Beyond the oracle: Opportunities of piecewise differentiation

Abstract:

For more than 30 years much of the research and development in nonsmooth optimization has been pbkp_redicated on the assumption that the user provides an oracle that evaluates at any given x ∈ Rn 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: Rn → 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 e-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 absnormal form.

Año de publicación:

2020

Keywords:

Fuente:

scopus