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

Abstract

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 $x_{\ast }$ where $f$ is discontinuous and whose objective function value $f(x_{\ast })$ 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 $x_{\ast }$. 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.

10 pages, 1 figure