Duality and Saddle Points
Please see the attached file for the fully formatted problem.
I am working on a way to find the minimum of a function J(Y) with the constraint set
C = {X E R^N such that gt(x) =<0 Vi E [1,n]}
Let L(Y, mu) = J(Y) + SIGMA m --> i = 1 muigi(Y) be the lagrangean of the problem.
I am having trouble proving the following theorem :
If is a saddle point of L, then and X minimises J on C.
Reciprocal:
If J and the gi are convex and differentiable in X, then for every solution X, there exists such that is a saddle point of the lagrangean L.
I would be grateful for a full proof of both the first and second parts of this theorem, in particular for the reciprocal.
Attachments
Solution Preview
Please see the attached file for the complete solution.
Thanks for using BrainMass.
Duality and saddle points
I am working on a way to find the minimum of a function J (Y) with the constraint set
Let be the langragean of the problem.
I am having trouble proving the following theorem :
If is a saddle point of L, then and X minimises J on C.
Reciprocal:
If J and the gi are convex and differentiable in X, then for every solution X( I think it should be an optimal solution) , there exists such that is a saddle point of the langrangean L.
Proof. First we need to write the primal problem and its Lagrangian Dual problem.
Primal Problem ...
Solution Summary
A theorem involving a Lagrangean and saddle points is proven. The solution is detailed and well presented.