- $f(x,y)=x^2-y^2$ subject to single constraint $g(x,y)=1-x-y=0$
For this question, I understand that the Constraint Qualification holds, since, rank of $D(g(x,y))=1$ everywhere. Solving Lagrange would suggest that no critical points exist.
For this example, my book states, "Since the constraint qualification holds everywhere, it must be the case that global maxima and global minima fail to exist in problem".
I cannot seem to conceptually grasp the meaning for this statement. In context of this, how are we supposed to find global maxima or minima? Please help me understand the significance of constraint qualification for global maxima or minima.
In your example, the set defined by the constraint is unbounded, so is possible that no global extremum exists. Taking $y = 1 - x$, then $$f(x,y) = x^2 - (1 - x)^2 = 2x - 1$$ and obviously this is the case.