On "A Homogeneous Interior-Point Algorithm for Non-Symmetric Convex Conic Optimization"

In a recent paper, Skajaa and Ye proposed a homogeneous primal-dual interior-point method for non-symmetric conic optimization. The authors showed that their algorithm converges to $\varepsilon$-accuracy in $O(\sqrt{\nu}\log \varepsilon^{-1})$ iterations, where $\nu$ is the complexity parameter associated with a barrier function for the primal cone, and thus achieves the best-known iteration complexity for this class of problems. However, an earlier result from the literature was used incorrectly in the proofs of two intermediate lemmas in that paper. In this note, we propose new proofs of these results, allowing the same complexity bound to be established.