Can we find vectors which which maximise one function while simultaneously minimise another function?

Question

For two known matrices $H_p$ and $H_s$, both of dimensions $N \times N$, Can we find two $N \times 1 $ vectors $X_0$ and $X_1$, such that $\| H_p (X_1-X_0)^2 \|$ is maximised and simultaneously $\lVert H_s (X_1-X_0)^2 \rVert$ is minimised? Known constraints are N= total sum of elements of vectors $X_0$ and $X_1$ ($1^TX_0 + 1^TX_1=P$, $1^T$ is row vector of all 1's), (and $P_0$= sum of elements of $X_0$ and $P_1$ = sum of elements of $X_1$), and all elements of $X_1$ and $X_0$ are assumed to be positive. I am also interested in knowing how the solution would change if some thresholds (ex. $\epsilon$) is put to the problem (ex. $\| H_p (X_1-X_0)^2 \| \geq \epsilon ?$)