Sometimes, the constraint is not a compact set. As a result, the local minimum may not be global.
For example, $ f=x^2+y^3$ subject to constraint $ x+y=4/3$. Using Lagrange multiplier method, I calculated local minimum at $(x,y)=(\frac23,\frac23)$. But I don't know what to do next.
I cheated a bit and looked at Wolfram Alpha's plot that shows that there is no global minimum subject to the constraint.
Is there any way to get this result (no global minima) without graphing?