Suppose I derived Fenchel conjugate of a function. My goal is to check if my solution is right. Suppose the steps are not available any more and only the final solution is present.
Is there any verification procedure (or a certificate) by which I can make sure that the solution is right or wrong? Any numerical or analytical procedure is useful.
Or perhaps any tools available for computing Fenchel conjugate of a non differentiable function?
Suppose you are given $f$, and have a candidate function $g$ that might be equal to $f^*$ (the the fenchel conjugate of $f$), but you're not sure. You can try checking numerically that: $$f(x) + g(y) \ge \langle x,y\rangle$$ for various values of $x,y$. Any counterexample to the inequality is certificate that $g \ne f^*$. To prove that $g=f^*$, you need to show the inequality is satisfied for all $x,y$, and that for each $y$ there is an $x$ such that the inequality is tight.