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I'm trying to minimize the following:

$P(x_{k+1}) = f(x_k) + \nabla f (x_k) \cdot (-h_k \nabla f(x_k)) + \frac{1}{2}(-h_k \nabla f(x_k))^T Hf(x_k) (-h_k \nabla f(x_k))$

I know that minimizing the quadratic form is by solving $Ax = b$ i.e. $x = A^{-1}b$, so how can I choose $h_k$ to minimize $P$.

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  • What are you assumptions on $f$? Is it convex and real-valued? Also, this is just semantics, but the correct terminology is that you're minimizing $f$. The method you minimize $f$ is by implementing the Newton method you've written above. It doesn't make a ton of sense to minimize an iteration! – Zim Dec 18 '20 at 22:01

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