I am attempting to solve the following optimization problem with the following three dimensional function $D(x,y,z)$:
Find the point $(x,y,z)$ such that the function $D(x,y,z)$ is a maximum and that the gradient $\mid \nabla D(x,y,z) \mid$ around a given distance $r$ around that point is a minimum.
What is the best way to approach this problem analytically? Am looking at Hessian methods but would like to find an explicit solution as well. I am essentially trying to find a point in space such that the value of the function is maximal and the magnitude of the gradient around a specified radius is minimal.
This question is posed within the field of image processing. Specifically, we are looking to measure the value of a 2D slice of $D$ in the plane where $D$ is maximum. We want to avoid large fall-off of $D$ as well, and that is why we are looking at the gradient. If the modulation of the gradient is too large, then we will look for another plane to measure at.
Thank you in advance.
The problem with this question (as it is currently posed) is that the gradient looks the same for all local optima.
Claim: If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is differentiable and $x \in \mathbb{R}^n$ is a local maximum, then $\nabla f(x) = 0$.
In other words, you can't really distinguish between the gradients of various local optima.