Direct versus indirect methods in structural optimization

Mathematical programming methods are among the most powerful optimization techniques. These techniques may be separated into direct and indirect methods. Of the direct methods of attack on general nonlinear inequality constrained problems, the largest class is the method of feasible directions. Of the indirect methods, the interior penalty function appears to be the most reliable one while the variable metric method seems to be an extremely powerful algorithm. This paper presents a comparison between the results obtained using Zoutendijk's method of feasible directions and the method of interior penalty function coupled with the variable metric method as a minimizing algorithm. A considerable improvement in convergence has been achieved by considering each push-off factor as a linear function of the corresponding active constraint. A comparison of the half-step vs full-step search procedure is presented. Also a comparison between the use of either the normalized or the non-normalized gradients is illustrated. A discussion of the linear vs quadratic interpolations of a constraint function in search for a bound point is presented. An initial step length based on a present decrement of objective function is used. The two algorithms are demonstrated with elastic design of a 25-bar space tower, a 3-bay single-storey frame and a double-bay double-storey rigid jointed plane frame. Data on the differences in the optimal designs obtained from different starting points are reported.