Enhancement of sandwich algorithms for approximating higher dimensional convex Pareto sets
Rennen,G. van Dam,E.R. den Hertog,D.
Rennen,G.
van Dam,E.R.
den Hertog,D.
Abstract
In many fields, we come across problems where we want to optimize several conflicting objectives simultaneously. To find a good solution for such multiobjective optimization problems, an approximation of the Pareto set is often generated. In this paper, we consider the approximation of higher-dimensional convex Pareto sets using sandwich algorithms. We extend higher-dimensional sandwich algorithms in three different ways. First, we introduce the new concept of adding dummy points to the inner approximation of a Pareto set. By using these dummy points, we can determine accurate inner and outer approximations more efficiently, i.e., using less time-consuming optimizations. Second, we introduce a new method for the calculation of an error measure that is easy to interpret. Third, we show how transforming certain objective functions can improve the results of sandwich algorithms and extend their applicability to certain nonconvex problems. To show the effect of these enhancements, we make a numerical comparison using four test cases, including a four-dimensional case from the field of intensity-modulated radiation therapy. The results of the different cases show that we can achieve an accurate approximation using significantly fewer optimizations by using the enhancements.
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Appeared earlier as CentER DP 2009-52
Date
2011
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Rennen, G, van Dam, E R & den Hertog, D 2011, 'Enhancement of sandwich algorithms for approximating higher dimensional convex Pareto sets', INFORMS Journal on Computing, vol. 23, no. 4, pp. 493-517. https://doi.org/10.1287/ijoc.1100.0419