Engineering Optimization

31Response Surface Aberrations

31.1 Introduction

The term surface aberrations refers to features of the OF response to DVs, features of the topology, that make it difficult for optimization algorithms. Finding the optimum of an unconstrained quadratic surface is a trivial exercise. However, for many applications several types of OF features cause difficulty for the algorithm. These are easily visualized in the three dimension of a 2‐D application (OF response to DV1 and DV2) but translate to 1‐D and N‐D applications. The following illustrations show the 3‐D net and/or 2‐D contour map patterns.

Unlike many classic test functions, most of the illustrations here are not simple functions that are contrived by the mind of man to reveal an aberration to challenge an optimizer. Most of these examples naturally appeared in my applications and are described in the following sections.

31.2 Cliffs (Vertical Walls)

Function 41, “parallel pumps,” has level discontinuities (cliffs, or vertical walls) in the 3‐D graph of the OF response to the DVs. This is illustrated in Figure 31.1.

Figure 31.1 3‐D view of a surface with cliffs (level discontinuities).

The contour plot of Figure 31.2 also reveals the level discontinuities as multiple collinear contours.

Figure 31.2 Contour map of a surface ...