First- and Second-Order Necessary Conditions Via Exact Penalty Functions

Kaiwen Meng, Xiaoqi Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)


In this paper, we study first- and second-order necessary conditions for nonlinear programming problems from the viewpoint of exact penalty functions. By applying the variational description of regular subgradients, we first establish necessary and sufficient conditions for a penalty term to be of KKT-type by using the regular subdifferential of the penalty term. In terms of the kernel of the subderivative of the penalty term, we also present sufficient conditions for a penalty term to be of KKT-type. We then derive a second-order necessary condition by assuming a second-order constraint qualification, which requires that the second-order linearized tangent set is included in the closed convex hull of the kernel of the parabolic subderivative of the penalty term. In particular, for a penalty term with order $$\frac{2}{3}$$23, by assuming the nonpositiveness of a sum of a second-order derivative and a third-order derivative of the original data and applying a third-order Taylor expansion, we obtain the second-order necessary condition.
Original languageEnglish
Pages (from-to)720-752
Number of pages33
JournalJournal of Optimization Theory and Applications
Issue number3
Publication statusPublished - 1 Jun 2015


  • KKT condition
  • Nonlinear programming problem
  • Regular subdifferential
  • Second-order necessary condition
  • Subderivative

ASJC Scopus subject areas

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

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