Differential Inclusions, Control and Optimization 20 (2000) 7-26
doi: 10.7151/dmdico.1001

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PENALTY/BARRIER PATH-FOLLOWING IN LINEARLY CONSTRAINED OPTIMIZATION

Christian Grossmann

Institute of Numerical Mathematics
Dresden University of Technology, D-01062 Dresden, Germany
e-mail: grossm@math.tu-dresden.de

Abstract

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 [,], 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.

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

1991 Mathematics Subject Classification: 49M15, 65K10, 65H10, 90C25.

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Received 13 November 1999
Revised 26 January 2000