widely in applications in part due to the difficulty and cost involved with computing them. Three primary algorithms for constructing exact $G$-optimal designs are presented in the literature: the coordinate exchange (CEXCH), a genetic algorithm (GA), and the relatively new $G$-optimal via $I_\lambda$-optimality algorithm ($G(I_\lambda)$-CEXCH) which was developed in part to address large computational cost. Particle swarm optimization (PSO) has achieved widespread use in many applications, but to date, its broad-scale success notwithstanding, has seen relatively few applications in optimal design problems. In this paper we develop an extension of PSO to adapt it to the optimal design problem. We then employ PSO to generate optimal designs for several scenarios covering $K = 1, 2, 3, 4, 5$ design factors, which are common experimental sizes in industrial experiments. We compare these results to all $G$-optimal designs published in last two decades of literature. Published $G$-optimal designs generated by GA for $K=1, 2, 3$ factors have stood unchallenged for 14 years. We demonstrate that PSO has found improved $G$-optimal designs for these scenarios, and it does this with comparable computational cost to the state-of-the-art algorithm $G(I_\lambda)$-CEXCH. Further, we show that PSO is able to produce equal or better $G$-optimal designs for $K= 4, 5$ factors than those currently known. These results suggest that PSO is superior to existing approaches for efficiently generating highly $G$-optimal designs.