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I have question. I have a Riemmanian manifold $\mathcal{M}$, like an $n$-dimensional regular surface in $\mathbb{R}^n$. And I have a smooth scalar field defined on this manifold $f:\mathcal{M} \to \mathbb{R}$ (positive function).

My actual problem is discretized but I'm trying to generalize it, I don't have much experience with optimization on manifold but I do know about smooth manifolds and riemannian manifolds.

I was wondering if the problem was more suitable for a standard optimization problem (if you have like a single chart maybe you take $f \circ \phi_{\alpha} : \mathbb{R}^n \to \mathbb{R}$ than this would cast in a standard optimization problem), or if problems like this (where you know in advance you have a manifold with some metric) are actually better tackled in this form. The idea would be to work out an algorithm at the last.

I have few references about optimization on manifolds and I'm slowly reading through them, it seems most of the time you have a closed form description of your manifold which actually seems to enable closed form expressions of gradients, hessians and retractions etc. In my case instead I don't really have that description (because I have a discretized mesh).

Is there any benefit you can highlight maybe? (Please let me know if I can give more details).

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  • $\begingroup$ What are you trying to optimize? What is the role of the Riemann metric? The Fermat principle works on smooth manifold. If $x_0$ is a local minimum of $f$ then $x_0$ is a critical point of $f$, i.e., $df(x_0)=0$. Also at a critical point there is a well defined concept of Hessian, and if this Hessian is positive definite then that critical point is a local min. $\endgroup$
    – Liviu Nicolaescu
    Commented Jun 4, 2020 at 11:09
  • $\begingroup$ @LiviuNicolaescu As an example I'm trying to optimize curvature, I have this manifold and I want to find the point whose curvature is maximum (or a local optimizer). $\endgroup$
    – user8469759
    Commented Jun 4, 2020 at 12:18
  • $\begingroup$ Here again, there are many flavors of curvature: the curvature tensor, the sectional curvatures, the Ricci curvature and the scalar curvature. Only the last is a scalar quantity and, usually, only scalar quantities can be optimized. Without any assumption on the manifold or the quantity you cannot expect a precise answer. $\endgroup$
    – Liviu Nicolaescu
    Commented Jun 4, 2020 at 12:44
  • $\begingroup$ And again, given my assumption (i.e. $f : \mathcal{M} \to \mathbb{R}$) in your list you can pick the scalar curvature. I think I've given already a stricter assumption already, i.e. regular submanifolds in $\mathbb{R}^n$, or isn't this enough? $\endgroup$
    – user8469759
    Commented Jun 4, 2020 at 13:23
  • $\begingroup$ Regular submanifold is no assumption at all since by Whitney's embedding says that manifold can be properly embedded as a regular submanifold. Is this manifold compact? If it is not compact, then maybe there is no absolute maximum or there could be many local maxima and no absolute maxiumum, or there could be an absolute maximum. A special case of your question is: given a function $f:\mathbb{R}\to\mathbb{R}$ find its maximum. possible answers to this question are all possible in the more general case, and a bit more. $\endgroup$
    – Liviu Nicolaescu
    Commented Jun 4, 2020 at 14:26

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