  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.
