The convergence rate of the penalty function method

Abstract

THE closeness of the solution of the problem with a penalty to the solution of the original problem is estimated at a conditional minimum. It is shown how to match the accuracy of the discovery of the minimum in the problem with a penalty function with the value of the coefficient of the penalty function. THE closeness of the solution of the problem with a penalty to the solution of the original problem is estimated at a conditional minimum. It is shown how to match the accuracy of the discovery of the minimum in the problem with a penalty function with the value of the coefficient of the penalty function. The idea of the penalty function method-the reduction of problems at a conditional extremum to problems without constraints by the introduction of a penalty on the infringement of constraints - has been known for a long time and has been discussed by many authors. At the present time there are numerous papers devoted to various modifications of the method, their applications to particular problems etc, (references specially devoted to this method can be found in the book [l], or in [2], section 12). However, the theoretical study of the method is far from complete, since the investigation is usually confined to the proof of its convergence, without estimating the rate of convergence. One of the few exceptions is [3] in which the closeness of the solution of the problem with a penalty to the solution of the original problem is estimated. However, in [3l only the convex case is considered (that is, these results are not applicable to non-linear constraints of the equality type), and also the estimate of closeness was there obtained with respect to a functional and to the constraints, but not with respect to the actual variables.Another important question left open is the following.In the practical realization of the penalty function method the discovery of the exact solution of each problem at an unconditional minimum is impossible. How must the accuracy of the solution of this auxiliary problem be chosen, in order that, on the one hand, the computing time is not increased too greatly, and on the other, the convergence of the method is not destroyed? Finally, it is known that in the penalty function method we obtain approximations for the dual variables (Lagrange multipliers) simultaneously. But the accuracy of these approximations has not been investigated. In the present paper an answer to these questions is given for the case of constraints given by equations and the simplest form of penalty.