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There are $n$ multiplication coefficients for which optimized values are searched. There is a restriction that the multiplication coefficients are not allowed to be negative and must have a sum of $1.0$:

$\sum_{i =0}^n c_i = 1.0$

The coefficients are multiplied by an individual vector and summed up to a new vector:

$\pmb v^{sum} = \sum_{i =0}^n c_i \cdot \pmb v_i$

Finally, the mean squared error between $\pmb v^{sum}$ and a vector $\pmb u$ is formed:

$mse = \frac{1}{m}\sum_{i = 0}^m (u_i - v^{sum}_i)^2$

So I want to find optimized $c_i$ values for which the $mse$ is minimized.

What's the best way to do that?

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  • $\begingroup$ Are the $c_n$ required to be nonnegative? $\endgroup$ – Rob Pratt Nov 20 '19 at 13:28
  • $\begingroup$ @RobPratt Yes, good hint. I have edited my text. $\endgroup$ – Anne Bierhoff Nov 20 '19 at 13:47
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Geometrically, this is a projection onto the convex hull of the $\mathbf{v}_i$. You can solve it as a quadratic programming problem.

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  • $\begingroup$ Thanks for your answer. I saw that ordinary least squares are a part of quadratic programming. Would ordinary least squares be a possible way to solve the problem? $\endgroup$ – Anne Bierhoff Nov 20 '19 at 18:14
  • $\begingroup$ This is ordinary least squares with some side constraints on the regression coefficients. $\endgroup$ – Rob Pratt Nov 20 '19 at 18:26
  • $\begingroup$ Currently, I have a hard time trying to translate the problem into a quadratic programming problem. So I created a new question: math.stackexchange.com/questions/3454546/… $\endgroup$ – Anne Bierhoff Nov 28 '19 at 13:58

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