(adsbygoogle = window.adsbygoogle || []).push({}); I wasn't sure into which category I should post this, so feel free to move it into a more appropriate place.

As part of my work I'm solving a system of nonlinear equations, of a usual form:

$$\vec{F}(\vec{X})=\begin{pmatrix}F_1(X_1, X_2, \cdots X_N) \\ F_2(\cdots) \\ \vdots \\ F_N(\cdots)\end{pmatrix}=\vec{0}$$

There's lots of literature about solving systems of nonlinear equations, and lots of discussion about "how Newton-Raphson methods can fail" and "how to counter it", usual methods being to combine the N-R step with a gradient-descent step, obtained by scalarising the function usually something like

$$f(\vec{X}) = ||\vec{F}(\vec{X})||^2$$

However, I'm struggling to find any literature which makes it explicit (either with proofs, or worked examples, in 2D would probably be the most useful) that you can't just minimise ##f(\vec{X})## in the first place, and that it becomesnecessaryto move to methods like N-R which explicitly handle the dimensionality and simultaneity. I mean, Iknowthat that doesn't tend to work, and I would usually hand-wave it away by making reference to the extra structure of ##\vec{F}(\vec{X})## which is lost by "flattening" things down.

If I have some free time this week, I intend to construct a pair of explicit functions of 2 variables, which definitely have regions where they cross the F=0 axis, and then some contour plots of both functions along with the L2-norm combination, and then graph some N-R and Grad-Desc directions at some sample x,y values. I hope that doing so might make things more explicit that in some cases the ##f(\vec{X})## approach will lead you astray where the ##\vec{F}(\vec{X})## behaves.

But until then, I'd really appreciate if anyone here happens to know of any relevant literature, that they provide me some citations or links.

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# Failure of Optimisation for Nonlinear Equation Systems

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