The relation between pseudonormality and quasiregularity in constrained optimization
We consider optimization problems with equality, inequality, and abstract set constraints. We investigate the relations between various characteristics of the constraint set related to the existence of Lagrange multipliers. For problems with no abstract set constraint, the classical condition of quasi regularity provides the connecting link between the most common constraint qualifications and existence of Lagrange multipliers. In earlier work, we introduced a new and general condition, pseudonormality, that is central within the theory of constraint qualifications, exact penalty functions, and existence of Lagrange multipliers. In this paper, we explore the relations between pseudonormality, quasi regularity, and existence of Lagrange multipliers, and show that, unlike pseudonormality, quasiregularity cannot play the role of a general constraint qualification in the presence of an abstract set constraint. In particular, under a regularity assumption on the abstract constraint set, we show that pseudo normality implies quasiregularity. However, contrary to pseudonormality, quasiregularity does not imply the existence of Lagrange multipliers, except under additional assumptions.
pseudonormality quasiregularity lagrange multipliers constraint qualifications quasi normality