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In the Trust Region Method we approximate a given function $f$ by a quadratic model $q_k(p)=f(x_k)+\nabla{f(x_k)}^Tp+\frac{1}{2}p^T\nabla^2{f(x_k})p$. Now, we want to minimize the function within a trusted region so we look at:

$\min q_k(p) \text{ subject to } \|p\|\leq\Delta$,

where $\Delta >0$ is the radius of the trusted region. If we have the case $\|p\|=\Delta$, what does this practically mean?

Thank you for your time!

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    Think of the trust region as a sort of step size. Instead of picking a direction and then finding a step size, you are selecting a maximum step size ($\Delta$ above) and finding the direction (well, next iterate, really) that works best. So, setting $\|p\| = \Delta$ is basically like fixing the step size. One presumes that allowing $\leq$ instead of $=$ permits a better search?2012-07-23

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