I am looking at SVM, and reviewing Lagrange multiplier. Let's say with the constraint function $$g(x,y) = x^2 + y^2 -1 > 0, $$ I am maximizing $$f(x,y) = x + y -1 .$$ Intuitively, since the constraint function does not provide a finite constrained region, there is no solution, and $f(x,y)$ can infinitely increase.

However, if I just plainly proceed with Lagrange multiplier calculation, $$L(x,\lambda) = (x + y -1) +\lambda(x^2 + y^2 -1)$$ I get the solution: $$y,x = -\sqrt{2}/2, \lambda = 1/\sqrt{2}$$ And this still satisfies KKT conditions although the solution is wrong. $$g(x) => 0 , \lambda => 0, \lambda g(x) = 0.$$

I expected that I would have some result that conflicted KKT conditions, but nothing was violated.

Does this mean that Lagrange multiplier under KKT conditions can produce a solution that cannot be solved, without giving any indication of intractability?

If functions are complicated, what's the systematic way to confirm that the optimization with inequality constraint functions is solvable using Lagrange multiplier?