Logarithmically homogeneous barrier ($F(ts)=F(s)-\nu \log t$)

A cone barrier with this scaling law satisfies $⟨\nabla F(s),s⟩=-\nu$ and $\nabla F(ts)=t^{-1}\nabla F(s)$. Therefore a central path with $y=-\mu \nabla F(s)$ has primal-dual gap $\nu \mu$. A product of orthant and Lorentz cone canonical barriers adds their parameters.

For a logarithmically homogeneous barrier on a proper cone, the convex conjugate with reversed argument defines a barrier on the interior dual cone. The inverse gradient relation makes $y=-\mu \nabla F(s)$ equivalent to $s=-\mu \nabla F_{†}(y)$. This gives a joint primal-dual central path without assuming the barrier is identical on a self-dual cone.