A performance analysis of an irreversible regenerative magnetohydrodynamic (MHD) power cycle operating at maximum power for a constant-velocity and a constant-Mach-number type of MHD generator has been carried out. The obtained results are evaluated together with the results of performance analysis at maximum thermal efficiency, and thus the upper and lower bounds of the optimal operating region in terms of power and thermal efficiency are determined. The considered MHD power cycle model includes irreversibilities coming from the electrical losses at the MHD generator, frictions at the compressor and pressure losses in the heater, the cooler and the regenerator. The effects of these major irreversibilities and regeneration application on the general and optimal performance are investigated. The analysis presented in this study may provide a theoretical basis for the optimal design and operating conditions for regenerative MHD power cycles.