{"title":"Counterexample and an additional revealing poll step for a result of “analysis of direct searches for discontinuous functions”","authors":"","doi":"10.1007/s10107-023-02042-3","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>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 <span> <span>\\(f: \\mathbb {R}\\rightarrow \\mathbb {R}\\)</span> </span> 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 <span> <span>\\((x_k)_{k \\in \\mathbb {N}}\\)</span> </span> converging to a point <span> <span>\\(x_*\\)</span> </span> where <em>f</em> is discontinuous, lower semicontinuous and whose objective function value <span> <span>\\(f(x_*)\\)</span> </span> is strictly less than <span> <span>\\(\\lim _{k\\rightarrow \\infty } f(x_k)\\)</span> </span>. Moreover the dDSM generates trial points in only one of the continuity sets of <em>f</em> near <span> <span>\\(x_*\\)</span> </span>. 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. </p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"1 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Programming","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10107-023-02042-3","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍：
Mathematical Programming publishes original articles dealing with every aspect of mathematical optimization; that is, everything of direct or indirect use concerning the problem of optimizing a function of many variables, often subject to a set of constraints. This involves theoretical and computational issues as well as application studies. Included, along with the standard topics of linear, nonlinear, integer, conic, stochastic and combinatorial optimization, are techniques for formulating and applying mathematical programming models, convex, nonsmooth and variational analysis, the theory of polyhedra, variational inequalities, and control and game theory viewed from the perspective of mathematical programming.
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