Over the last several years fixed-structure multiplier versions of mixed structured singular value (MSSV) theory have been developed and have led to the development of linear matrix inequalities (LMIs) for the analysis of robust stability and performance. These LMIs have in turn led to the development of bilinear matrix inequalities BMIs for the synthesis of robust controllers. The BMI formulation in practice requires the multiplier to lie in the span of a stable basis, potentially introducing significant conservatism. This paper uses the LMI approach to MSSV analysis to develop an approach to robust controller synthesis that is based on the stable factors of the multipliers and does not require the multipliers to be restricted to a basis. It is shown that this approach requires the solution of non-linear matrix inequalities (NMIs). A continuation algorithm is presented for the solution of NMIs. This algorithm may be used to solve NMIs in general and its usefulness is not limited to the robust controller synthesis problem alone. The primary computational burden of the continuation algorithm is the solution of a series of LMIs. The use of this algorithm is demonstrated by designing a robust controller for a benchmark problem.