An irreversible cycle model of the Brayton heat engine is established, in which the irreversibilities resulting from the internal dissipation of the working substance in the adiabatic compression and expansion processes and the finite-rate heat transfer in the regenerative and constant-pressure processes are taken into account. The power output, and efficiency of the cycle are expressed as functions of temperatures of the working, substance and the heat sources, heat transfer coefficients, pressure ratio, regenerator effectiveness, and total heat transfer area including the heat transfer areas of the regenerator and other heat exchangers. The regenerative criteria are given. The power output, is optimized for a given efficiency. The general optimal performance characteristics of, the cycle are revealed. The optimal performance of the Brayton heat engines with and without regeneration is compared quantitatively. The advantages of using the regenerator are expounded. Some important parameters of an irreversible regenerative Brayton heat engine, such as the temperatures of the working substance at different states, pressure ratio, maximum value of the pressure ratio, regenerator effectiveness and ratios of the various heat transfer areas to the total heat transfer area of the cycle, are further optimized. The optimal relations between these parameters and the efficiency of the cycle are presented by a set of characteristic curves for some assumed compression and expansion, efficiencies. The results obtained may be helpful to the comprehensive understanding of the optimal performance of the Brayton heat engines with and without regeneration and play a theoretical instructive role for the optimal design of a regenerative Brayton heat engine.