The papers in the present Study deal with quasidifferentiable functions, i.e. functions which are directionally differentiable and such that at each fixed point the directional derivative as a function of direction can be expressed as the difference of two convex positively homogeneous functions. It turns out that uasidifferentiable functions form a linear space closed with respect to all 'differentiable' operations and (very importantly) with respect to the operations of taking the point-wise maximum and minimum. Many properties of these functions have been discovered, and we are now in a position to speak about Quasidifferential Calculus. But the importance of quasidifferentiable functions is not simply based on the results obtained so far. We can foresee a much greater role for these functions since (as far as the first-order properties are concerned) all directionally differentiable Lipschitzian functions can be approximated by quasidifferentiable functions. This is due to the fact that the directional derivative of any directionally differentiable Lipschitzian function can be approximated to within any given accuracy by the difference of two convex positively homogeneous functions.
Read more at http://ebookee.org/Quasidifferential-Calculus_1005301.html#ZeUqySJYlhgVdfo7.99