  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.
