On a Variant of the Method of Han and Powell with Continuously Differentiable Penalty Function

Abstract

In the last 15 years, many algorithms have been developed to solve the general nonlinear programming problem:

$$\begin{array}{*{20}c} {(NLP)\,\min \,f(x)} \\ {x\varepsilon \mathbb{R}^n \,s.t.\,g_i (x)\, = \,0.\,,\,i\, = \,1,\, \ldots \,,\,m_e } \\ {g_i (x)\, \leqq \,0\,,\,i\, = \,m_e \, + \,1,\, \ldots \,,\,m} \\ \end{array}$$

In tests performed by Schittkowski [12], the method of Han and Powell ([4], [5], [10], [11]) proved to be quite successful. However, some theoretical and practical drawbacks led Schittkowski to introduce a new augmented Lagrangian-type penalty function ([131) .This paper deals with a modification in the penalty parameter choice of Schittkowski, which potentially keeps these parameters at a lower level, thus improving the overall speed of the algorithm. Section 1 introduces the original method of Han/Powell and discusses some drawbacks that led to modifications by Schittkowski. These modifications are presented briefly in section 2. Section 3 deals with problems arising from Schittkowski’s penalty parameter choice, and introduces a new method which is able to overcome the problem. Section 4 presents the concept of restoration directions, which potentially also leads to an improvement of Schittkowski’s method. A numerical example illustrating the effects of the various modifications is given in section 5, and finally, section 6 contains some summary results of comparative tests performed by the author.