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How to get the derivative of this convex quadratic

  1. Oct 17, 2011 #1
    [itex]\frac{d}{dx}f(x)=\frac{d}{dx}[ \frac{1}{2}x_{}^{T}Qx-b_{}^{T}x][/itex]

    how to get this derivative, what is the answer? is there textbook describe it?
     
  2. jcsd
  3. Oct 17, 2011 #2

    Bacle2

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  4. Oct 23, 2011 #3
    This equation is analogous to a second degree polynomial, so I think the derivative is simply Qx - b.
     
  5. Oct 24, 2011 #4

    HallsofIvy

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    Not quite. Because multiplication of matrices is not commutative, taking the derivative you have to "symmetrize" Q- the derivative is [itex][(Q+Q^T)/2]x- b[/itex].

    Look at the simple two dimensional situation:
    [tex]\frac{1}{2}\begin{bmatrix}x & y \end{bmatrix}\begin{bmatrix}a & b \\ c & d \end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}- \begin{bmatrix}e & f\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}[/tex]
    [tex]= \frac{1}{2}ax^2+ \frac{b+ c}{2}xy+ \frac{1}{2}dy^2- ex- ey[/tex]

    The "derivative with respect to x", since x is a vector, is the gradient:
    [tex]\begin{bmatrix}ax+ \frac{b+c}{2}y- e & dy+ \frac{b+c}{2}x- f\end{bmatrix}[/tex]
    [tex]= \begin{bmatrix}a & \frac{b+c}{2} \\ \frac{b+c}{2} & d\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}- \begin{bmatrix}e & f\end{bmatrix}[/tex]
     
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