Optimizate objective function in matrix

1.Min12xTAx+xTb+c

Equals to solve:

12(A+AT)x=−b

2.Min12xTAx+xTb+c

subject to:

A′x=b′

Equals to solve:

⎡⎣12(A+AT)A′A′T0⎤⎦[xλ]=[−bb′]

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

3.Min∥Ax+b∥2

Equals to solve:

Ax=−b

4.Min∑i=1m∥Aixi+bi∥2

Equals to sovle:

Aixi=−bi,∀i=1,2,⋯,m

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

note:∥v∥≥0

5.Min∑i=1m∥Rpi−qi∥2

where R is a rotation matrix

Equals to solve:

the orthogonal part in polar decomposition of the matrix qipTi

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