I see a similar problem in How do I fit a constrained regression in R so that coefficients total = 1?

Specifically, my model is $Y_i= \pi_1 X_1+\pi_2 X_2 +...+ \pi_K X_K +\epsilon_i$ with $\pi_k \ge 0$ and $\sum_k \pi_k =1$ . That is $\pi$'s are probabilities. and I need to minimize $\sum_i(Y_i -\pi_1 X_{i1}-\pi_2 X_{i2} -...- \pi_K X_{iK})^2$. However, the number of observations $n$ are few, $n<<K$. The question is how do I minimize so that only a few of $\pi_k$ are $\gt 0$. The answer based on quadratic programming don't take sparseness into account. Do I need to do some sort of regularization?

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    $\begingroup$ How do you define "a few of [the] $\pi_k$"? Could the number of non-zero coefficient estimates be equal to $n$? $\endgroup$
    – jbowman
    May 14 at 1:43
  • $\begingroup$ It should be $n$ or less. $\endgroup$ May 14 at 3:53


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