Generalization of the Derivative where the linear approximation is defined to lie below the function, for Convex functions, at least..

See here

Subgradient method

Subgradient

Note that here they define the subgradient as a vector that defines a tangent plane (note that a hyperplane can always be uniquely specified by its gradient (and a point on it)). This is not a local definition (unlike the definition of subderivative above), and so existence of subgradient at every point implies convexity

https://www.stats.ox.ac.uk/~lienart/blog_opti_basics.html