Journal of Optimization Theory and Applications

, Volume 114, Issue 2, pp 287–343

Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization

Authors

  • D.P. Bertsekas
    • Department of Electrical Engineering and Computer ScienceMassachusetts Institute of Technology
  • A.E. Ozdaglar
    • Department of Electrical Engineering and Computer ScienceMassachusetts Institute of Technology
Article

DOI: 10.1023/A:1016083601322

Cite this article as:
Bertsekas, D. & Ozdaglar, A. Journal of Optimization Theory and Applications (2002) 114: 287. doi:10.1023/A:1016083601322

Abstract

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.

pseudonormalityinformative Lagrange multipliersconstraint qualificationsexact penalty functions

Copyright information

© Plenum Publishing Corporation 2002