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I don't know if there exist a simple solution. But I have the following problem: I have a vector positive $m \in (\mathbb{R}^{+})^n$ and a set of vectors (which can have negative composants) $(v_1, v_2, \cdots, v_M)$ each of this vectors are in $\mathbb{R}^n$. All theses vectors are normalize to 1.

I want the linear combinaisaon of $v^i$ with $i\in [1,M]$ which its absolute value vectors is closest to $m$. ie I want to minimize this quantity:

$$ L(\beta^i) = \sum_{k=1}^{n} \left\Vert m_k - \left\Vert \sum_{i=1}^M \beta^i v^i_k \right\Vert \right\Vert $$

and get the $\beta^i$. This feel like an optimization probleme but I have no clue where to start to solve this problem (numerically in the general case)

sailx
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  • There are no restriction on the $\beta^i$, nor on the sign of $L$. Let $\beta^i\rightarrow\infty$ whereby $L\rightarrow-\infty$. Also, the vector structure of $m$ is lost when you sum the components. $m$ could be replaced by a scalar. – Asdf Sep 25 '18 at 20:45
  • @Asdf I guess $L$ is a function $\beta_i$ and not a constant that can arbitrarily tend to $-\infty$. – YukiJ Sep 26 '18 at 06:40
  • @Asdf sorry, i forgot to mention that. But I came up with the function to minimize with the probleme states above. I'm not looking for a solution where $L -> -\infty$ but rather a solution where $L-> 0$. For the vectors structure, the sum apply to both term so i don't understand why you say the vector structure is lost (I put parenthesis to be clearer) – sailx Sep 26 '18 at 07:37
  • You can write $L=\sum_{k=1}^n m_k-\sum_{k=1}^n|\sum_{i=1}^M\beta^iv_k^i|=x-\sum_{k=1}^n|\sum_{i=1}^M\beta^iv_k^i|$, where $x=\sum_{k=1}^n m_k$. From here it should be clear that the values of individual $m_k$ have no impact on $L$. – Asdf Sep 26 '18 at 11:54
  • Read about least squares problems on Wikipedia to get some context: https://en.wikipedia.org/wiki/Least_squares – Asdf Sep 26 '18 at 11:56
  • Ok, sorry, I didn't wrote the optimization probelm correctly. Sorry about that. Waht I want is construct beta such that the vectors $|\sum_{i=1}^M \beta^i v_k^i|$ is closeast to the vector $m$. I guess the function i really want to optimize is more something like $\sum_{k=1}^{n} (m_k - |\sum_{i=1}^M \beta^i v^i_k |)^2$ where I do care about the particularity of individual $m_\k$ – sailx Sep 26 '18 at 12:15
  • sorry for rejecting your edit. But i it was not exactly the right way. I have try to put it in the right form – sailx Sep 26 '18 at 17:46

1 Answers1

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Calling

$$ v^i = {v^i_1,\cdots,v^i_n}^{\top}\\ \beta = (\beta_1,\cdots,\beta_n)^{\top}\\ e_k = (0_1,\cdots, 1_k,\cdots, 0_n)^{\top}\\ V = [v^1,v^2,\cdots, v^m] $$

and assuming

$$ f(\beta) = \sum_{k=1}^n\left\lVert m_k-\left\lVert V.\beta.e_k\right\rVert^2\right\rVert^2 $$

we have the extrema conditions

$$ \frac{\partial f}{\partial \beta_k} = \sum_{k=1}^n\left(m_k-\left\lVert V.\beta.e_k\right\rVert^2\right)\left(V.\beta.e_k\right)V.e_k = 0 $$ This is a nonlinear equations system with $m$ equations and $m$ unknowns.

Cesareo
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