Smoothing techniques for exact penalty methods

摘要

In this paper numerical methods based on exact penalties for the treatment of nonlinear programming problems in finite dimensions as well as in Hilbert spaces are under considered. The advantage of exact penalties is that under suitable conditions already for sufficiently large, but finite penalty parameters optimal solutions of the original constrained problem are obtained. However, as a rule, exact penalty functions are not differentiable. This restricts the applicability or at least the efficiency of the methods, like Newton's method, for solving the generated unconstrained problems. To overcome this drawback various smoothing techniques are proposed in the literature and intensively studied till now. Smoothing techniques replace the original penalty functions by differentiable smooth ones. First, an overview of different types of smoothing of exact penalties known from the literature are presented and its approximation properties investigated. Among all the various types the method based upon the smoothing {formula} for |t| with the smoothing parameter s → 0+ appears preferable because of its arbitrarily often differentiability and its low growth behavior. This convergence behavior of this method is analyzed in the case its application to finite dimensional optimization problem. In particular, by means of implicit function technique first order convergence could be obtained. Under the second order sufficiency condition these improved convergence results with respect to the smoothing parameter are derived and its extension to optimal control problems discussed. Numerical examples in finite dimensional as well as discretized optimal control cases show the proved convergence order.