Constraint Qualifications

Abstract

We have seen from the last chapter that if a point solves a nonlinear programming problem then the Fritz John conditions hold, provided that the candidate point under investigation belongs to the interior of X. We also shoved that if the lagrangian multiplier of the objective function is positive then the Fritz John conditions reduce to the Kuhn-Tucker conditions. In this chapter we will develop in detail various conditions which guarantee positivity of the lagrangian multiplier of the objective function. These conditions are known as constraint qualifications since they only involve the constraints. Section 6.1 below is devoted for problems with inequality constraints while Section 6.2 treats both equality and inequality constraints. Finally a constraint qualification is presented in Section 6.3 in order to obtain necessary optimality criteria of the minimum principle type where the requirement x_o ε int X is relaxed. The emphasis in this chapter is on the various relationships among different constraint qualifications. The reader may refer to [5, 28] for a more exhaustive discussion of the different constraint qualifications.