Discussion of a Family of Merit Functions

Optimization

Again we consider the sequence of proposed solutions using our merit function coupled with the optimization. As before, we denote by x_c the current best solution for the "true" problem f(x). But now x_t is the trial solution proposed by the merit function m rather than by the approximation a.

• Iteration 1: x_c = 0.20, x_t = -0.50
• Iteration 2: x_c = 0.20, x_t = 0.50
• Iteration 3: x_c = 0.20, x_t = 1.50
• Iteration 4: x_c = 0.20, x_t = -0.20
• Iteration 5: x_c = 0.20, x_t = 1.20
• Iteration 6: x_c = 1.20, x_t = 1.30
• Iteration 7: x_c = 1.30, x_t = 1.40
• Iteration 7: x* = 1.30

It is the case that because we now assign "merit" to sampling in regions for which we have no information about the objective, we are slower to improve the best known solution to the optimization problem. For this example, x_c is not replaced until Iteration 6 whereas, earlier, working only with the approximation a, we replaced x_c by Iteration 3.

On the other hand, using the merit function---rather than just the approximation--enforces a better "spread" of the sample points throughout the design region. The beneficial effects of this are apparent by Iteration 5. Earlier, when we did not make use of the merit function, with the exception of the two end points, all our sampling occurred in the basin with a local minimizer at 0; thus, the behavior of the objective in the rest of the design space is largely unknown. However, when we use the merit function to select candidate designs, by Iteration 5 even the approximation indicates that a global minimizer should be in the region near 1.2; the merit function m only further emphasizes this prediction.

Note that the approximation a still predicts the local minimizer at 0--and the merit function m emphasizes this prediction too. If we wished to search for other minimizers, we certainly have adequate predictive ability to do so. But now the global minimizer dominates. Having identified it, we are content to stop.