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We present a method for the optimal design of superconducting magnet systems for magnetic resonance imaging (MRI). The method integrates a linear-programming technique with the finite-element method (FEM) to calculate minimum-volume coil configurations subject to magnetic field homogeneity constraints for MRI systems that contain general nonaxisymmetric magnetic yoke structures. The method rapidly converges and only requires a small number of iterations and FEM analyses to be performed. In particular, the method is well suited for magnet design problems that necessitate large 3-D FEM models. We demonstrate the method with the optimal design of an open and compact 0.5 T yoked biplanar magnet assembly considered for use in an integrated medical linear accelerator and MRI system. In particular, the coil configuration for this magnet design is constructed from a MgB2 high-temperature superconducting material that operates in a conduction-cooled cryogen-free environment.