BigPicture Lagrangian, KKT, Duality

Question

I have a question regarding the big-picture in the field: Lagrangian, Duality, KKT, sufficient, necessary conditions.

1) Duality is a concept: “The solution to the dual problem provides a lower bound to the solution of the primal”

2) The concept of Lagrangian is a strategy to find local maxima, minima with equality constraints.

3) KKT concept: Is the generalized idea of the Lagrangian – includes inequality constraints.

4) Lagrange Duality: Transfers any problem into a convace (the dual) problem --> uses the idea/proves of 1.

Question 1: The KKT and the Lagrange-Dualty concept use both the Lagrangian right? The Lagrange-Dualty: $ g = inf(x)> L(x,u,v)$

Question2: KKT: Regarding necessary and sofficient conditions: If I can prove, that the problem is convex - a KKT point = global optimum right? If the problem is not convex, i have to compare all KKT points and the lowest (in a min. problem) is the local, not a global minimum? Or do I have to prove if Hessian is positiv(semi) definit?

Thank you very much for your help!

Answer 1

Question 1: Yes you can sort of see KKT as an extension of Langrange multipliers if that helps.

Question 2: Nope. A point can be a KKT point without being optimal. We say that the KKT conditions are necessary but not always sufficient. You need some additional constraint qualifications to hold here, which are kind of boring and technical in nature and usually highly overkill. Convexity and the existance of some interior point is usually fine.