  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.
