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I would like to estimate the optimal weights where weights are all positive and add up to $1$. The most basic problem of this is as follows:

Regression $y = \beta_1x_1 + \beta_2x_2$ with constraints $\beta_1 + \beta_2 = 1, \beta_1, \beta_2 > 0$.

Try

I would make up the following loss,

$$ \sum (y_i - \beta_1 x_1 - \beta_2 x_2)^2 + \lambda (\beta_1 + \beta_2 - 1) $$

where $\lambda$ : lagrange multiplier.

But I'm not sure this optimization problem is related to any statistical problem.

moreblue
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    Also see [this post](https://stats.stackexchange.com/questions/41168/constrained-regression-in-r-coefficients-positive-sum-to-1-and-non-zero-interc) – user20160 Mar 07 '19 at 00:58

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