I am trying to get into convex optimization. But I could not find a clear explanation on why we dirsturb the objective function $f:\mathbb{R}^d \rightarrow \mathbb{R}$ like this:
$$F(x,u): \mathbb{R}^d\times\mathbb{R}^m\rightarrow \mathbb{R} \\ F(x,0)=f(x)$$
I suspect that it helps us to achieve strong duality for a disturbed version of the original problem, which certainly be better than nothing.