In modeling physical systems it is sometimes useful to construct border bases
of 0-dimensional polynomial ideals which are contained in the ideal generated
by a given set of polynomials. We define and construct such subideal border
bases, provide some basic properties and generalize a suitable variant of the
Buchberger-Moeller algorithm as well as the AVI-algorithm to the subideal
setting. The subideal version of the AVI-algorithm is then applied to an actual
industrial problem.