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 Custódio 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 (x_k)_k ∈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→∞ 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.