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We describe a new interactive learning-oriented method called Pareto navigator for nonlinear multiobjective optimization. In the method, first a polyhedral approximation of the Pareto optimal set is formed in the objective function space using a relatively small set of Pareto optimal solutions representing the Pareto optimal set. Then the decision maker can navigate around the polyhedral approximation and direct the search for promising regions where the most preferred solution could be located. In this way, the decision maker can learn about the interdependencies between the conflicting objectives and possibly adjust one's preferences. Once an interesting region has been identified, the polyhedral approximation can be made more accurate in that region or the decision maker can ask for the closest counterpart in the actual Pareto optimal set. If desired, (s)he can continue with another interactive method from the solution obtained. Pareto navigator can be seen as a nonlinear extension of the linear Pareto race method. After the representative set of Pareto optimal solutions has been generated, Pareto navigator is computationally efficient because the computations are performed in the polyhedral approximation and for that reason function evaluations of the actual objective functions are not needed. Thus, the method is well suited especially for problems with computationally costly functions. Furthermore, thanks to the visualization technique used, the method is applicable also for problems with three or more objective functions, and in fact it is best suited for such problems. After introducing the method in more detail, we illustrate it and the underlying ideas with an example.

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Multi-objective optimization problems are often solved by a sequence of parametric single-objective problems, so-called scalarizations. If the set of nondominated points is finite, the entire nondominated set can be generated in this way. In the bicriteria case it is well known that this can be realized by an adaptive approach which requires the solution of at most 2|Z N | − 1 subproblems, where Z N denotes the nondominated set of the underlying problem and a subproblem corresponds to a scalarized problem. For problems with more than two criteria, no methods were known up to now for which the number of subproblems depends linearly on the number of nondominated points. We present a new procedure for finding the entire nondominated set of tricriteria optimization problems for which the number of subproblems to be solved is bounded by 3|Z N | − 2, hence, depends linearly on the number of nondominated points. The approach includes an iterative update of the search region that, given a (sub-)set of nondominated points, describes the area in which additional nondominated points may be located. If the ε-constraint method is chosen as scalarization, the upper bound can be improved to 2|Z N | − 1.