Tight upper bound: Given $x,y,z \in \mathbb{R}^n$; $\alpha, \beta> 0$, then $\alpha \| x-y\|_2^2 + \beta \| z - y \|_2^2\leq ?$

Solution 1:

If you meant to find the bound to hold for all possible value of $x,y,z,$, which would mean $\beta,\gamma$ would be independent of $x,y,z$, then the best you can do is only $\gamma = \alpha$ and $\mu = \beta,$ because you can take $x = z$ and write you write your inequality as: $$\|x-z\|^2\geq \dfrac{\alpha-\gamma}{\gamma}\|x-y\|^2+\dfrac{\beta - \mu}{\gamma}\|y-z\|^2 \,\,\,(1).$$

But $x,y,z$ are fixed, then $\gamma,\mu$ obviously must be chosen to be dependent on them to make sense. However, in that case the sense in which you can do "better" is a bit vague. You can get tighter $\gamma$ by by letting $\mu = \beta$ and then taking: $$\gamma = \dfrac{\alpha\|x-y\|^2}{\|x-y\|^2+\|x-z\|^2} < \alpha.$$ Or you can sacrifice $\gamma$ and then get a better $\mu$ and so on.

If you have some unified sense of cost that combines $C = C(\gamma,\mu),$ then you can turn this into an optimization problem: $$\min C(\gamma,\mu)\quad\text{subject to } (1)$$