Multi objective optimization

In multi-objective optimization or Pareto optimization, there is no longer only one objective function that needs to be optimized, but several objectives must be considered simultaneously. Since the different goals are usually opposite, there is also no optimal point as a solution. The result is now a solution set. It is defined by the fact that at each point in this set, one objective function can only be improved by worsening another. According to its founder Vilfredo Pareto (1848-1923), this separation of inefficient and unworkable solutions is called the Pareto front or the Pareto border.

Of course, it would be ideal to investigate the entire Pareto front. However, since this can be very time-consuming, it is common practice to combine the individual goals of interest into a single common goal function using weighting factors. Thus, there is again only one goal that can be determined with the known optimization methods. The solution found in this way corresponds to a point on the Pareto front. The choice of weighting factors, however, is inherent in a certain subjectivity, which provides a correspondingly arbitrary result. It could be that a slight variation of the factors leads to a significant improvement of one target value, while the other deteriorates only marginally. This can be countered by carrying out the optimization with different weighting factors. This gives you several points on the Pareto front and you can evaluate their shape and progression.

A similar point cloud on or near the Pareto front is also obtained by an evolutionary algorithm. Since this method is used to search the entire design space globally, the cloud of individuals moves from generation to generation towards the Pareto front. The more the algorithm converges, one measure of this is the fitness of the entire generation, the closer the points are to the Pareto front.