# BigPicture Lagrangian, KKT, Duality

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!