Penalty Functions

This state of the art review starts with a discussion of the classical (Courant) penalty function and of the various theoretical results which can be proved. The function is used sequentially and numerical results are disappointing; the reasons for this are explained, and cannot be alleviated by extrapolation. These difficulties are apparently overcome by using the multiplier (augmented Lagrangian) penalty function, and the theoretical background and practical possibilities are described. However the current approach for forcing convergence has its disadvantages and there is scope for more research. Most interest currently centres on exact penalty functions, and in particular the l 1 exact penalty function. This is a nonsmooth function so cannot be minimized adequately by current techniques for smooth functions. However it is very useful as a criterion function in association with other techniques such as sequential QP. A thorough description of the l 1 penalty function is given, which aims to clarify the first and second order conditions associated with the minimizing point. There are some disadvantages of using non-smooth penalty functions including the existence of curved grooves which can be difficult to follow, and the possibility of the Maratos effect occurring. Suggestions for alleviating these difficulties are discussed. These effects have recently caused more emphasis in the search for suitable smooth exact penalty functions, and a survey of research in this area, and the theoretical and practical possibilities, is given. Finally some of the many other ideas for penalty functions are discussed briefly.