Approximating a Second-Order Directional Derivative for Nonsmooth Convex Functions
Jean‐Baptiste Hiriart‐Urruty · SIAM Journal on Control and Optimization · 1982
For a lower-semicontinuous convex function f, the approximate second-order directional derivative $(d,\delta ) \mapsto f''_\varepsilon (x_0 ;d,\delta )$ is defined through the $\varepsilon $-directional derivative $f'_\varepsilon (x;d)$. The function $v_d :x \mapsto v_d (x) = f'_\varepsilon (x;d)$ is, for all $\varepsilon > 0$, locally Lipschitz on inf (domf) and, at those points where it is not differentiable, $v_d$ admits a directional derivative $v'_d (x_0 ;\delta )$ for all $\delta $, which we precisely denote by $f''_\varepsilon (x_0 ;d,\delta )$. The objective of the present work is two-fold: to classify all the possible differentiability properties of $v_d$ according to the behavior of the function $\lambda \mapsto f(x_0 + \lambda d)$ on $R_ + $, and to study the existence or nonexistence of the limit of $f''_\varepsilon (x_0 ;d,\delta )$ when $\varepsilon \to 0^ + $.