In the common polynomial regression model of degree m we consider
the problem of determining the $D$- and $D_1$-optimal designs subject to
certain constraints for the $D_1$-efficiencies in the models of degree $m - j,
m - j + 1,\dots, m + k(m > j \geq 0, k \geq 0 \text{given})$.We present a
complete solution of these problems, which on the one hand allow a fast
computation of the constrained optimal designs and, on the other hand, give an
answer to the question of the existence of a design satisfying all constraints.
Our approach is based on a combination of general equivalence theory with the
theory of canonical moments. In the case of equal bounds for the
$D_1$-efficiencies the constrained optimal designs can be found explicitly by
an application of recent results for associated orthogonal polynomials.