This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-order output-feedback synthesis problems of nth-order linear time-invariant (LTI) systems with n(u) (respectively, n(y)) independent inputs (respectively, outputs), The algorithm is based on a "cone complementarity" formulation of the problem and is guaranteed to produce a stabilizing controller of order m less than or equal to n - max(n(u), n(y)), matching a generic stabilizability result of Davison and Chatterjee [7], Extensive numerical experiments indicate that the algorithm finds a controller with order less than or equal to that predicted by Kimura’s generic stabilizability result (m less than or equal to n-n(u)-n(y)+1). A similar algorithm can be applied to a variety of control problems, including robust control synthesis.