Hi all,

I currently have a minimization problem I am trying to solve that involves the norm of an absolute value function (see below), where both Z and alpha are fixed and have non-negative entries.

I know that directly plugging this in will not be DCP compliant as the norm expects an affine function. My thinking at this point is that I need to rewrite the optimization using appropriate constraints, but I am not sure of the right constraints and have only considered one basic (but wrong) form of the problem that CVX at least accepts shown below.

```
%set seed
rng(123)
% sample size, number of 1st/2nd level vars
n = 100;
p = 30;
q = 2;
% external data
z1 = [ones(5,1); zeros(p-5, 1)];
z2 = [zeros(5,1); ones(10, 1); zeros(p - 15, 1)];
Z = [z1, z2];
% 1st level coef
b0 = 0.1;
b = [normrnd(0.2, 0.1, [5,1]); normrnd(0.5, 0.1, [5,1]);
normrnd(-0.5, 0.1, [5,1]); normrnd(0, 0.1, [p - 15,1])];
% simulate predictors and outcome
mu_x = zeros(p, 1);
sigma_x = diag(ones(p, 1));
X = mvnrnd(mu_x, sigma_x, n);
y = b0 + X * b + normrnd(0, 1, [100, 1]);
%{
Problem we are interested in solving
minimize_betas (1/2n)||y - X * betas||^2 + (lambda/2)||abs(betas) - Z * alphas||^2
Z and alphas have non-negative entries, lambda > 0
%}
% fix alphas / lambda
alphas = [0.1; 0.3];
lambda = 0.1;
% fit in cvx (need to figure out how to represent abs(betas))
cvx_begin
variables betas(p) betas_abs(p);
minimize(square_pos(norm(y - X*betas)) / (2*n) + lambda * square_pos(norm(betas_abs - Z * alphas)) / 2);
subject to
abs(betas) <= betas_abs;
cvx_end
[betas, betas_abs]
% are we getting the right coef. (do all abs(betas) == betas_abs) --> no
disp(sum(round(abs(betas), 4) == round(betas_abs, 4)) == p);
```

Thanks for any assistance you might be able to provide!