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December 8th, 2009, 02:40 AM  #1 
Newbie Joined: Dec 2009 Posts: 1 Thanks: 0  Optimization Problem with Constraint
I have a (maybe trivial?) question on constrained optimization. Assume that I have the following maximization problem: \max_{x,y} 2x^{2}+3xy3y^{2} 10x10y, subject to 2x^{2}+3xy10x >= 0. I setup the Lagrangian and I get the following first order conditions with the lagrange multiplier \lambda. 4x + 3y  10 + \lambda (4x + 3y 10) = 0, 3x 6y  10 + \lambda (3x) = 0. By the Kuhn Tucker conditions, we know that if \lambda > 0, then the constraint is binding. However, from the first order conditions, we can see that the multiplier \lambda is negative! How can this be? Am I doing things wrong here? I know that this could mean that the constraint is always nonbinding. But what if we changed the constraint to 2x^{2}+3xy10x \geq 100. This clearly has an effect on the problem. So does this mean that the regularity condition (constraint qualification) is violated? Please help! Thank you 

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