Is there any smart way to maximize the function $f:\mathbb{R}^2_{++} \to \mathbb{R} $, $(x,y)\mapsto xy$ constrained to a given convex subset of $\mathbb{R}^2_{++}$ ?
Does exist some remarkable cases in which there are smart solutions to this problem?
(note: $\mathbb{R}^2_{++}:=\{(x,y)\in \mathbb{R}^2: x\geq 0 \land y \geq 0 \} $)