Upper bound on difference of function value

Question

I have the function of form $f(X) = -2A^TXZ + 4XZX^TXZ$ where $Z,A,X,Y \in R^{n \times n}$. I wanted to get an upper bound on $\|f(X_1)-f(X_2) \|^2_F$ such that $\|f(X_1)-f(X_2) \|^2_F \leq L \|X_1 - X_2 \|^2_F$ where $L$ is a constant and $n\geq 2$. I am able to get bound if I just consider the first term of the $f$ by using sub-multiplicative property of frobenious norm, but I am not able to get the bound if I include the second term.

Answer 1

In general, there is no such bound even for $n=1$ (or, if you insist on $n=2$, for diagonal matrices), because the second term is quadratic and a function $x^2$ is not Lipschitz.