An augmented Lagrangian approach with a variable transformation in nonlinear programming

Liwei Zhang, Xiaoqi Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

2 Citations (Scopus)

Abstract

Tangent cone and (regular) normal cone of a closed set under an invertible variable transformation around a given point are investigated, which lead to the concepts of θ- 1-tangent cone of a set and θ- 1-subderivative of a function. When the notion of θ- 1-subderivative is applied to perturbation functions, a class of augmented Lagrangians involving an invertible mapping of perturbation variables are obtained, in which dualizing parameterization and augmenting functions are not necessarily convex in perturbation variables. A necessary and sufficient condition for the exact penalty representation under the proposed augmented Lagrangian scheme is obtained. For an augmenting function with an Euclidean norm, a sufficient condition (resp., a sufficient and necessary condition) for an arbitrary vector (resp., 0) to support an exact penalty representation is given in terms of θ- 1-subderivatives. An example of the variable transformation applied to constrained optimization problems is given, which yields several exact penalization results in the literature.
Original languageEnglish
Pages (from-to)2095-2113
Number of pages19
JournalNonlinear Analysis, Theory, Methods and Applications
Volume69
Issue number7
DOIs
Publication statusPublished - 1 Oct 2008

Keywords

  • Augmented Lagrangian
  • Duality
  • Exact penalty representation
  • Normal cone
  • Subderivative
  • Subdifferential
  • Tangent cone

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Mathematics(all)

Cite this