A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems
Multi-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute <i>N</i> solutions along the Pareto set/front of a given MOP...
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MDPI AG
2020-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/10/1822 |
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| author | Lourdes Uribe Johan M Bogoya Andrés Vargas Adriana Lara Günter Rudolph Oliver Schütze |
| author_facet | Lourdes Uribe Johan M Bogoya Andrés Vargas Adriana Lara Günter Rudolph Oliver Schütze |
| author_sort | Lourdes Uribe |
| collection | DOAJ |
| description | Multi-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute <i>N</i> solutions along the Pareto set/front of a given MOP. In this work, we propose and discuss the set based Newton methods for the performance indicators Generational Distance (GD), Inverted Generational Distance (IGD), and the averaged Hausdorff distance <inline-formula><math display="inline"><semantics><msub><mi mathvariant="sans-serif">Δ</mi><mi>p</mi></msub></semantics></math></inline-formula> for reference set problems for unconstrained MOPs. The methods hence directly utilize the set based scalarization problems that are induced by these indicators and manipulate all <i>N</i> candidate solutions in each iteration. We demonstrate the applicability of the methods on several benchmark problems, and also show how the reference set approach can be used in a bootstrap manner to compute Pareto front approximations in certain cases. |
| first_indexed | 2024-03-10T15:33:02Z |
| format | Article |
| id | doaj.art-f505febf83fc43c3922df28230c584d1 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T15:33:02Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f505febf83fc43c3922df28230c584d12023-11-20T17:30:16ZengMDPI AGMathematics2227-73902020-10-01810182210.3390/math8101822A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set ProblemsLourdes Uribe0Johan M Bogoya1Andrés Vargas2Adriana Lara3Günter Rudolph4Oliver Schütze5Instituto Politécnico Nacional, Mexico City 07738, MexicoDepartamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 N. 40-62, Bogotá D.C. 111321, ColombiaDepartamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 N. 40-62, Bogotá D.C. 111321, ColombiaInstituto Politécnico Nacional, Mexico City 07738, MexicoDepartment of Computer Science, TU Dortmund University, 44227 Dortmund, GermanyDepartment of Computer Science, Cinvestav-IPN, Mexico City 07360, MexicoMulti-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute <i>N</i> solutions along the Pareto set/front of a given MOP. In this work, we propose and discuss the set based Newton methods for the performance indicators Generational Distance (GD), Inverted Generational Distance (IGD), and the averaged Hausdorff distance <inline-formula><math display="inline"><semantics><msub><mi mathvariant="sans-serif">Δ</mi><mi>p</mi></msub></semantics></math></inline-formula> for reference set problems for unconstrained MOPs. The methods hence directly utilize the set based scalarization problems that are induced by these indicators and manipulate all <i>N</i> candidate solutions in each iteration. We demonstrate the applicability of the methods on several benchmark problems, and also show how the reference set approach can be used in a bootstrap manner to compute Pareto front approximations in certain cases.https://www.mdpi.com/2227-7390/8/10/1822multi-objective optimizationNewton methodperformance indicator Δpgenerational distanceinverted generational distanceset based optimization |
| spellingShingle | Lourdes Uribe Johan M Bogoya Andrés Vargas Adriana Lara Günter Rudolph Oliver Schütze A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems Mathematics multi-objective optimization Newton method performance indicator Δp generational distance inverted generational distance set based optimization |
| title | A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems |
| title_full | A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems |
| title_fullStr | A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems |
| title_full_unstemmed | A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems |
| title_short | A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems |
| title_sort | set based newton method for the averaged hausdorff distance for multi objective reference set problems |
| topic | multi-objective optimization Newton method performance indicator Δp generational distance inverted generational distance set based optimization |
| url | https://www.mdpi.com/2227-7390/8/10/1822 |
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