Optimizate objective function in matrix

1.Min12xTAx+xTb+cMin \frac{1}{2}x^TAx+x^Tb+c
Equals to solve:

12(A+AT)x=−b

\frac{1}{2}(A+A^T)x = -b

2.Min12xTAx+xTb+cMin\frac{1}{2}x^TAx+x^Tb+c
subject to:
A′x=b′A'x=b'
Equals to solve:

⎡⎣12(A+AT)A′A′T0⎤⎦[xλ]=[−bb′]

\begin{bmatrix} \frac{1}{2}(A+A^T) &A'^T\\ A' &0 \end{bmatrix}\begin{bmatrix} x \\ \lambda \end{bmatrix}=\begin{bmatrix} -b \\ b' \end{bmatrix}

whose application can be seen in http://blog.csdn.net/seamanj/article/details/51769868

3.Min∥Ax+b∥2Min\lVert Ax + b\rVert^2
Equals to solve:

Ax=−b

Ax = -b

4.Min∑i=1m∥Aixi+bi∥2Min\displaystyle\sum_{i=1}^{m}\lVert A_ix_i + b_i\rVert^2
Equals to sovle:

Aixi=−bi,∀i=1,2,⋯,m

A_ix_i=-b_i, \forall i=1,2,\cdots,m

⎡⎣⎢⎢⎢⎢⎢A1A2⋱Am⎤⎦⎥⎥⎥⎥⎥⎡⎣⎢⎢⎢⎢x1x2⋮xm⎤⎦⎥⎥⎥⎥=⎡⎣⎢⎢⎢⎢−b1−b2⋮−bm⎤⎦⎥⎥⎥⎥

\begin{bmatrix} A_1 &&&\\&A_2&&\\&&\ddots&\\&&&A_m \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix}=\begin{bmatrix} -b_1 \\ -b_2 \\ \vdots \\ -b_m \end{bmatrix}

note:∥v∥≥0\lVert v\rVert \geq0

5.Min∑i=1m∥Rpi−qi∥2Min\displaystyle\sum_{i=1}^{m}\lVert Rp_i - q_i\rVert^2
where RR is a rotation matrix
Equals to solve:
the orthogonal part in polar decomposition of the matrix qipTiq_ip_i^T

[Meshless Deformations Based on Shape Matching]
[Example-Based skeleton Extraction]

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Optimizate objective function in matrix

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