CONSTRAINT QUALIFICATIONS IN NONSMOOTH OPTIMIZATION: CLASSIFICATION AND INTER-RELATIONS

The aim of this paper is to systematically study various known and some new constraint qualifications for a nonsmooth optimization problem constrained by inequality constraints where the functions involved are locally Lipschitz continuous. We classify the constraint qualifications into four levels by using the inclusion relations among the cones of interior constrained directions, feasible directions, attainable directions, tangent directions, and locally constrained directions. Numerous inter-relationships between the constraint qualifications are summarized schematically. We further discuss the nature of various cones of the feasible set by assuming the constraint functions to be semilocally convex, and establish the equivalence among some of the constraint qualifications.