On an extended lagrange claim

Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

3 Citations (Scopus)


Lagrange once made a claim having the consequence that a smooth function f has a local minimum at a point if all the directional derivatives of f at that point are nonnegative. That the Lagrange claim is wrong was shown by a counterexample given by Peano. In this note, we show that an extended claim of Lagrange is right. We show that, if all the lower directional derivatives of a locally Lipschitz function f at a point are positive, then f has a strict minimum at that point.
Original languageEnglish
Pages (from-to)685-688
Number of pages4
JournalJournal of Optimization Theory and Applications
Issue number3
Publication statusPublished - 1 Mar 2001


  • Directional derivatives
  • Lipschitz continuous functions
  • Minimum points

ASJC Scopus subject areas

  • Management Science and Operations Research
  • Applied Mathematics
  • Control and Optimization


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