Considerations of a Penalty Shifting Method for Nonlinear Programming Problems

Abstract

Ordinary Lagrange function and penalty function methods are typical methods for solving nonlinear programing problems with constraints. The application of the ordinary Lagrange function method is in general limited to convex programing problems. The penalty function method can be applied to nonconvex programing problems, but may suffer from numerical difficulties. The penalty shifting method is superior to the above two methods, since it can be applied to nonconvex programing problems without suffering from numerical difficulties. In this paper two geometric interpretations for the penalty shifting method are mainly given, so that its characteristics may be more clarified. First, the correction rules of two parameters and the relation between one of the parameters and the existence of a saddle point are investigated by the ordinary geometric interpretation. Secondly, the operation of the parameters is also investigated by a new geometric interpretation, and finally, the result for a nonconvex programing problem is illustrated by two geometric interpretations.