We study some fundamental asymptotic properties associated with the well-posedness of constrained optimization problems. Emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic Karush-Kuhn-Tucker (KKT) optimality conditions. Unlike the subdifferential approach used by Auslender, Cominetti, and Crouzeix, we use a residual function approach that is closely tied to the theory of error bounds; this approach handles constraints explicitly and allows the effective treatment of infeasible, stationary sequences. The asymptotic KKT conditions provide an asymptotic optimality certificate for inequality constrained programs. Specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with Holderian minima are discussed. An application of the results to the convergence of a family of Newton-type iterative descent methods involving singular quasi-Newton matrices for solving a constrained minimization problem is also presented.