$D$-optimal designs for a multidimensional polynomial model

For a multidimensional polynomial regression model, the problem of constructing a $D$-optimal design on a symmetric (relative to the origin) design region is investigated. An approach based on the symmetry property is proposed, which reduces the computational complexity of constructing the optimal design. For the case of an asymmetric region, an algorithm for constructing $D$-optimal designs for a multidimensional quadratic polynomial model without an intercept term is developed. This model has significant practical importance and can be used in various applications, such as calculating the lifespan of road pavement based on different factors.