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 (Vicente and Custódio Math Program 133:299–325, 2012). The counterexample involves an objective function  \(f: \mathbb {R}\rightarrow \mathbb {R}\) 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  \((x_k)_{k \in \mathbb {N}}\) converging to a point  \(x_*\) where f is discontinuous, lower semicontinuous and whose objective function value  \(f(x_*)\) is strictly less than  \(\lim _{k\rightarrow \infty } f(x_k)\) . Moreover the dDSM generates trial points in only one of the continuity sets 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 introduces a modification of the dDSM that allows, in usual cases, to recover the properties broken by the counterexample.