KKT in a nutshell graphically

Question

Objective

Confirm if the understanding of KKT is correct or not. Seek for further explanation and confirmations on KKT.

Reposting original at Cross Validated as suggested to seek answers from Math side.

Background

Trying to understand KKT conditions, especially the complementary one, which always pops up out of blue in SVM articles. I do not need list of abstract formulas but do need a concrete, intuitive, and graphical explanation.

Question

If P, which minimizes the cost function f(X), is inside the constraint (g(P) >= 0), it is the solution. It seems KKT is not relevant in this case.

It seems KKT says if P is not inside the constraint, then the solution X should satisfy below in the picture. Is it KKT all about or do I miss other important aspects?

Other clarifications

• Should f(x) be convex for KKT to apply?
• Should g(x) be linear for KKT to apply?
• Should λ be necessary in λ*g(X) = 0? Why g(X) = 0 or g(Xi) = 0 is not enough?

References

• Lagrange Multipler KKT condition
• Does every gutter point in SVM have positive multiplier?
• http://fnorio.com/0136Lagrange_method_of_undetermined_multipliers/Lagrange_method_of_undetermined_multipliers.html