Directional Derivative and Subdifferential

For a convex function, we define the directional derivative—a positively homogeneous, convex function—the derivative in each direction from a vector. The conjugate of the directional derivative is the indicator of the subdifferential. We study the local behavior of a convex function by translating its epigraph so $$f\left ( \mathbf {0}\right ) =0$$ and studying the local behavior at the origin. The conjugate is nonnegative, and the subdifferential is the vectors such that the conjugate is zero. Because the function is convex, one can express the derivative as the value function of a perturbed convex optimization: the derivative in direction x is $$\inf _{y}\left ( y\star f\right ) \left ( \boldsymbol {x}\right ) \!.$$ The objective function is the epi-multiplication of the function. The perturbation is the direction, and the choice variable is the multiplying factor. One obtains the directional derivative as the choice variable rises to infinity. We obtain Fermat's rule: a vector minimizes a proper, closed, convex function if and only if the directional derivative there is nonnegative.