Karush–Kuhn–Tucker systems: Regularity conditions, error bounds and a class of Newton-type methods

We consider optimality systems of Karush–Kuhn–Tucker (KKT) type, which arise, for example, as primal–dual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newton-type methods for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for KKT systems, such as quasi-regularity or semistability (equivalently, the R0-property). Error bounds are useful, among other things, for identifying active constraints and developing efficient local algorithms. We propose a family of local Newton-type algorithms. This family contains some known active-set Newton methods, as well as some new methods. Regularity conditions required for local superlinear convergence compare favourably with convergence conditions of nonsmooth Newton methods and sequential quadratic programming methods.