The first-order <subgradient inequality> for a differentiable <convex function> gives
$$
f(v)\geq f(w)+\langle\nabla f(w),v-w\rangle
$$
and, after interchanging $v$ and $w$,
$$
f(w)\geq f(v)+\langle\nabla f(v),w-v\rangle.
$$
Adding and rearranging yields
$$
\boxed{\langle\nabla f(v)-\nabla f(w),v-w\rangle\geq0.}
$$
Thus the <gradient> $F=\nabla f$ is a <monotone operator>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 339 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-339/2/a/solution