Two-step framework for multi-objective optimization, Report no. LiTH-ISY-R-3043

In many real-world optimization applications there are often a number of conflicting objective functions that are all important to optimize. The purpose of multiobjective optimization (MOO) is to give the decision maker (DM) an understanding of how these functions are conflicting and the possibility to choose an appropriate trade-off between them. There are multiple methods for solving MOO problems but the focus in this paper is on interactive methods. When the size and complexity of the MOO problem grows the time needed to find a solution is too long to yield a pleasant experience for the DM. In this paper, a method to replace the original MOO problem with an approximation is suggested to speed up the process. The approximation is created and used in a two-step framework which makes it possible to investigate the Pareto frontier in real-time and that can handle nonlinear and non-convex MOO problems with m objective functions. The first step generates a number of samples of the complete Pareto frontier which is sparse but dense enough for the approximation. The second step is an interactive tool for the DM to use to continuously and in real-time navigate on the approximated Pareto set in both objectiveand decision space. The tool is used to investigate the Pareto frontier and to find a preferred solution. A method of decomposing the approximated set into simplices has been developed using Delaunay triangulation. This method is able to make a good approximation for sets that are non-convex. The method is also able to handle disconnected sets and holes. This makes it possible to change the feasible region in both decisionand objective space. The framework is demonstrated on three example problems that show the functionality and performance of the implemented framework.