Beyond the Oracle: Opportunities of Piecewise Differentiation

Andreas Griewank () and Andrea Walther ()
Additional contact information
Andreas Griewank: Humboldt University zu Berlin, Department of Mathematics
Andrea Walther: Humboldt University zu Berlin, Department of Mathematics

Chapter Chapter 10 in Numerical Nonsmooth Optimization, 2020, pp 331-361 from Springer

Abstract: Abstract 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.

Date: 2020
References: Add references at CitEc
Citations: View citations in EconPapers (1)

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-34910-3_10

Ordering information: This item can be ordered from
http://www.springer.com/9783030349103

DOI: 10.1007/978-3-030-34910-3_10

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().