The idea is to find out whether 2nd law efficiency optimization may be a suitable trade-off between maximum work output and maximum 1st law efficiency designs for a regenerator gas turbine engine operating on the basis of an open Brayton cycle. It is shown that an ideal Brayton-type engine with or without a regenerator cannot operate at fully reversible limit; regime of zero entropy generation. Hence, the 2nd law efficiency of the power cycle is defined as the ratio of the 1st law efficiency to the efficiency of the ideal power cycle. For an ideal cycle, minimization of entropy production is equivalent to maximization of the system thermal efficiency. Except at regenerator effectiveness of 50 percent, at which maximum work output, maximum 1st law efficiency, and minimum entropy generation become identical, no relation is observed between these three optimization criteria for an irreversible engine. A design region is established within which pressure ratio of a real engine must lie between optimum pressure ratios corresponding to maximum work output and maximum 1st law efficiency, respectively, as the upper and the lower limits of this region. The results indicate that the 2nd law efficiency optimization is approximately equivalent to maximum work output design when regenerator effectiveness is between 0.78 and 0.82. For the regenerator effectiveness beyond 0.82, the 2nd law efficiency optimization may be considered as a trade-off between optimized work output and 1st law efficiency. However, if the effectiveness value happens to be less than 0.78, the 2nd law efficiency optimization may give no useful design information.