Enhancement of sandwich algorithms for approximating higher dimensional convex Pareto sets

Research output: Contribution to journalArticleScientificpeer-review

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.
Original languageEnglish
Pages (from-to)493-517
JournalINFORMS Journal on Computing
Volume23
Issue number4
DOIs
Publication statusPublished - 2011

Fingerprint

Radiotherapy
Multiobjective optimization
Pareto
Approximation
Enhancement
Objective function
Therapy
Optimization problem
Radiation
Multi-objective optimization

Cite this

@article{09fec7dbfd17483597614fc816851fce,
title = "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 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.",
author = "G. Rennen and {van Dam}, E.R. and {den Hertog}, D.",
note = "Appeared earlier as CentER DP 2009-52",
year = "2011",
doi = "10.1287/ijoc.1100.0419",
language = "English",
volume = "23",
pages = "493--517",
journal = "INFORMS Journal on Computing",
issn = "1091-9856",
publisher = "INFORMS Inst.for Operations Res.and the Management Sciences",
number = "4",

}

Enhancement of sandwich algorithms for approximating higher dimensional convex Pareto sets. / Rennen, G.; van Dam, E.R.; den Hertog, D.

In: INFORMS Journal on Computing, Vol. 23, No. 4, 2011, p. 493-517.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Enhancement of sandwich algorithms for approximating higher dimensional convex Pareto sets

AU - Rennen, G.

AU - van Dam, E.R.

AU - den Hertog, D.

N1 - Appeared earlier as CentER DP 2009-52

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

U2 - 10.1287/ijoc.1100.0419

DO - 10.1287/ijoc.1100.0419

M3 - Article

VL - 23

SP - 493

EP - 517

JO - INFORMS Journal on Computing

JF - INFORMS Journal on Computing

SN - 1091-9856

IS - 4

ER -