Counterexample and an additional revealing poll step for a result of "analysis of direct searches for discontinuous functions"

This note provides a counterexample to a theorem announced in the last part of the paper (Vicente and Custodio Math Program 133:299-325, 2012). The counterexample involves an objective function f : R -> 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 (xk) (k is an element of 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 ->infinity) f(xk) . 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.