Let us say I am supposed to solve the convex optimality problem in the following form. $$\min f(x)\\ \quad\quad\quad g_{j}(x)\le 0,\forall j\in J\\ \quad\quad\quad h_{k}(x)=0,\forall k \in K. $$ I will solve it using lagrange function and I will obtain some KKT points.
$1.$ If I get only one KKT point I can say that it is a global minimum because for a convex problem KKT conditions are not only necessary but also sufficient conditions not only for a local optimality but also for a global optimality.
$2.$ If I get two or more KKT points I can say that all of them are global minimum, because the points are different, but they have the same optimal value.
$3.$ What about if I get no KKT point? I guess I cannot say that the problem has no solution but I have to use some constraint qualification. However which one should I use? And if the constraint qualification is satisfied, what it means? How to find solution using the constraint qualification?
