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

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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.

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