If $A \text{ and } B $ are two equivalent optimization problem. The $\epsilon-$optimal value, $A_\epsilon^*$, of $A$ was obtained, can we claim that $| B^* - A_\epsilon^* | = |A^* - A_\epsilon^*|$, or more generally $dist(B^* , A_\epsilon^*) = dist(A^* , A_\epsilon^*)$ where $B^*$ is the optimal value of problem $B$ and $A^* = B^*$ (since they are equivalent) is the optimal value of problem $A$, and $dist$ is any distance measure?
To narrow the question, we may assume that both optimization problems are convex.