Lagrange multipliers and calmness conditions of order p

Xiaoqi Yang, Z. Q. Meng

Research output: Journal article publicationJournal articleAcademic researchpeer-review

11 Citations (Scopus)


In this paper, by assuming that a non-Lipschitz penalty function is exact, new conditions for the existence of Lagrange multipliers are established for an inequality and equality-constrained continuously differentiable optimization problem. This is done by virtue of a first-order necessary optimality condition of the penalty problem, which is obtained by estimating Dini upper-directional derivatives of the penalty function in terms of Taylor expansions, and a Farkas lemma. Relations among the obtained results and some well-known constraint qualifications are discussed.
Original languageEnglish
Pages (from-to)95-101
Number of pages7
JournalMathematics of Operations Research
Issue number1
Publication statusPublished - 1 Feb 2007


  • Dini directional derivative
  • Generalized calmness condition
  • Lagrange multiplier
  • non-Lipschitz penalty function

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics
  • Management Science and Operations Research


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