In this talk we examine the sharpness of regularity estimates associated with the Ladyzhenskaya-Prodi-Serrin conditions for the 3D Navier-Stokes equations. These conditions assert that the solution $u$ remains smooth on $[0,T]$ if $$\int_0^T \|u\|_{L^q}^p \, dt<\infty$$ for $\frac{2}{p}+\frac{3}{q}\leq 1$ and $q>3$. At the same time, the rate of growth of the Lebesgue norm is subject to the bound $$\frac{d}{dt}\|u\|_{L^q} \leq c \|u\|_{L^q}^{3\frac{q-1}{q-3}}.$$ If this a priori bound is attained on $[0,T]$, the integral criterion fails, implying blow-up. To probe whether this bound is sharp, we formulate an optimization problem where one maximizes the growth rate of the $L^q$-norm subject to suitable constraints. This problem is solved numerically using a Riemannian conjugate gradient approach, producing candidates for initial data that may lead to singularity formation in the Navier-Stokes equations. The numerical results show that the bound is sharp.