  The optimal currents on arbitrarily shaped radiators with respect to the
minimum quality factor Q are found using a simple and efficient procedure. The
solution starts with a reformulation of the problem of minimizing quality
factor Q as an alternative, so-called dual, problem. Taking advantage of modal
decomposition and group theory, it is shown that the dual problem can easily be
solved and always results in minimal quality factor Q. Moreover, the
optimization procedure is generalized to minimize quality factor Q for embedded
antennas, with respect to the arbitrarily weighted radiation patterns, or with
prescribed magnitude of the electric and magnetic near-fields. The obtained
numerical results are compatible with previous results based on composition of
modal currents, convex optimization, and quasi-static approximations; however,
using the methodology in this paper, the class of solvable problems is
significantly extended.
