Let a function $f:X\rightarrow \mathbb{R}^n$, where $X \subseteq \mathbb{R}^n$, satisfy the Lipschitz continuity condition
$$ \|f(x) - f(y) \| \leq L \|x-y\|, \quad \forall x,y\in X $$
where $L\geq 0$ is the Lipschitz constant of $f$ and $\|\cdot\|$ is, for instance, a Euclidean norm. If $f$ is differentiable, it is well-known that
$$ L = \max_{x\in X} \sigma_{\max}\left(\frac{\partial f}{\partial x}\right) $$
where $\sigma_{\max}(F)$ denotes the maximum singular value of a matrix $F$. This can be solved using any nonlinear numerical optimization solver.
I recently saw a generalized Lipschitz condition in this paper, which is
$$ (f(x)-f(y))^T Q (f(x)-f(y)) \leq (x-y)^T R (x-y), \quad \forall x,y\in X $$
where $Q = Q^T > 0$ and $R = R^T > 0$ are some positive definite matrices.
Questions:
Given $f$, is there a way to compute the matrices $Q > 0$ and $R>0$?
Given $f$ and $Q > 0$, is there a way to compute the matrix $R > 0$?