The Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, introduced more than 70 years ago, is a hallmark of colloidal particle modeling. For highly charged particles in the dilute regime, it is often supplemented by Alexander's prescription (Alexander et al. in J Chem Phys 80:5776, 1984) for using a renormalized charge. Here, we solve the problem of the interaction between two charged colloids at finite ionic strength, including dielectric mismatch effects, using an efficient numerical scheme to solve the nonlinear Poisson-Boltzmann (NPB) equation with unknown boundary conditions. Our results perfectly match the analytical predictions for the renormalized charge by Trizac and coworkers (Aubouy et al. in J Phys A 36:5835, 2003). Moreover, they allow us to reinterpret previous molecular dynamics (MD) simulation results by Kreer et al. (Phys Rev E 74:021401, 2006), rendering them now in agreement with the expected behavior. We furthermore find that the influence of polarization becomes important only when the Debye layers overlap significantly.