The approximate Determinantal Assignment Problem

The Determinantal Assignment Problem (DAP) has been introduced as the unifying description of all frequency assignment problems in linear systems and it is studied in a projective space setting. This is a multi-linear nature problem and its solution is equivalent to finding real intersections between a linear space, associated with the polynomials to be assigned, and the Grassmann variety of the projective space. This paper introduces a new relaxed version of the problem where the computation of the approximate solution, referred to as the approximate DAP, is reduced to a distance problem between a point in the projective space from the Grassmann variety Gm(Rn). The cases G2(Rn) and its Hodge-dual Gn−2(Rn) are examined and a closed form solution to the distance problem is given based on the skew-symmetric matrix description of multivectors via the gap metric. A new algorithm for the calculation of the approximate solution is given and stability radius results are used to investigate the acceptability of the resulting perturbed solutions.