Engineering Optimization

23Multiple Optima

23.1 Introduction

One of the several attributes of optimization applications that can lead an optimization astray is the existence of multiple optima, multiple local minima. If an optimizer trial solution is in the vicinity of local optima, then it will seek the bottom of that local hole, and not climb out to see the deeper hole next door. The concept is illustrated in Figure 23.1.

Figure 23.1 Multiple optima (a) discontinuous (b) continuum values.

Source: Rhinehart (2016b). Reproduced with permission of Wiley.

Each figure illustrates three local minima. Figure 23.1a could illustrate delivery costs for a shipping company as it changes with business size. With small‐sized service, buy a small truck. The more you ship, the less the truck cost/delivery. However, when shipping volume exceeds the small truck capacity, you need to either trade it for a larger truck or buy a second small truck. This extra capital increases cost/delivery of the first increment over the small truck capacity but permits you to move toward larger volumes. The OF has local optima with discontinuities.

In contrast, Figure 23.1b shows an OF as a continuous function with multiple optima. It also has three optima, one at the feasible DV left boundary.

In each case, one optimum is clearly the global best. This is called the global optima. The others are local optima.

Also, in each case the ...