When an inequality is not of the form CONVEX $\leq$ CONCAVE

Question

One of the most important rules in using CVX is that a ''less-than inequality constraint'' must be of the form

\begin{align} \textbf{CONVEX function} \leq \textbf{CONCAVE function} ~~~(1) \end{align}

However, it is not always easy for one to get such an inequality. In many cases, it might be of the form CONVEX function $\leq$ CONVEX function or CONCAVE function $\leq$ CONCAVE function. For these cases, we may transform them into the form (1), but what kind of methods should we use? Could you please suggest several widely used theorems/lemmas/methods to overcome the issue?

I've got the following problem as an example: $\| \textbf{a}^T\textbf{x} \|^2 \leq \| \textbf{B}\textbf{y} \|^2$ is the constraint with the right hand side (RHS) being convex function. Given that $\textbf{a}$ and $\textbf{B}$ are constant vector and matrix, what should we manipulate the inequality to make it suitable for the DCP ruleset of CVX?

Answer 1

You can't. The set

$S=\left\{ (x,y)\; |\;\; \| a^{T}x \|^{2} \leq \| By \|^{2} \right\}$

simply isn't convex in general. You'll have to back up and find different constraints if you want to use convex optimization.