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Consider an iterative algorithm of the form $x^{t+1} = x^t - \eta g(x^t)$. (..if necessary feel free to assume that a function $L$ is explicitly known such that $g = \frac{\partial L}{\partial x}$..). Suppose that now I want to show that there exists a point $x^*$ within some pre-determined ball which is a fixed point for this iterative algorithm.

  • I would like to know what methods are generally known to show such a thing in cases when the function $g$ is too complicated to solve for its roots?

  • Is there a way to argue for the existence of such fixed points starting from proving some Lyapunov-like properties for $L$?

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  • $\begingroup$ (1) Isn't your question just asking how to show that the given function $g$ has a root $x^*$ in some predetermined ball? (As $x^*$ is a fixed point of your algorithm if and only if $x^*$ is a root of $g$.) (2) What is the type of $x^t$? Is $x^t\in \mathbb{R}$? $\endgroup$ – Neal Young Jun 9 '17 at 16:02
  • $\begingroup$ Yes, you can think of it this way. I want to know if there are ways to use the Lypunaov kind of thinking with $L$ to somehow circumvent the usually hard problem of showing roots of $g$. You can assume $x^t \in \mathbb{R}^n$ for some $n$. $\endgroup$ – gradstudent Jun 9 '17 at 17:02

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