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    Duality and Saddle Points

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    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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    https://brainmass.com/math/graphs-and-functions/duality-saddle-points-16444

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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 Summary

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

    $2.49

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