Waveform optimization is a crucial step in the design of a multiple-input multiple-output system. This letter considers the joint optimization of constant modulus waveforms and mismatched (or matched) receive filters to suppress the auto- and cross correlations using the minimax (ℓ_∞-norm) design criterion. For practical waveform length and system size, the waveform design problem becomes quite challenging due to the large problem size (more than IO⁵ unimodular complex variables and IO⁷ nonlinear constraints). In addition to the large size, this problem is nonconvex, nonsmooth, and as such, cannot be handled effectively by the existing waveform design algorithms or off-the-shelve optimization tools. This letter develops an efficient primal-dual type algorithm with low per-iteration complexity to solve this problem. Numerical comparison shows that the waveforms based on the minimax design outperform those obtained from the existing ℓ₂-norm design by 4-5 dBs in terms of peak sidelobe levels.