Looking for a very particular kind of non-convex functions

Question

I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously,

1. It should be at least twice differentiable.

2. It should have a unique global minima at the origin.

3. It should have a few local minima which are not global. (preferably symmetrically positioned around the global-minima/origin) If it has saddle points then thats okay too.

4. The norm of the gradients of the function be bounded by some constant.

5. It should be gradient Lipschitz.

Feel free to let me know if this is not an appropriate question for MO! Just that I tried quite a bit but couldn't come up with one such function!

Answer 1

For $\lambda\in\mathbf{R}$, $a,x\in\mathbf{R}^n$, let $$f_{\lambda;a}(x) = -\exp(-\lambda\|x-a\|^2)$$ Then linear combinations of $f_{\lambda;a}$ can satisfy these conditions, where $a$ ranges over your chosen minima. E.g., if $n=2$, $$f_{6;(0,0)}+ f_{1;(1,0)}+ f_{1;(0,1)}$$ will have a global minimum at the origin and two symmetric local minima, as in these plots and computations.

Answer 2

I think you can start with a convex function with unique global minimum in zero with all extra properties you want to add on the gradient. Let call the set of theses functions $\mathcal F$. Then for $f,g\in \mathcal{F}$ build $f + \alpha (1 - cos(g)) $ where $\alpha$ is a real parameter

An example

Answer 3

$$f(x,y)~=~ \frac{-1}{1+ x^2 + y^2}-\cos (a\,x) \,\cos (b\,y)\,e^{-\frac{x^2 + y^2}{c^2}}$$ for $a,b,c \in \mathbb{R}^+$, where $c $ is sufficiently large.