On self-concordant barriers for generalized power cones

In the study of interior-point methods for nonsymmetric conic optimization and their applications, Nesterov (Optim Methods Softw 27(4–5): 893–917, 2012) introduced a 4-self-concordant barrier for the power cone, also known as Koecher’s cone. In his PhD thesis, Chares (Cones and interior-point algorithms for structured convex optimization involving powers and exponentials, PhD thesis, Universite catholique de Louvain, 2009) 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 paper, we prove Chares’ conjecture. As a byproduct of our analysis, we derive a self-concordant barrier for a nonnegative power cone.