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Kaisa Miettinen: Introduction to Some Methods and Applications of Nonlinear Multiobjective Optimization

In multiobjective optimization, the goal is to find the best possible solution in the presence of several, conflicting objectives. Multiobjective optimization is important because most real-life problems contain more than one objective and they should be considered simultaneously.
Using mathematical tools we can define a set of nondominated or Pareto optimal solutions where none of the objective values can be improved without impairing at least one of the other objective values. Because vectors cannot be ordered completely, we need some some additional preference information from an expert, a decision maker, to find the most preferred Pareto optimal solution as the final solution to be implemented.
Multiobjective optimization methods can be classified according to the role of the decision maker in the solution process. We review different types of methods and summarize their strengths and weaknesses. We pay most attention to interactive methods, where the decision maker takes actively part in the solution process and directs the search for the final solution according to her/his desires and hopes. This enables the decision maker to gain insight about the interdependencies of the conflicting objectives and learn about one's own preferences. Among the methods to be discussed are the classification-based NIMBUS method, the Nautilus method which enables decision making without trading-off and Pareto Navigator for computationally expensive problems.
Finally, we study some complex real-life problems that have been successively solved by applying multiobjective optimization software and demonstrate how useful it is to consider several objectives simultaneously.
The problems are related, for example, to chemical engineering processes, optimal shape design and wastewater treatment. To conclude, we collect some experiences.

Keywords: Multiple objective optimization, nonlinear optimization, MCDM, decision support, interactive methods, NIMBUS, Nautilus, Pareto Navigator, software