Abstract: Two approaches to reducing experimental costs associated with measurements are considered for a multivariate quadratic regression model without an intercept. The first approach picks an appropriate design region, for which the optimal design has a smaller number of support points within its class. For a two-dimensional model, the corresponding boundaries of the regions that determine the structure and form of optimal designs as well as the designs themselves are derived explicitly in analytical form. In the general case, the author proposes an algorithm for constructing designs with a reduced number of support points. The second approach uses alternative designs with a sufficiently high efficiency instead of optimal designs, while significantly reducing the number of support points.