I am solving this problem but it gives homogeneous problem detected and it gives the optimal value as 0. Why is it the case? Any help would be highly appreciated.

Here X{p} is a vector.

cvx_begin
variable x§

f = dot(alpha_v,X{p}.^2) + dot(beta_v,X{p})+transpose(v)*X{p}+rho/2*(norm(Nabla{p}*X{p}))^2;
minimize f
subject to
l(p) <= norm(X{p},1) <= u(p)
cvx_end

Homogeneous problem detected; solution determined analytically.
Status: Solved
Optimal value (cvx_optval): +0

Can you show a complete, but minimally-sized, reproducible example? Apparently, the problem is simplified enough by CVX’s processing that it does not need to call the solver.

I don’t understand why l(p) <= norm(X{p},1) would be accepted by CVX, because it si a non-convex constraint. Even if l(p) were zero, CVX still would reject it, even though in that case it would be a vacuous constraint which some optimization modeling system could eliminate, but CVX does not.

I am very dubious that the output you show corresponds to the program you show. So my question is, how is this program accepted by CVX?

Yes you are right. When I do it as a separate cvx example, it gives the following error:
Disciplined convex programming error:
Invalid constraint: {constant} <= {convex}
Though I don’t understand how it is a non-convex constraint? Moreover, it does not accept norm(x)^2 too in the objective function. Why is that?

This is actually a convex problem. Or more specifically, with this input data, can be rewritten as a convex problem.

First of all, norm.(..)^2 needs to be replaced by square_pos(norm(...)) .

Second of all, as I wrote in my previous reply:

I don’t understand why l(p) <= norm(X{p},1) would be accepted by CVX, because it si a non-convex constraint. Even if l(p) were zero, CVX still would reject it, even though in that case it would be a vacuous constraint which some optimization modeling system could eliminate, but CVX does not.

Indeed, CVX does not accept this. But given that l is the zero vector, that constraint can be removed as vacuous. Therefore, the problem can be rewritten as below, and it is successfully solved. But it is not a homogeneous problem; and CVX solves it by calling a solver. not analytically.

cvx_begin
variable x(n)
f = dot(alpha_v,x.^2) + dot(beta_v,x)+transpose(v)*x+rho/2*square_pos(norm(Nabla*x));
minimize f
subject to
norm(x,1) <= u(1);
cvx_end`

In the future, please make sure that the output you show corresponds to the program you show, which obviously was NOT the case in your post.