Multi-objective Optimization - Solving A Multiobjective Optimization Problem

Solving A Multiobjective Optimization Problem

As there usually exist multiple Pareto optimal solutions for multiobjective optimization problems, what it means to solve such a problem is not as straightforward as it is for a single objective optimization problem. Therefore, different researchers have defined the term "solving a multiobjective optimization problem" in various ways. This section summarizes some of them and the contexts in which they are used. Many methods convert the original problem with multiple objectives into a single objective optimization problem. This is called a scalarized problem. If scalarization is done carefully, Pareto optimality of the solutions obtained can be guaranteed.

Solving a multiobjective optimization problem is sometimes understood as approximating or computing all or a representative set of Pareto optimal solutions. This is done, e.g., in and.

When decision making is emphasized, the objective of solving a multiobjective optimization problem is referred to supporting a decision maker in finding the most preferred Pareto optimal solution according to his/her preferences. This is followed e.g., in and. The underlying assumption is that one solution to the problem must be identified to be implemented in practice. Here, a human decision maker (DM) plays an important role. (S)he is expected to be an expert in the problem domain.

The most preferred solution can be found using different philosophies. In, multiobjective optimization methods are divided into four classes. In so-called no preference methods, no decision maker is expected to be available, but a neutral compromise solution is identified without preference information. The other classes are so-called a priori, a posteriori and interactive methods and they all involve preference information from the decision maker in different ways.

In a priori methods, preference information is first asked from the decision maker and then a solution best satisfying these preferences is found. In a posteriori methods, a representative set of Pareto optimal solutions is first found and then the decision maker must choose one of them. In interactive methods, the decision maker is allowed to iteratively search for the most preferred solution. In each iteration of the interactive method, the decision maker is shown Pareto optimal solution(s) and (s)he can tell how the solution(s) could be improved. The information given by the decision maker is then taken into account while generating new Pareto optimal solution(s) for the decision maker to study in the next iteration. In this way, the decision maker learns about the feasibility of his/her wishes and can concentrate on solutions that are interesting to him/her. The decision maker may stop the search whenever he/she wants to. More information and examples of different methods in the four classes are given in the following sections.