Thm [KKT conditions] Suppose the local minimizer $ { x ^ {\ast} }$ is also a regular point of $ { \widetilde {\mathscr {F}} (x ^ {\ast}) . }$ That is, the collection of gradients
What is the point of KKT conditions for constrained optimization? In other words, how is the best way to use them. I have seen examples in different contexts, but miss a short overview of the proce...
Since stationarity of $ (X', y_i')$ alone is sufficient for its equality-constrained problem, whereas inequality-constrained problems require all KKT conditions to be fulfilled, it is not surprising that fulfilling some of the KKT conditions for $ (X, y_i)$ does not imply fulfilling the condition for $ (X', y_i')$.
For a convex problem, all the KKT points automatically satisfy the saddle point conditions - that's for the same reason that critical points of a convex function are automatically global minimizers. Anything that satisfies the saddle point conditions, in any kind of problem, is a global minimum, and this we can prove by just a bit of messing around with inequalities.
My concern is how to state the KKT conditions, in general. That is, do I need to discern the set of active constraints ahead of time to setup the KKT conditions?
The KKT conditions are more restrictive and thus shrink the class of points (from those satisfying the Fritz John conditions) that must be tested for optimality. The additional restriction with KKT is that the Lagrange multiplier on the gradient of the objective function cannot be zero.
Is it necessary that a convex optimization problem will satisfy the regularity condition? I understand that it is not necessary for a convex optimization problem to satisfy the KKT condition. But if Slater's condition is satisfied, then KKT is satisfied. So what is the difference between Slater's condition and regularity condition?