[en] A numerical technique is presented to optimize the design of quadrupole magnetic coil systems, such as presently used in tandem mirror fusion experiments. The optimization is carried out with respect to various plasma physics properties (e.g. MHD stability and reduced radial transport of particles) which depend on the magnetic field geometry. The coils are treated as filaments of infinitesimal thickness and the magnetic field is computed in the paraxial limit. Constraints on the coil shape and on the magnetic geometry can be incorporated into the method. It is demonstrated that coil solutions can be found which match given paraxial field profiles or which directly minimize a scalar functional of these profiles. As an interesting application of the latter technique, a filamentary coil design for a nearly omnigenous anchor is obtained