In post-optimal analysis, when I already have an optimal solution (unique solution). I can add a new constraint and obtain multiple optimal solutions, but only if the new constraint eliminates the previous optimal solution.
My question is: Can multiple optimal solutions be obtained by conserving the previous optimal solution?
The intuition says that the new constraint has to define the optimal point, making the optimal solution degenerate and thus the gradient perpendicular to the newly added constraint. However, I tried to find examples in $\mathbb R^2$ and $\mathbb R^3$ without success, I believe the answer to my question is no, but I do not see a way to prove it formally (with reduced costs for example).
I am new in operations research, so any help to satisfy my curiosity is welcome.
The answer is no; adding a new constraint cannot make a unique optimizer non-unique without removing the original optimal point.
The reason is that adding more constraints will only reduce the size of the feasible set (the set of points that satisfy all of the constraints). Suppose we're trying to minimize the function in question. If we're looking at, say, a minimization problem, then the original optimizer already has a lower function value than every other point in the feasible set before adding more constraints. As long as the minimizing point isn't removed, it will also have a lower function value than any other point in the feasible set after adding more constraints. Since adding another constraint can't add any new points to the feasible set, the original optimizer thus stays as a unique optimizer.
It also happens that for this property, it doesn't actually matter that the optimization problem is a linear programming problem; the exact same thing will happen with any function optimization problem where you have a unique optimizer.