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Duality and Saddle Points

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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.

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Solution Summary

A theorem involving a Lagrangean and saddle points is proven. The solution is detailed and well presented.

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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 provided by:
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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