Actively shielded coil configurations offer an attractive design alternative as magnetic resonance imaging (MRI) magnets. Arriving at an optimized design requires an understanding of the interactions between the various performance requirements. A unique design approach, based on the multistage Monte Carlo method, is suggested which allows for a thorough search for optimized designs within the bounds of the multidimensional design space created by the competing requirements and imposed spatial restrictions. This design space is shown to contain a multiplicity of different coil configurations, each possessing a sharply defined minimum in the resulting stray field level, field inhomogeneity, or amount of conductor. Optimization techniques that depend on local gradients would tend, in such a case, to be trapped in the many local minima and fail to locate a global minimum. Several examples are discussed to highlight the advantages of the proposed design approach.