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.
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Audet, C., Bouchet, PY. & Bourdin, L. Counterexample and an additional revealing poll step for a result of “analysis of direct searches for discontinuous functions”. Math. Program. (2024). https://doi.org/10.1007/s10107-023-02042-3
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DOI: https://doi.org/10.1007/s10107-023-02042-3