### G-2022-56

# Erratum, counterexample and an additional revealing poll step for a result of ``Analysis of direct searches for discontinuous functions''

## , , and BibTeX reference

This note provides a counterexample to a theorem announced in the last part of the paper *Analysis of direct searches for discontinuous functions*, Mathematical Programming Vol. 133, pp.299-325, 2012.
The counterexample involves an objective function `\(f:{\mathbb{R}}\to{\mathbb{R}}\)`

which satisfies all the assumptions required by the theorem but contradicts some of its conclusions.
A corollary of this theorem is also affected by this counterexample.
The main flaw revealed by the counterexample is the possibility that a directional direct search method (dDSM) generates a sequence of trial points `\((x_k)_k\)`

converging to a point `\(x_*\)`

where `\(f\)`

is discontinuous and whose objective function value `\(f(x_*)\)`

is strictly less than `\(\lim_{k\to\infty} f(x_k)\)`

.
Moreover the dDSM generates no trial point in one of the two branches of `\(f\)`

near `\(x_*\)`

.
This note also investigates the proof of the theorem to highlight the inexact statements in the original paper.
Finally this work concludes with a modification of the dDSM that allows to recover the properties broken by the counterexample.

Published **December 2022**
,
9 pages