Optimization analyses of thermoelectric generators operation is of importance both for practical applications and theoretical considerations. Depending on the desired goal, two different strategies are possible to achieve high performance: through optimization one may seek either power output maximization or conversion efficiency maximization. Recent literature reveals the persistent flawed notion that these two optimal working conditions may be achieved simultaneously. In this article, we lift all source of confusion by correctly posing the problem and solving it. We assume and discuss two possibilities for the environment of the generator to govern its operation: constant incoming heat flux, and constant temperature difference between the heat reservoirs. We demonstrate that, while power and efficiency are maximized simultaneously if the first assumption is considered, this is not possible with the second assumption. This latter corresponds to the seminal analyses of Ioffe who put forth and stressed the importance of the thermoelectric figure of merit $ZT$. We also provide a simple procedure to determine the different optimal design parameters of a thermoelectric generator connected to heat reservoirs through thermal contacts with a finite and fixed thermal conductance.
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