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How could I solve the following optimization problem using MATLAB or an other way?

Given ${E}^{1}, {A}^{21}, {A}^{22}, {C}^{1}, {A}^{12}$

$ \underset{{C}^{2}, {E}^{2}}{min} {\left \| {C}^{2}{E}^{1} - {A}^{21} \right \|}_{F}^{2} + {\left \| {C}^{2}{E}^{2} - {A}^{22} \right \|}_{F}^{2} + {\left \| {C}^{1}{E}^{2} - {A}^{12} \right \|}_{F}^{2} $

I know the approximated linear solution, assuming $ {A}^{22} $ is negligible is given by:

$ {C}^{2} = {A}^{21} {{E}^{1}}^{T} {\left( {E}^{1} {{E}^{1}}^{T} \right)}^{-1}, {E}^{2} = {\left( {\begin{bmatrix} {C}^{1}\\ {C}^{2} \end{bmatrix}}^{T} \begin{bmatrix} {C}^{1}\\ {C}^{2} \end{bmatrix} \right)}^{-1} {\begin{bmatrix} {C}^{1}\\ {C}^{2} \end{bmatrix}}^{T} \begin{bmatrix} {A}^{12}\\ {A}^{22} \end{bmatrix} $

Yet it would be nice to know how to solve it accurately. Thanks.

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    Derivative o$f$ a $f$unction always lies in the domain o$f$ the function. This means that the derivative of your objective has to be a matrix. If you want to translate this into a vector, then rewrite your objective function in terms of the individual matrix components. Then take derivative w.r.t each component.2012-08-09

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Since your problem is unconstrained you can us fminunc to solve it in MATLAB (given an initial starting point $x_0$). Perhaps, you can use your approximate solution as $x_0$.

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    Yet it is a function of 2 variables - C2 and E2.2012-08-05
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OK, it turns out the best way to solve it is using MATLAB's 'lsnonlin' function. Now, I just need to compute the Jacobain of the function relative to C2, E2 and the Frobenius Norm.