Legendre dual cone barrier
= Legendre dual cone barrier
{c}
{title2=$F_\dagger(y)=F^*(-y)$}
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 $s=-\mu\nabla F_\dagger(y)$. This gives a joint primal-dual <central path> without assuming the barrier is identical on a self-dual cone.