<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>0122-7483</journal-id>
<journal-title><![CDATA[Universitas Scientiarum]]></journal-title>
<abbrev-journal-title><![CDATA[Univ. Sci.]]></abbrev-journal-title>
<issn>0122-7483</issn>
<publisher>
<publisher-name><![CDATA[Facultad de Ciencias de la Pontificia Universidad Javeriana de Bogotá.]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0122-74832018000300333</article-id>
<article-id pub-id-type="doi">10.11144/javeriana.sc23-3.otpc</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[On the Pareto compliance of the averaged hausdorff distance as a performance indicator]]></article-title>
<article-title xml:lang="es"><![CDATA[Sobre la sujeción de Pareto para la distancia promedio de hausdorff como indicador de desempeño]]></article-title>
<article-title xml:lang="pt"><![CDATA[Sobre a sujeição de Pareto para a distância média de hausdorff como indicador de desempenho]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Vargas]]></surname>
<given-names><![CDATA[Andrés]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Salcedo-Reyes]]></surname>
<given-names><![CDATA[Juan Carlos]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Pontificia Universidad Javeriana  ]]></institution>
<addr-line><![CDATA[Bogotá ]]></addr-line>
<country>Colombia</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>12</month>
<year>2018</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>12</month>
<year>2018</year>
</pub-date>
<volume>23</volume>
<numero>3</numero>
<fpage>333</fpage>
<lpage>354</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_arttext&amp;pid=S0122-74832018000300333&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_abstract&amp;pid=S0122-74832018000300333&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_pdf&amp;pid=S0122-74832018000300333&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract The averaged Hausdorff distance &#916; p is an inframetric, recently introduced in evolutionary multiobjective optimization (EMO) as a tool to measure the optimality of finite size approximations to the Pareto front associated to a multiobjective optimization problem (MOP). Tools of this kind are called performance indicators, and their quality depends on the useful criteria they provide to evaluate the suitability of different candidate solutions to a given MOP. We present here a purely theoretical study of the compliance of the &#916;  p -indicator to the notion of Pareto optimality. Since &#916;  p is defined in terms of a modified version of other well-known indicators, namely the generational distance GDp , and the inverted generational distance IGDp , specific criteria for the Pareto compliance of each one of them is discussed in detail. In doing so, we review some previously available knowledge on the behavior of these indicators, correcting inaccuracies found in the literature, and establish new and more general results, including detailed proofs and examples of illustrative situations.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen La distancia promedio de Hausdorff &#916;p es una inframétrica recientemente introducida en optimización multiobjetivo evolutiva (EMO) como una herramienta para medir la optimalidad de aproximaciones finitas al frente de Pareto asociado con un problema de optimización multiobjetivo (MOP). Presentamos aquí un estudio puramente teórico sobre la sujeción del indicador &#916;p a la noción de optimalidad de Pareto. Puesto que &#916;p está definida en términos de una versión modificada de otros indicadores bien conocidos como lo son la distancia generacional GD p y la distancia generacional invertida IGD p , discutimos en detalle criterios específicos para la sujeción de tipo Pareto de cada uno de ellos. Adicionalmente, presentamos una revisión del comportamiento previamente conocido de estos indicadores, corrigiendo imprecisiones que se encuentran en la literatura y establecemos resultados nuevos y más generales, incluyendo pruebas detalladas y ejemplos ilustrativos.]]></p></abstract>
<abstract abstract-type="short" xml:lang="pt"><p><![CDATA[Resumo A distância média de Hausdorff &#916;p é uma inframétrica introduzida recentemente em otimização multiobjetivo evolucionária (EMO) como uma ferramenta para medir a otimalidade de aproximações finitas para o frente de Pareto associado com um problema de optimização multiobjetivo (MOP). Apresentamos aqui um estudo puramente teórico sobre a sujeição do indicador &#916;p à noção de otimalidade de Pareto. Desde &#916;p é definido em termos de uma versão modificada de outros indicadores bem conhecidos, tais como a distância geraçional GD p &#8712; a distância geraçional invertida IGD p , discutimos em detalhes critérios específicos para a sujeição de Pareto de cada um deles. Além disso, apresentamos uma revisão do comportamento previamente conhecido desses indicadores, corrigindo imprecisões encontradas na literatura &#8712; estabelecemos novos &#8712; mais gerais resultados, incluindo testes detalhados &#8712; exemplos ilustrativos.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[averaged Hausdorff distance]]></kwd>
<kwd lng="en"><![CDATA[generational distance]]></kwd>
<kwd lng="en"><![CDATA[inverted generational distance]]></kwd>
<kwd lng="en"><![CDATA[multiobjective optimization]]></kwd>
<kwd lng="en"><![CDATA[Pareto optimality]]></kwd>
<kwd lng="en"><![CDATA[performance indicator]]></kwd>
<kwd lng="es"><![CDATA[distancia promedio de Hausdorff]]></kwd>
<kwd lng="es"><![CDATA[distancia generacional]]></kwd>
<kwd lng="es"><![CDATA[distancia generacional invertida]]></kwd>
<kwd lng="es"><![CDATA[optimización multiobjetivo]]></kwd>
<kwd lng="es"><![CDATA[optimalidad de Pareto]]></kwd>
<kwd lng="es"><![CDATA[indicador de desempeño]]></kwd>
<kwd lng="pt"><![CDATA[distância média de Hausdorff]]></kwd>
<kwd lng="pt"><![CDATA[distância geraçional]]></kwd>
