Subgradient

A subgradient of a convex function $f$ at $x$ is a vector $g$ such that $f(y)\ge f(x)+g^T(y-x)$ for every $y$ in the domain.

A real convex function locally bounded on an open convex set has a continuous supporting linear functional at every interior point. Apply the dominated Hahn-Banach theorem to its sublinear directional derivative $p_x$ to obtain $l\le p_x$. Increasing secant slopes imply $p_x(z)\le f(x+z)-f(x)$ whenever the endpoint is in the domain. A local Lipschitz continuity bound gives both $l(y)\le L\Vert y\Vert$ and $-l(y)=l(-y)\le L\Vert y\Vert$, proving continuity.

A proper convex function $f$ is minimized at $x$ exactly when $0\in \partial f(x)$. For a differentiable convex term $g$ and another convex term $h$, the subdifferential sum rule gives the condition $0\in \nabla g(x)+\partial h(x)$.

The absolute value function has

A vector is a subgradient when its supporting affine function lies below the convex function.