22 Feb 2018

Log barrier function

We are given an oracle access to a function value \(f(x)\), its derivative \(\nabla f(x)\) and the hessian \(\nabla^2 f(x)\). Based on this provided values, we can easily find the value \(g(x) = - log f(x)\), derivative \(\nabla g(x)\) and hessian \(\nabla^2 g(x)\).

First, we use the chain rule: \(\begin{gather*} \nabla^2 (-log(f(x))) = - \nabla ( \nabla log (f(x))) = \nabla \frac{-\nabla f(x)}{f(x)} \end{gather*}\)

Next, we use the quotient rule: \(\begin{gather*} \frac{-\nabla^2f(x) f(x) + \nabla f(x) \nabla f(x)^T}{f(x)^2} = \frac{-\nabla^2f(x)}{f(x)} + \frac{\nabla f(x)\ \nabla f(x)^T}{f(x)^2} \end{gather*}\)

\[\begin{gather*} \nabla g(x) = \frac{-\nabla f(x)}{f(x)} \end{gather*}\] \[\begin{gather*} \nabla^2 g(x) = \frac{-\nabla^2f(x)}{f(x)} + \frac{\nabla f(x)\ \nabla f(x)^T}{f(x)^2} \end{gather*}\]