Beyond the Oracle: Opportunities of Piecewise Differentiation

For more than 30 years much of the research and development in nonsmooth optimization has been predicated on the assumption that the user provides an oracle that evaluates at any given x ∈ ℝ n $${\boldsymbol x} \in \mathbb {R}^n$$ the objective function value φ(x) and a generalized gradient g ∈ ∂φ(x) in the sense of Clarke. We will argue here that, if there is a realistic possibility of computing a vector g that is guaranteed to be a generalized gradient, then one must know so much about the way φ : ℝ n → ℝ $$\varphi : \mathbb {R}^n \to \mathbb {R}$$ is calculated that more information about the behavior of φ in a neighborhood of the evaluation point can be extracted. Moreover, the latter can be achieved with reasonable effort and in a stable manner so that the derivative information provided varies Lipschitz continuously with respect to x. In particular we describe the calculation of directionally active generalized gradients, generalized ε-gradients and the checking of first and second order optimality conditions. All this is based on the abs-linearization of a piecewise smooth objective in abs-normal form.