Say I have two optimization problems A and B.
If an optimal solution of A is feasible for b and vice versa:
Q 1) Is every optimal solution for A an optimal solution of B (and vice versa)?
Q 2) If the answer to Q1 is false, then is there at least one optimal solution of A that is also an optimal solution of B (and vice-versa)?
No to both questions, consider these problems:
A: $\min_x \{ x : 0 \leq x \leq 1\}$
B: $\min_x \{ -x : 0 \leq x \leq 1\}$