Fermat rule for convex minimization
= Fermat rule for convex minimization
For a proper <convex function> finite at $x$, global minimality is equivalent to $0\in\partial f(x)$. This follows immediately from the defining <subgradient inequality>. Combined with a <subdifferential sum rule>, it converts a convex minimization problem into an inclusion, for example the equation defining a <proximal operator>.