Optimal designs for two-level main effects models on a restricted design region

We develop $D$-optimal designs for linear main effects models on a subset of the $2^K$ full factorial design region, when the number of factors set to the higher level is bounded. It turns out that in the case of narrow margins only those settings of the design points are admitted, where the number of high levels is equal to the upper or lower bounds, while in the case of wide 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.