The Miles-Howard theorem states that a necessary condition for normal-mode instability in parallel, inviscid, steady stratified shear flows is that the minimum gradient Richardson number, $Rig,min$, is less than 1/4 somewhere in the flow. However, the non-normality of the Navier-Stokes and buoyancy equations may allow for substantial perturbation energy growth at finite times. We calculate numerically the linear optimal perturbations which maximize the perturbation energy gain for a stably stratified shear layer consisting of a hyperbolic tangent velocity distribution with characteristic velocity $U0\ast$ and a uniform stratification with constant buoyancy frequency $N0\ast$. We vary the bulk Richardson number $Rib$=$N0\ast$$2$$h\ast 2$/$U0\ast$$2$ (corresponding to $Rig,min$) between 0.20 and 0.50 and the Reynolds numbers $Re$=$U0\ast$$h\ast$/$v\ast$ between 1000 and 8000, with the Prandtl number held fixed at $Pr$=1. We find the transient growth of non-normal perturbations may be sufficient to trigger strongly nonlinear effects and breakdown into small-scale structures, thereby leading to enhanced dissipation and non-trivial modification of the background flow even in flows where $Rig,min$>1/4. We show that the effects of nonlinearity are more significant for flows with higher $Re$, lower $Rib$ and higher initial perturbation amplitude $E0$. Enhanced kinetic energy dissipation is observed for higher-$Re$ and lower-$Rib$ flows, and the mixing efficiency, quantified here by $ϵp$/($ϵp$+$ϵk$) where $ϵp$ is the dissipation rate of density variance and $ϵk$is the dissipation rate of kinetic energy, is found to be approximately 0.35 for the most strongly nonlinear cases.