Solving Optimisation Problem with Rank Constraint in MATLAB

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rihab
rihab 2015-11-20
评论： Torsten 2015-11-23
I have a typical least squares problem, i.e I have to find the value of x that minimizes norm of C∗x(:)−d. C is 180x16 matrix(C is rank deficient,i.e rank(C)=7), x is 4x4 matrix & d is 180x1 vector. However, I have a constraint that rank(x)=1. If x was a 16x1 vector and didn't have rank constraint, this problem could be easily solved by using y = pinv(C)*d in MATLAB. But since x is a matrix and has rank constraint, I am not able to proceed further. I would be grateful if someone provides me hint or suggestion to tackle this problem.

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Torsten
Torsten 2015-11-20
1. Solve y=pinv(C)*d
2. Determine the best rank-1 - approximation x to y as discussed in the previous thread:
http://www.mathworks.com/matlabcentral/answers/255420-representing-a-matrix-as-a-product-of-two-matrices-in-matlab
My guess is that x solves your original problem, but I'm not 100% certain.
Best wishes
Torsten.
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Torsten
Torsten 2015-11-23
In the last line of the code, you will have to replace
fun = abs(residuals)
by
fun = residuals'*residuals
or
fun = norm(residuals)
Best wishes
Torsten.

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