In this note the problem of restricted complexity stability margin maximization (RCSNM) for single-input-single-output (SISO) plants affected by rank one real perturbations is considered. This problem amounts to maximizing the real 12 parametric stability margin over an assigned class of restricted complexity controllers, which are described by rational transfer functions of fixed order with coefficients depending affinely on some free parameters. It is shown that the RCSMM problem, which is nonconvex in general, can be approached by means of convex optimization methods. Specifically, a lower bound of the stability margin, whose maximization can be accomplished via linear matrix inequality (LMI) techniques, is developed.