Ch. Grossmann

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 of Newtons 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.

A counterexample to Rosen's decomposition method

In this paper an example is presented showing that Rosens decomposition method can be convergent to some nonoptimal solution. In this an example is presented showing that Rosens decomposition method can be convergent to some nonoptimal solution.

Eine Erweiterung der verfahren der zulässigen richtungen

An extension of feasible direction methods is proposed permitting nonfeasible points to be used as starting points. The principle is described at a P1-method. Furthermore to reduce the computational effort a new “ϵ-technique” adapted t the given optimization problem is presented.

Ein überlinear konvergentes verfahren mit modifizierten Lagrangefunktionen

By means of augmented Lagrangians some dual optimization problem is constructed and the properties of this dual problem are investigated. An approximated Newtons method is applied to solve the senerated auxiliary problems. This paper deals with the investigation of the convergence of the presented algorithm.

Penalty methods in nonlinear programming (survey)

In this paper general interior and exterior penalty methods are considered and their essential properties are summarized. Penalty-shifting methods are also contained in this survey.