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Optimization inequality constraint

  1. Mar 5, 2008 #1
    1. The problem statement, all variables and given/known data

    Minimize 2x²+2y²-2xy-9y subject
    4x + 3y =,< 10 ,
    y - 4x² =,< -2
    x >,= 0
    and y >,= 0.


    I don't undersant this:

    "This equation has no nonnegative root, which contradicts a nonnegativity
    constraint."
    and how we solve
    -16x² + 2x + 17 + h2 = 0
     

    Attached Files:

  2. jcsd
  3. Mar 5, 2008 #2

    HallsofIvy

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    Do you know how to complete the square? That would seem to me to be the best way to solve that equation. (The "h2" here is supposed to be [itex]\lambda_2[/itex], one of the Lagrange multipliers.)
     
    Last edited: Mar 5, 2008
  4. Mar 6, 2008 #3
    complete the square on -16x² + 2x + 17 + h2 = 0 or -16x² + 2x + 17 = 0, because i found positive and negative root [ x=~+-1] on -16x² + 2x + 17=0 and i dont know how solve with λ2.
     
  5. Mar 6, 2008 #4

    HallsofIvy

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    Okay, since you are basically saying you do NOT know how to complete the square,
    [tex]-16x^2+ 2x+ 17= -16(x^2- (1/8)x)= -16(x^2- (1/8)x+ (1/256)- 256)+ 17= -16(x- 1/16)^2+ 17+ 1/16[/tex].
    So [itex]]-16x^2+ 2x+ 17+ \lambda_2[/itex] can only be equal to 0 if [itex]\lambda+ 17+ 1/16>0[/itex] or [itex]\lambda< -(17+ 1/16)[/itex]. That's what violates the "nonegativity constraint", that [itex]\lambda[/itex] can't be negative.
     
  6. Mar 7, 2008 #5
    why
    λ + 17 + 1/16 > 0?
     
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