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

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-01-08 DOI:10.1007/s10107-023-02042-3

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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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反例和 "不连续函数直接搜索分析 "结果的额外揭示性投票步骤

摘要本注释提供了论文最后一部分（Vicente and Custódio Math Program 133:299-325, 2012）中公布的一个定理的反例。该反例涉及一个目标函数（f: \mathbb {R}\rightarrow \mathbb {R}/），它满足定理所要求的所有假设，但与定理的某些结论相矛盾。该定理的一个推论也受到了这个反例的影响。这个反例揭示的主要缺陷是定向直接搜索法（dDSM）有可能产生一连串的试验点 \((x_k)_{k \in \mathbb {N}}\) 收敛到 f 不连续的点\(x_*\)、并且其目标函数值（f(x_*)）严格小于（f(x_k)）。此外，dDSM 只在在\(x_*\)附近的 f 的连续性集合中的一个集合中产生试验点。本注释还研究了定理的证明，以突出原论文中不精确的陈述。最后，本文介绍了对 dDSM 的修改，在通常情况下，它可以恢复被反例破坏的性质。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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