Directional Derivative and Subgradients

Definition 2.2.10 (Directional derivative) Let 5 be a nonempty convex set in %n, x° G S, and y be a nonzero vector such that (x° + Ay) S for sufficiently small and strictly positive A. The directional derivative of f(x) at the point x°, along the directiony, denoted as / (x°,y), is defined as the limit ( oo included) of 

Remark 1 The right-hand side of the above inequality (2.1) is a linear function in x and represents the first-order Taylor expansion of f(x) around x° using the vector d instead of the gradient vector of /(x) at x°. Hence, d is a subgradient of /(x) at x° if and only if the first-order Taylor approximation always provides an underestimation of /(x) for all x. 

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