$ {L}_{1} $ (L1) Norm Regularized Minimization with of Convex Function with Linear Equality Constraint Using ADMM Framework In section 6.3 of this note there is a method for minimizing a loss function with l1 regularization. i.e.
minimize $l(\bf{x})+\lambda||x||_1$
How can I add the equality constraint
$\sum\limits_{i} x_i =1$ for such a problem and perform the optimization ? 
 A: One approach could be to rewrite it as $$\sum_i x_i - 1 = 0$$ which motivates adding the following penalty term $$\lambda_S\left\|\sum_i x_i - 1\right\|_k = \lambda_S\|{\bf Sx} - 1\|_k$$ for some suitable $k$ where $\bf S$ is a matrix for the sum operator. Basically a dot product between x vector and a vector of ones. The equality will be true when the thing inside the norm is equal to 0. Which norm $k$ we choose ( and maybe the size of a scalar weight like $\lambda_S$ ) will determine how important that it is exactly 0.
A: You have 2 options here:


*

*Solve the problem using ADMM and transform $ \left\| x \right\|_{1} $ into $ \left\| z \right\|_{1} $ (Classic LASSO ADMM). In the $ x $ update step add projection onto the set by $ x = x - \frac{ \boldsymbol{1}^{T} x - 1 }{n} \boldsymbol{1} $ which is the orthogonal projection onto the set.

*Solve a different ADMM when adding $ I_{\mathcal{S}} \left( z \right) $ which is the Indicator function of the set $ \mathcal{S} $ as defined above. This means you'll have ADMM which on one iteration solve LASSO problem with reagridng to $ x $ (Actually LASSO with Tikhonov Regularization, which is called Elastic Net Regularization) and on the other, regarding $ z $ you will have a projection operation (As in (1)).


Clearly the first approach is much easier.
If you want a code, let me know.
