Improving Penalty Functions for Structural Optimization


Department of Civil Engineering,Sharif University of Technology


Abstract. New penalty functions, which have better convergence properties, as compared to the
commonly used exterior and interior penalty functions, have been proposed in this paper. The convergence
behavior and accuracy of ordinary penalty functions depend on the selection of appropriate penalty
parameters. The optimization of ordinary penalty functions is accomplished after several rounds of
optimization where, at each round a di erent but xed value of penalty parameter is used. While some
useful hints and rules for the selection of suitable penalty parameter values have been provided by di erent
authors, the objective of this paper has been to improve this procedure by including the penalty parameter
in the design vector, so that it can be modi ed during the optimization, automatically, in order to improve
the convergence characteristics. This can also help accomplish optimization in only one round, which is
of considerable importance when it is desired to solve a constrained problem by using genetic algorithms.
The proof of convergence to the optimum solution of the proposed functions is also included in the paper.
Ten-bar and three-bar truss examples are used for illustration through which the convergence of ordinary
and new functions are evaluated and compared. The results show that the new penalty functions can
outperform the ordinary functions, especially in combination with genetic algorithms.