This article reports the thermodynamic optimization based on the maximum power (MP) and maximum power density (MPD) criteria for an irreversible Atkinson heat-engine model which includes internal irreversibility resulting from the adiabatic processes. The power density, power output, and thermal efficiency are obtained by introducing the isentropic temperature ratio of the compression process, cycle temperature ratio, and the compression and expansion efficiencies. Optimal performance and design parameters of the Atkinson cycle are obtained analytically for the MP conditions and numerically for the MPD conditions. The results at MPD conditions are compared with those results obtained by using the MP and maximum thermal efficiency criteria. The effects of the cycle temperature ratio and irreversibilities on the general and optimal performances are investigated. It is shown that for the Atkinson cycle, a design based on the MPD conditions is more advantageous from the point of view of engine sizes and thermal efficiency.