A variational procedure due to Ponte Castañeda et al. [Phys. Rev. B 46, 4387 (1992)] is used to determine three-point bounds and other types of estimates for the effective response of strongly nonlinear composites with random microstructures. The variational procedure makes use of estimates for the effective properties of "linear comparison composites" to generate corresponding estimates for nonlinear composites. Several equivalent forms of the variational procedure are derived. In particular, it is shown that the mean-field theory of Wan et al. [Phys. Rev. B 54, 3946 (1996)], which also makes use of a linear comparison composite, together with a certain "decoupling approximation," leads to results that are precisely identical to those that can be obtained from the earlier variational procedure. Finally, three-point bounds and other estimates are computed for power-law composites with cell-type microstructures, and the results are compared with random resistor network simulations available from the literature.