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Math Help - A Question on Constrained Optimization

  1. #1
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    A Question on Constrained Optimization

    I have a (maybe trivial?) question on constrained optimization. Assume that I have the following maximization problem:

    <br />
\max_{x,y} -2x^{2}+3xy-3y^{2} -10x-10y,<br />
    subject to
    <br />
-2x^{2}+3xy-10x \geq 0.<br />

    I setup the Lagrangian and I get the following first order conditions with the lagrange multiplier \lambda.
    <br />
-4x + 3y - 10 + \lambda (-4x + 3y -10) = 0,<br />
    <br />
3x -6y - 10 + \lambda (3x) = 0. <br />

    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?

    Please help!
    Thank you
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Justin Lo View Post
    I have a (maybe trivial?) question on constrained optimization. Assume that I have the following maximization problem:

    <br />
\max_{x,y} -2x^{2}+3xy-3y^{2} -10x-10y,<br />
    subject to
    <br />
-2x^{2}+3xy-10x \geq 0.<br />

    I setup the Lagrangian and I get the following first order conditions with the lagrange multiplier \lambda.
    <br />
-4x + 3y - 10 + \lambda (-4x + 3y -10) = 0,<br />
    <br />
3x -6y - 10 + \lambda (3x) = 0. <br />

    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?

    Please help!
    Thank you
    The unconstrained maxima of the objective is in the interior of the feasible region so the constraint is not binding. That is at the maximum of the unconstrained problem:

    <br />
-2x^{2}+3xy-10x > 0.<br />



    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    The unconstrained maxima of the objective is in the interior of the feasible region so the constraint is not binding. That is at the maximum of the unconstrained problem:

    <br />
-2x^{2}+3xy-10x > 0.<br />



    CB
    Well, does that mean I do not need to do a feasibility check of the unconstrained maxima?

    Because to me, it feels like we can also say that the unconstrained maxima of the objective is outside of the feasible region, right?
    <br />
-2x^{2}+3xy-10x < 0<br />

    Also, if I changed the constraint to
    <br />
-2x^{2}+3xy-10x \geq 300.<br />
    the first order conditions will still look the same, but clearly, the optimum has violated the constraint.

    Please give me some pointers
    Last edited by Justin Lo; December 3rd 2009 at 09:13 AM.
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