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Math Help - How to get the solution of this simple optimization problem?

  1. #1
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    Question How to get the solution of this simple optimization problem?

    How to get the solution of this simple optimization problem?

    <br />
minimize~~\Phi = x_1^2 + x_2^2<br />
    <br />
subject~to~~x_1 +2x_2 = 5<br />

    I konw we may use Moore-Penrose inverse like,

    <br />
Ax = b => x = (A^TA)^{-1}A^Tb<br />

    but what are A and x now?

    if  A = (1~~2) and A^T = (1~~2)^T then there is no inverse of  A^TA. I am quite confused how to formulate these A and x to do the matrix multiplication.


    Thanks a lot
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  2. #2
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    Quote Originally Posted by ggyyree View Post
    How to get the solution of this simple optimization problem?

    <br />
minimize~~\Phi = x_1^2 + x_2^2<br />
    <br />
subject~to~~x_1 +2x_2 = 5<br />

    I konw we may use Moore-Penrose inverse like,

    <br />
Ax = b => x = (A^TA)^{-1}A^Tb<br />

    but what are A and x now?

    if  A = (1~~2) and A^T = (1~~2)^T then there is no inverse of  A^TA. I am quite confused how to formulate these A and x to do the matrix multiplication.


    Thanks a lot
    "Moore-Penrose" is much too advanced for me ("Moore" isn't so bad but I cringe when I hear "Penrose") so I would do it with Lagrange multipliers (I much better with "Lagrange").

    You want to minimize F(x,y)= x^2+ y^2 subject to the constraint G(x,y)= x+ 2y= 5. The gradients are \nabla F= 2x\vec{i}+ 2y\vec{j} and \nabla G= \vec{i}+ 2\vec{j}. Now we need to find (x,y) so that 2x\vec{i}+ 2y\vec{j}= \lambda(\vec{i}+ 2\vec{j}). That means we must have 2x= \lambda and 2y= 2\lambda. Dividing the second equation by the first give \frac{y}{x}= 2 or y= 2x. Putting that into the constraint x+ 2y= 5, x+ 2(2y)= x+ 4x= 5x= 5 or x= 1, y= 2.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by ggyyree View Post
    How to get the solution of this simple optimization problem?

    <br />
minimize~~\Phi = x_1^2 + x_2^2<br />
    <br />
subject~to~~x_1 +2x_2 = 5<br />
    Why not solve the constraint for x_1 in terms of x_2 and substitute into the objective to reduce this to a 1-D optimisation??

    x_1=5-2x_2, then:

     <br />
\Phi=5x_2^2-20x_2+25<br />

    CB
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