On constraint qualifications: Motivation, design and inter-relations

Abstract

Constraint qualication (CQ) is an important concept in nonlinear programming. This paper investigates the motivation of introducing constraint qualications in developing KKT conditions for solving nonlinear programs and provides a geometric meaning of constraint qualications. A unied frame-work of designing constraint qualications by imposing conditions to equate the so-called "locally constrained directions" to certain subsets of "tangent di-rections" is proposed. Based on the inclusion relations of the cones of tangent directions, attainable directions, feasible directions and interior constrained directions, constraint qualications are categorized into four levels by their rel-ative strengths. This paper reviews most, if not all, of the commonly seen con-straint qualications in the literature, identies the categories they belong to, and summarizes the inter-relationship among them. The proposed framework also helps design new constraint qualications of readers' specic interests.