Pseudonormality and a lagrange multiplier theory for constrained optimization

D. P. Bertsekas, A. E. Ozdaglar

Research output: Contribution to journalArticlepeer-review

56 Scopus citations


We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the Fritz-John theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new properties, pseudonormality and quasinormality, emerge as central within the taxonomy of interesting constraint characteristics. In the case where there is no abstract set constraint, these properties provide the connecting link between the classical constraint qualifications and two distinct pathways to the existence of Lagrange multipliers: one involving the notion of quasiregularity and the Farkas lemma, and the other involving the use of exact penalty functions. The second pathway also applies in the general case where there is an abstract set constraint.

Original languageEnglish (US)
Pages (from-to)287-343
Number of pages57
JournalJournal of Optimization Theory and Applications
Issue number2
StatePublished - Aug 2002
Externally publishedYes


  • constraint qualifications
  • exact penalty functions
  • informative Lagrange multipliers
  • pseudonormality

ASJC Scopus subject areas

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


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