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A recently developed numerical algorithm for optimizing quadrupole magnetic coil systems is applied to the tandem mirror design problem. By adjusting filamentary coil shapes, the magnetic field structure minimizing radial particle transport is found, subject to a stability constraint. Two configurations are examined: a MARS-like reactor design and an unconventional TMX-like system with a nonaxisymmetric central cell. In the latter case, a number of nearly omnigenous geometries with various stability properties are investigated.