Enhancement of Sandwich Algorithms for Approximating Higher Dimensional Convex Pareto Sets

Abstract

In many fields, we come across problems where we want to optimize several conflicting objectives simultaneously. To find a good solution for such multi-objective optimization problems, an approximation of the Pareto set is often generated. In this paper, we consider the approximation of Pareto sets for problems with three or more convex objectives and with convex constraints. For these problems, sandwich algorithms can be used to determine an inner and outer approximation between which the Pareto set is 'sandwiched'. Using these two approximations, we can calculate an upper bound on the approximation error. This upper bound can be used to determine which parts of the approximations must be improved and to provide a quality guarantee to the decision maker. In this paper, we extend higher dimensional sandwich algorithms in three different ways. Firstly, 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. Secondly, we introduce a new method for the calculation of an error measure which is easy to interpret. The combination of easy calculation and easy interpretation makes this measure very suitable for sandwich algorithms. Thirdly, we show how transforming certain objective functions can improve the results of sandwich algorithms and extend their applicability to certain non-convex problems. The calculation of the introduced error measure when using transformations will also be discussed.

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 (IMRT). The results of the different cases show that we can indeed achieve an accurate approximation using significantly fewer optimizations by using the enhancements.

Keywords: Convexity, є-efficiency, ε-Pareto optimality, Geometric programming, Higher dimensional, Inner and outer approximation, IMRT, Pareto set, Multi-objective optimization, Sandwich algorithms, Transformations

JEL Classification: C61