  Hybrid automatic repeat 1 request (HARQ) is a key enabler for ultrareliable
communications. This paper optimizes transmit power for the initial
transmission and the subsequent retransmissions of HARQ with either incremental
redundancy or Chase combining, aiming to minimize the expected energy
consumption given the target outage probability and the target latency. The
main challenge is due to the fact that the outage probability is a complicated
function of the power variables which are nested in successive convolutions.
The existing works mostly use a classic upper bound to approximate the outage
probability by assuming unbounded transmit power, then convert the original
problem to a geometric programming (GP) problem. In contrast, we propose a
novel and much tighter upper bound by taking the practical power limit into
consideration. The new bound and the resulting new GP method are further
extended to a broader group of channel models with various fading, multiple
antennas, and multiple receivers. As shown in simulations, the GP method based
on the new bound significantly outperforms the existing strategies that either
fix transmit power or optimize power by the classic bounding technique.
