I am trying to perform a Lagrange constraint problem for a simple set of linear equations (I realize this can be solved by substitution) but I'm curious why/how the Lagrange method is failing and I'm getting a conflicting Lagrange multiplier (terminology?). In my example, x and y are constants, S represents the constraint, Psi represents the function.
$$
\Psi(a,b)=(a+b)*x+b*y
$$
$$
S(a,b)=a+b=0.5
$$
$$
\nabla\Psi(a,b)=\lambda\nabla S(a,b)
$$
$$
\frac{\partial\Psi}{\partial a}=\lambda\frac{\partial S}{\partial a}
$$
$$
x =\lambda*1
$$
$$
\frac{\partial\Psi}{\partial b}=\lambda\frac{\partial S}{\partial b}
$$
$$
x+y =\lambda*1
$$
Best Answer
Part of the problem is that your constraint is a line, this isn't a compact set (closed and bounded), so you aren't guaranteed that your function will have an extreme value on the set.
Furthermore your function, being linear, is just going to grow/decay monotonically as you traverse the constraint. So there won't be any critical points for the method to detect.
Another way to think of this,
The gradient of your function is a constant vector, i.e., independent of $a$ and $b$. The gradient of the constraint function is also constant. The method of Lagrange multipliers answers the question "when are these vectors parallel". If the two constant vectors are not parallel then the method cannot give you an answer.