On self-concordant barriers for generalized power cones

In the study of interior-point methods for nonsymmetric conic optimization and their applications, Nesterov introduced the power cone, together with a 4-self-concordant barrier for it. In his PhD thesis, Chares found an improved 3-self-concordant barrier for the power cone. In addition, he introduced the generalized power cone, and conjectured a nearly optimal self-concordant barrier for it. In this short note, we prove Chares’ conjecture. As a byproduct of our analysis, we derive a self-concordant barrier for a high-dimensional nonnegative power cone.