Optimisation along a curve on a surface

[Vasileios]

Hi everyone,

First of all I am not sure if I have chosen the right category for this posting but this looked the most reasonable out.
I have a problem that I would like to solve but I am not sure where to look for answers. It seems like something other people might have worked with before but I am not sure what these types of problems are called.


(Please do excuse the abuse of any notation since I am only learning now about diff. geometry and such).

Basically it is an optimization problem. I have a 2d surface y= F(σ,κ) in ℝ3 and on that surface I have a curve γ. I do not have a parameterisation for the curve, but it is continuous and I can sample it at any point P (see illustration).
http://img694.imageshack.us/img694/9179/surfacekq.png



My task is:

Given a random initial point P0 on γ, to find the minimum of F along γ.
I can only move along this (blue) curve γ, backwards or forwards. So the only thing I can adjust is my step length λ.

Now of course I can use various heuristics to solve this, but there must be a more elegant way to solve this problem and someone must have seen these class of problems before.

I do have a closed form analytic expression for the whole F(σ,κ) as well as the first and second partial derivatives at each point. Also note that I have expressions for geodesics, arc elements, Riemmanian metric etc.

Can someone point me to the right direction?

Many thanks
V.

 

[Vasileios]

I do have a closed form analytic expression for the whole F(σ,κ) as well as the first and second partial derivatives at each point. Also note that I have expressions for geodesics, arc elements, Riemmanian metric etc.

Hi just a further clarification.
The actual objective surface F is defined on a Riemmanian manifold W of constant Gaussian curvature. So it is more like in the picture
http://img840.imageshack.us/img840/2443/manifoldm.jpg But I can still only move along a line. I cannot chose a direction to get to the global optimum of F, bhttps://www.physicsforums.com/editpost.php?do=editpost&p=3737946ut only the "global optimum" of the curve γ.I guess I could try to look at this problem in the easier case for a Euclidean space and then try the manifold approach. But still, the fact that I can only move on a pre-defined curve gives me a bit of trouble.