Problem 1. Minimize ||Ax-b||_1^2

is it correct to solve it as follows:

mu = 10; [x,out] = tfocs_SCD([],{A,-b},prox_linf^2,mu,[],[],opt,[])

No, unfortunately the duals don’t work that way: if f and g are conjugates, then f^2 and g^2 are not necessarily conjugates. Also, TFOCS is not smart enough to recognize that prox_linf^2 means you want the prox of linf^2.

if it is not possible to solve this one. how I can solve

Minimize ||Ax-b||_1 ?

for examlpe, can I use

mu = 10; [x,out] = tfocs_SCD([],{A,-b},prox_linf,mu,[],[],opt,[]) ???

Yes, off the top of my head, that looks fine. For problem 1, since there are no other terms, it doesn’t matter if the l1 norm is squared or not, so this should be equivalent to your original problem. (If you do add other terms and really need the l1 norm squared, then first verify that it is convex, and then work out how to compute the proximity operator of the dual)

Problem 2. Minimize ||Ax-b||_D^2 + ||x||_1^p

when p >= 1 and ||x||_D = sqrt(x^T D x).

For this, follow the examples for the LASSO problem that we have in the release and see `solver_LASSO.m`

. To deal with the g(x)=\|Ax-b\|_D^2 term (which is differentiable), you just need to write a small function that calculates its gradient, i.e \nabla_x g(x) = A^TD(Ax-b). Or, use \tilde{A} = D^{1/2}A and \tilde{b} = D^{1/2}b and then run the LASSO solver with these terms. You can use TFOCS.m (you don’t need to use TFOCS_SCD.m) since you can solve the primal problem.

Problem 3. Minimize ||Ax-b||_D^2 + ||Bx||_2^2 + ||x||_2^2 + t^T |x|, wehre t = (t1,t2, …, tn).

The first three terms are all differentiable, so combine these into one function that calculates the gradient. The last term is just a weighted l1 norm term, so use prox_l1(t).

Problem 4. ||Ax-b||_2^2 + ||W x||_1 Where W \in R^{m \times n}

Try modifying solver_sBPDN_W.m for this one.