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.
© BrainMass Inc. brainmass.com February 24, 2021, 2:22 pm ad1c9bdddfhttps://brainmass.com/math/graphs-and-functions/duality-saddle-points-16444
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.