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

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