I have a general question about the Karush-Kuhn-Tucker-Theorem. Let's assume that we want to maximize the following funtion: $$f(x)$$ subject to some constraints $$g_i(x ) \leq 0$$ Using the objective function $f(x)$ and the constraints I get the following Lagrangian Function $$Z(x)=f(x)+\sum_i \lambda_i g_i(x)$$
Let's assume further that the function $Z(x)$ can not be solved algebraically because it's a function of higher order. Instead I'm using a program (e. g. gradient method) to determine a value for $x^*$ that maximizes $f(x)$ under the constraints $g_i(x)$. Let's assume also that all Karush-Kuhn-Tucker-conditions are satisfied ($f(x)$ is concave; $g_i(x)$ is convex....)
Now my question: Is $x^*$ a value that gives a global maximum for $f(x)$ or is $x^*$ just the value that gives a local maximum in the possible range under the constraints $g_i(x)$? And if the Karush-Kuhn-Tucker-Conditions are satisfied is $x^*$ the only value which I have to consider? Maybe I have to add that my main concern has to do with the higher order of my function $f(x)$ and therefore more than one optimal value for $x^*$ (but not under the constraints).
Thank you for your answer.