Optimal designs for $K$-factor two-level models with first-order interactions on a symmetrically restricted design region

We develop $D$-optimal designs for linear models with first-order interactions on a subset of the $2^K$ full factorial design region, when both the number of factors set to the higher level and the number of factors set to the lower level are simultaneously bounded by the same threshold. It turns out that in the case of narrow margins the optimal design is concentrated only on those design points, for which either the threshold is attained or the numbers of high and low levels are as equal as possible. In the case of wider margins the settings are more spread and the resulting optimal designs are as efficient as a full factorial design. These findings also apply to other optimality criteria.