I do not understand the point of the dogleg method. If I have an estimate of the Hessian available, why wouldn't I just take a (Quasi)Newton step and stop it at the boundary?
My problem is:
$min_{p} \quad f(x) + p^{T} \nabla f(x) + \frac{1}{2} p^{T} \nabla^{2} f(x) p\\ s.t. ||p||_{2} \leq \Delta$
The dogleg method says to solve this by solving for $\tau$:
$|| p_{u} + (\tau-1)(p_{b}-p_{u})||_{2}^{2} = \Delta^{2}$
Where:
$p_{u} = -\frac{g^{T}{g}}{g^{T}Bg}g\\ p_{b} = -B^{-1}g\\ g \approx \nabla f(x)\\ B \approx \nabla^{2} f(x)$
But isn't $p_{b}$ simply the (Quasi)Newton step? Why wouldn't I just always do something like:
$p = -\frac{\Delta}{||B^{-1} g||}B^{-1} g$
How is the dogleg method any better? Especially when the trust region radius $\Delta$ is large enough that there is some curvature but the 2nd order model still provides a good fit, I see no point of using the dogleg approach.
Where should I use the dogleg method over the (Quasi)Newton's Step?
The idea is if you're far away from the minimum, taking steps in steepest descent direction $g$ usually converges fastest. When close to the minimum the Gauss-newton steps converge faster. Dogleg is using a linear combination of the two to try to achieve a reasonable tradeoff