On constraint qualifications: Motivation, design and inter-relations

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.