Local structure of feasible sets in nonlinear programming, part I: Regularity

Abstract

Given a feasible point for a nonlinear programming problem, we investigate the structure of the feasible set near that point. Under the constraint qualification called regularity, we show how to compute the tangent cone to the feasible set, and to produce feasible arcs with prescribed first and second derivatives. In order to carry out these constructions, we show that a particular way of representing the feasible set (as a system of equations with constrained variables) is particularly useful. We also give fairly short proofs of the first-order and second-order necessary optimality conditions in very general forms, using the arc constructions mentioned above.