General primal-dual penalty/barrier path-following Newton methods for nonlinear programming

In the present article rather general penalty/barrier-methods are considered, that define a local continuously differentiable primal-dual path. The class of penalty/barrier terms includes most of the usual techniques like logarithmic barriers, SUMT, quadratic loss functions as well as exponential penalties, and the optimization problem which may contain inequality as well as equality constraints. The convergence of the corresponding general primal-dual path-following method is shown for local minima that satisfy strong second-order sufficiency conditions with linear independence constraint qualification (LICQ) and strict complementarity. A basic tool in the analysis of these methods is to estimate the radius of convergence or Newton's method depending on the penalty/barrier-parameter. Without using self-concordance properties convergence bounds are derived by direct estimations of the solutions of the Newton equations. Parameter selection rules are proposed which guarantee the local convergence of the considered penalty/barrier-techniques with only a finite number of Newton steps at each parameter level. Numerical examples illustrate the practical behavior of the proposed class of methods.