A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems

Lourdes Uribe, Johan M Bogoya, Andrés Vargas, Adriana Lara, Günter Rudolph and Oliver Schütze
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Lourdes Uribe: Instituto Politécnico Nacional, Mexico City 07738, Mexico
Johan M Bogoya: Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 N. 40-62, Bogotá D.C. 111321, Colombia
Andrés Vargas: Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 N. 40-62, Bogotá D.C. 111321, Colombia
Adriana Lara: Instituto Politécnico Nacional, Mexico City 07738, Mexico
Günter Rudolph: Department of Computer Science, TU Dortmund University, 44227 Dortmund, Germany
Oliver Schütze: Department of Computer Science, Cinvestav-IPN, Mexico City 07360, Mexico

Mathematics, 2020, vol. 8, issue 10, 1-29

Abstract: 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 N 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 Δ p 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 N 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.

Keywords: multi-objective optimization; Newton method; performance indicator ?p; generational distance; inverted generational distance; set based optimization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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