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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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You don't get to set $\|p\|$ arbitrarily; what you're supposed to do is to find that $p$ satisfying your constraint... it can happen that the $p$ needed will have $\|p\|=\Delta$, of course. –  J. M. Jul 22 '12 at 15:29
    
You are right. I'll rephrase: if we have the case that $||p||=\Delta$, what does this mean? –  Chris Jul 22 '12 at 18:00
    
Clarifications of the question should be edited into the question; otherwise people have to delve into the comments to properly understand the question. –  joriki Jul 22 '12 at 18:42
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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? –  copper.hat Jul 23 '12 at 0:43
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