With concave objective and convex constraint, are points satisfying the KKT conditions global or local maxima?

Question

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.

Answer 1

Since $f(x)$ is concave and $g(x)$ is convex, any local maximum of $f(x)$ in the region $\{ x : g_i(x) \leq 0 \}$ is automatically a global maximum of $f(x)$ subject to $g_i(x) \leq 0$.

Furthermore, if $f(x)$ is strictly concave, then the global maximum of $f(x)$ subject to $g_i(x) \leq 0$ is guaranteed to be unique (if it exists).

See this Wikipedia article.

Answer 2

I'm very sorry if my answer was misleading and I think it is entirely my fault for not making my point clear enough. I think I need a little bit more space so I have to "answer my own question".

The point I was trying to make is the following: We have a maximization problem and the following conditions are satisfied:

(1) $f(x)$ is concave

(2) $g(x)$ is convex

(3) the point $x^*$ is a solution to the maximization problem and satisfies the Karush-Kuhn-Tucker conditions (the one you mentioned in your comment with $∇f=λi∇gi∇f=λi∇gi , gi≤0gi≤0, λi≥0λi≥0, giλi=0giλi=0 $)

then $x^*$ gives a global maximum for $f(x)$ and is therefore the optimal (and best) solution for the optimization problem. (This is exactly the statement made by Chiang and Wainwright, p. 424 in "Fundamental Methods of Mathematical Economics" (I was trying to find an online version of the book but couldn't find one)). They go on and say that "(1), (2) and (3) are sufficient to say that $x^*$ is an optimal solution"

The problem is that even when $f(x)$ is of higher order (and has maybe more than one solution) how do I know that the value $x^*$ is the optimal solution for my maximization problem and gives a global maximum?

About your last comment on "really" understanding global and local optima: A local maximum for $f(x^*)$ is the highest value for $f$ in the immediate neighborhood of $x^*$. But I can't be sure if there is a value for $x$ which is far away from $x^*$ that gives a higher value for $f$. On the other hand a global maximum is the highest value for $f$ in the entire range or domain of $x^*$ and there is no value for $x$ that gives e higher $f$. This does not necessarily mean that $x^*$ is unique.