The use of negative penalty functions in constrained variational problems

A major drawback of the penalty function method in solving constrained variational problems is the difficulty in choosing suitable penalty parameters that are large enough to effect a constraint but small enough to avoid computational problems. This problem steins from the fact that the appropriate penalty value is normally found by a trial and error process by gradually increasing the parameter until the results converge numerically. In this paper, it is shown that the constrained solution for the deflection of a beam is bounded by results from mathematical models using positive and negative penalty parameters. If this is true in general, any error due to the use of penalty functions in solving a variational problem may be determined and controlled by using positive and negative values for the penalty parameter. Copyright (C) 2002 John Wiley Sons, Ltd.