Function of symmetry character for comparing two optimization methods

[i_a_n]

First refer to this question:Is this function "$h$" symmetric of the plane $x=y$? - Mathematics Stack ExchangeThe main problem is to find an appropriate (objective) function (a curve) to compare the implicit filtering method and Nelder-Mead method for optimization. The backgrounds of these methods are not important here. But you need to know according to the features of the two methods and in order to compare (by writing computer codes to verify the process of finding the minimum), the function is better to be this kind:It is very smooth and possibly not differentiable but only Lipschitz continuous are the methods comparable.So it should be like the kind of function at the page first I let you refer to. A piecewise function about the symmetric plane or axis. (Like the example $h=\left\{\begin{matrix}
x^2-y^2, x<y\\ y^2-x^2,x\geq y
\end{matrix}\right.$ (for $0\leq x,y\leq1$),$\frac{\partial f}{\partial x}=2x$ when $x<y$ and $\frac{\partial f}{\partial x}=-2x$ when $x\geq y$ so $\lim_{x\rightarrow \frac{1}{2}^-}=1$ but $\lim_{x\rightarrow \frac{1}{2}^+}=-1$ ).
Also, for this function we are finding, **the minimum should appear at the symmetry plane(axis)**, the discontinuity places. Like, a function that is symmetric to the plane $y=x$ and the minimum also appears at $y=x$ is fine. (Notice, the minimum of the example function at the page first I let you refer to is not at $y=x$.) But it's even perfect if we can find a function that is symmetric about a parabola, like $y=x^2$. (And my question here is how to find such functions which are symmetric about a parabola?)What are the possible functions satisfying those? How you construct one?

 

[Aaron Bex]

One possible function that is symmetric about a parabola is the following:

$f(x,y)=\begin{cases}
x^2-y^2 & x<y \\
y^2-x^2 & x\geq y
\end{cases}
$

This function is continuous and differentiable, and has a minimum at the symmetry plane $y=x^2$.