Penalty/barrier path-following in linearly constrained optimization

Grossmann, Christian

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 20 (2000) no. 1, pp. 7-26.

Résumé

In the present paper rather general penalty/barrier path-following methods (e.g. with p-th power penalties, logarithmic barriers, SUMT, exponential penalties) applied to linearly constrained convex optimization problems are studied. In particular, unlike in previous studies [1,11], here simultaneously different types of penalty/barrier embeddings are included. Together with the assumed 2nd order sufficient optimality conditions this required a significant change in proving the local existence of some continuously differentiable primal and dual path related to these methods. In contrast to standard penalty/barrier investigations in the considered path-following algorithms only one Newton step is applied to the generated auxiliary problems. As a foundation of convergence analysis the radius of convergence of Newton's method depending on the penalty/barrier parameter is estimated. There are established parameter selection rules which guarantee the overall convergence of the considered path-following penalty/barrier techniques.

Détail

Keywords: penalty/barrier, interior point methods, convex optimization

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