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∗ and a uniform stratification with constant buoyancy frequency N0∗. We vary the bulk Richardson number Rib=N0∗2h∗2/U0∗2 (corresponding to Rig,min) between 0.20 and 0.50 and the Reynolds numbers Re=U0∗h∗/v∗ 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 ϵkis the dissipation rate of kinetic energy, is found to be approximately 0.35 for the most strongly nonlinear cases.EPSR