I am studying numerical methods and NLP. I started with gradient based methods, newton methods and KKT conditions. I found the following sentence:
- A local minimum is found by solving KKT conditions, using iterative techniques based on Newton’s method.
In basic courses we used KKT conditions to calculate the minimum. Does the sentence above mean, that you calculate the "candidate - point" through numerical methods (gradient, Newton, quasi netwon..) and then you check if this particular point satisfy KKT?
Example:
$min: f(a,b) = 3a + 4b$
subject to
$k_1 (a,b) = -a^2 -b^2+2b \le 0 $
$k_2 (a,b) = a^2+b^2-4 \le 0 $
The KKT conditions are:
$3-2\lambda_1a+2\lambda_2 a = 0 $
$4+2\lambda_1(2-2b)+2\lambda_2b = 0 $
$\lambda_1(-a^2 -b^2+2b) = 0 $
$\lambda_1(a^2+b^2-4) = 0 $
$\lambda_1, \lambda_2 \ge 0 $
In basic courses we learned how to calculate based on this KKT conditions the KKT-points. But I am not sure, how to use numerical methods (Newton, steepest descent,...) to solve this KKT conditions (this are equations, there is no longer a $min ...$)?