Dimension reduction for optimal design problems with Kronecker product structure

This paper is motivated by the problem of optimal allocation of trials in multi‐environment crop variety testing with a large number of varieties. Optimizing the allocation of trials results in the minimization of a design criterion with a Kronecker product structure in the information matrix. We consider the Kronecker–Bayesian linear criterion, which generalizes this design problem and has the form of the trace of the inverse of a sum of two Kronecker products. We derive a new general formula for the inverse of the sum of two Kronecker products, and we use this result to rewrite the Kronecker–Bayesian criterion in the form of the compound Bayes risk criterion, which can be recognized as a sum of Bayesian linear criteria with the same moment matrix. Based on the convexity and differentiability of the Kronecker–Bayesian linear criterion, we establish optimality conditions for approximate designs. We also propose a dimension reduction approach that provides highly efficient approximations for optimal designs. The proposed method allows for the preselection of an upper bound on the efficiency loss, which is independent of the true optimal design. Optimal or highly efficient designs can be computed under any kind of additional linear constraints, such as cost constraints. We apply our results to the problem of optimizing the allocation of trials in multi‐environment crop variety testing, and we illustrate the behavior of the optimal designs by real data examples. Finally, we consider further applications of the general formula for computing the inverse of the sum of two Kronecker products in control theory or multivariate time series analysis.