A bit too long for a comment: Let's consider a optimization problem of the form $$ \min_x f(x)\quad \text{s.t.}\quad x\in C. $$

If we now phrase "symmetry" bit more abstract by the saying that you have a group $G$ acting on the set $C$ such that the objective is invariant under group action, i.e. for $g\in G$ we have $f(gx) = f(x)$ then you see that the set of minimizers is also invariant under the group action.

A bit too long for a comment: Let's consider a optimization problem of the form $$ \min_x f(x)\quad \text{s.t.}\quad x\in C. $$

If we now consider "symmetry" a bit more abstract by saying that you have a group $G$ acting on the set $C$ such that the objective is invariant under the group action, i.e. for $g\in G$ we have $f(gx) = f(x)$ then you see that the set of minimizers is also invariant under the group action.