| Abstract: | A well known approach to constrained optimization is via a sequenceof unconstrained minimization calculations applied to a penaltyfunction. This paper shown how it is posiible to generalizePowell's penelty function to solve constrained problems withboth equality and inequality constraints. The resulting methodsare equivalent to the Hestenes' method of multipliers, and ageneralization of this to inequality constraints suggested byRockafellar. Local duality results (not all of which have appearedbefore) for these methods are reviewed, with particular emphasison those of practical importance. It is shown that various strategiesfor varying control parameters are possible, all of which canbe viewed as Newton or Newton-like iterations applied to thedual problem. Practical strategies for guaranteeing convergenceare also discussed. A wide selection of numerical evidence isreported, and the algorithms are compared both amongst themselvesand with other penalty function methods. The new penalty functionis well conditioned, without singularities, and it is not necessaryfor the control parameters to tend to infinity in order to forceconvergence. The rate of convergence is rapid and high accuracyis achieved in few unconstrained minimizations.; furthermorethe computational effort for successive minimizations goes downrapidly. The methods are very easy to program efficiently, usingan established quasi-Newton subroutine for unconstrained minimization. |
