Nonlocal error bounds for piecewise affine functions

TL;DR

This paper provides a comprehensive analysis of nonlocal error bounds for nonconvex piecewise affine functions, improving existing results and establishing new necessary and sufficient conditions for their existence on various sets.

Contribution

It introduces new conditions for error bounds of piecewise affine functions, including on unbounded and finite union of polyhedral sets, enhancing understanding of their structure.

Findings

Any piecewise affine function has an error bound on any bounded set.

Provides verifiable sufficient conditions for error bounds on unbounded sets.

Derives necessary and sufficient conditions for error bounds on finite unions of polyhedral sets.

Abstract

The paper is devoted to a detailed analysis of nonlocal error bounds for nonconvex piecewise affine functions. We both improve some existing results on error bounds for such functions and present completely new necessary and/or sufficient conditions for a piecewise affine function to have an error bound on various types of bounded and unbounded sets. In particular, we show that any piecewise affine function has an error bound on an arbitrary bounded set and provide several types of easily verifiable sufficient conditions for such functions to have an error bound on unbounded sets. We also present general necessary and sufficient conditions for a piecewise affine function to have an error bound on a finite union of polyhedral sets (in particular, to have a global error bound), whose derivation reveals a structure of sublevel sets and recession functions of piecewise affine functions.