The problem of choosing the optimal design to estimate a regression function which can be well-approximated by a polynomial is considered, and two new optimality criteria are presented and discussed. Use of these criteria is illustrated by a detailed discussion of the case that the regression function can be assumed approximately linear. These criteria, which can be considered as compromises between the incompatible goals of inference about the regression function under an assumed model and of checking the model's adequacy, are found to yield designs superior in certain respects to others which have been proposed to deal with this problem, including minimum bias designs.