Why is CVXPY throwing a DCP error with cp.sqrt but no error with cp.norm

Question

I am trying to use CVXPY to optimise signal-to-noise-plus interference ratio (SINR) for a visible light communication (VLC) system. I have one of my SINR constraints stated as

\begin{equation} \mathbf{h}^{\textsf{T}}_{l,u}\mathbf{p}_{l} \geq \sqrt{\gamma_{u}\sum_{j\in \mathcal{L}\backslash\lbrace l\rbrace}\left(\mathbf{h}^{\textsf{T}}_{j,u}\mathbf{p}_{j}\right)^{{2}} + \sigma_{u}^{{2}}}. \end{equation}

If I use cp.sqrt(A), where A is the term inside the square root, my constraint violates DCCP rules but when I cast it with cp.norm(A), there seems to be no violations. May somebody please clarify to me why this is the case. I doubt that simply taking cp.norm(A) is correct. I have just started learning how to use CVXPY, I appreciate any help and guide.

Answer 1

Based on your experience, one of (ignoring the subscripts) h or p must be a variable. Therefore h'*p is affine. After introducing subscripts, you can form the norm of the vector of these individual affine terms in compliance with CVXPY's DCP rules, because the argument of norm is affine (vector). cp.norm of the appropriate vector is correct, but not cp.norm of the argument of the square root.

When norm is used, CVXPY will recognize this constraint as a Second Order Cone Constraint, and send it to the solver accordingly.

Entering the square root directly violates the DCP rules, because concave of convex (sqrt of sum of squares) is not allowed per DCP rules, because curvature of concave if convex is not clearly determined for arbitrary concave and convex functions.