In this work, we analyze hybrid ARQ (HARQ) protocols over the independent block fading channel. We assume that the transmitter is unaware of the channel state information (CSI) but has a knowledge about the channel statistics. We consider two scenarios with respect to the feedback received by the transmitter: i) ''conventional'', one-bit feedback about the decoding success/failure (ACK/NACK), and ii) the multi-bit feedback scheme when, on top of ACK/NACK, the receiver provides additional information about the state of the decoder to the transmitter. In both cases, the feedback is used to allocate (in the case of one-bit feedback) or adapt (in the case of multi-bit feedback) the power across the HARQ transmission attempts. The objective in both cases is the minimization of the outage probability under long-term average and peak power constraints. We cast the problems into the dynamic programming (DP) framework and solve them for Nakagami-m fading channels. A simplified solution for the high signal-to-noise ratio (SNR) regime is presented using a geometric programming (GP) approach. The obtained results quantify the advantage of the multi-bit feedback over the conventional approach, and show that the power optimization can provide significant gains over conventional power-constant HARQ transmissions even in the presence of peak-power constraints.