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This paper describes an optimization method for designing electromagnetic coil arrangements that generate uniform magnetic fields. The method can accurately and efficiently determine an arrangement that minimizes the volume of the coils. An important advantage of the method is that the solutions it generates are guaranteed to be global minimums. The method may be useful for designing large superconducting magnets, since their cost depends strongly on the amount of superconducting wire required. It is also shown that the method can be applied to minimize the power consumption of resistive magnets.