Automatic decrease of the penalty parameter in exact penalty function methods

This paper presents an analysis of the involvement of the penalty parameter in exact penalty function methods that yields modifications to the standard outer loop which decreases the penalty parameter (typically dividing it by a constant). The procedure presented is based on the simple idea of making explicit the dependence of the penalty function upon the penalty parameter and is illustrated on a linear programming problem with the $l_1$ exact penalty function and an active-set approach. The procedure decreases the penalty parameter, when needed, to the {\em maximal\/} value allowing the inner minimization algorithm to leave the current iterate. It moreover avoids unnecessary calculations in the iteration following the step in which the penalty parameter is decreased. We report on preliminary computational results which show that this method can require fewer iterations than the standard way to update the penalty parameter. This approach permits a better understanding of the performance of exact penalty methods.