<kwd lng="pt"><![CDATA[distância geraçional invertida]]></kwd>
<kwd lng="pt"><![CDATA[otimização multiobjetivo]]></kwd>
<kwd lng="pt"><![CDATA[otimalidade de Pareto]]></kwd>
<kwd lng="pt"><![CDATA[indicador de desempenho]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<label>[1]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bogoya]]></surname>
<given-names><![CDATA[JM]]></given-names>
</name>
<name>
<surname><![CDATA[Vargas]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Cuate]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<name>
<surname><![CDATA[Schütze]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A (p, q)-averaged Hausdorff distance for arbitrary measurable sets]]></article-title>
<source><![CDATA[Mathematical and Computational Applications]]></source>
<year>2018</year>
<volume>23</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>51</page-range></nlm-citation>
</ref>
<ref id="B2">
<label>[2]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Coello]]></surname>
<given-names><![CDATA[CAC]]></given-names>
</name>
<name>
<surname><![CDATA[Cortés]]></surname>
<given-names><![CDATA[NC]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Solving multiobjective optimization problems using an artificial immune system]]></article-title>
<source><![CDATA[Genetic Programming and Evolvable Machines]]></source>
<year>2005</year>
<volume>6</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>163-90</page-range></nlm-citation>
</ref>
<ref id="B3">
<label>[3]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Hansen]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<name>
<surname><![CDATA[Jaszkiewicz]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
</person-group>
<source><![CDATA[Evaluating the quality of approximations to the non-dominated set]]></source>
<year>1998</year>
<publisher-name><![CDATA[IMM, Department of Mathematical Modelling, Technical University of Denmark]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B4">
<label>[4]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Heinonen]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
</person-group>
<source><![CDATA[Lectures on analysis on metric spaces. Universitext]]></source>
<year>2001</year>
<publisher-name><![CDATA[Springer-Verlag, New York]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B5">
<label>[5]</label><nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[lermeier]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
</person-group>
<source><![CDATA[Nonlinear Multiobjective Optimization. A Generalized Homotopy Approach]]></source>
<year>2001</year>
<publisher-loc><![CDATA[Birkhauser ]]></publisher-loc>
</nlm-citation>
</ref>
<ref id="B6">
<label>[6]</label><nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ishibuchi]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
<name>
<surname><![CDATA[Masuda]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
<name>
<surname><![CDATA[Tanigaki]]></surname>
<given-names><![CDATA[Y]]></given-names>
</name>
<name>
<surname><![CDATA[Nojima]]></surname>
<given-names><![CDATA[Y]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Modified Distance Calculation in Generational Distance and Inverted Generational Distance]]></article-title>
<person-group person-group-type="editor">
<name>
<surname><![CDATA[Gaspar-Cunha]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Antunes]]></surname>
<given-names><![CDATA[CH]]></given-names>
</name>
<name>
<surname><![CDATA[Coello Coello]]></surname>
<given-names><![CDATA[CA]]></given-names>
</name>
</person-group>
<source><![CDATA[Evolutionary Multi-Criterion Optimization 8th International Conference EMO 2015, Guimarães, Portugal. Lecture Notes in Computer Science]]></source>
<year>2015</year>
<numero>9019</numero>
<issue>9019</issue>
<page-range>110-25</page-range></nlm-citation>
</ref>
<ref id="B7">
<label>[7]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Pareto]]></surname>
<given-names><![CDATA[V]]></given-names>
</name>
<name>
<surname><![CDATA[Montesano]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Zanni]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Bruni]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
<name>
<surname><![CDATA[Chipman]]></surname>
<given-names><![CDATA[JS]]></given-names>
</name>
<name>
<surname><![CDATA[McLureet]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
</person-group>
<source><![CDATA[Manual of Political Economy: A Critical and Variorum Edition]]></source>
<year>2014</year>
<publisher-name><![CDATA[Oxford University Press]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B8">
<label>[8]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Rudolph]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Schütze]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<name>
<surname><![CDATA[Grimme]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
<name>
<surname><![CDATA[Domínguez-Medina]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
<name>
<surname><![CDATA[Trautmann]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Optimal averaged Hausdorff archives for bi-objective problems: theoretical and numerical results]]></article-title>
<source><![CDATA[Computational Optimization and Applications]]></source>
<year>2016</year>
<volume>64</volume>
<page-range>589-618</page-range></nlm-citation>
</ref>
<ref id="B9">
<label>[9]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Schütze]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<name>
<surname><![CDATA[Esquivel]]></surname>
<given-names><![CDATA[X]]></given-names>
</name>
<name>
<surname><![CDATA[Lara]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Coello]]></surname>
<given-names><![CDATA[CA]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Using the averaged Hausdorff distance as a performance measure in evolutionary multiobjective optimization]]></article-title>
<source><![CDATA[IEEE Transactions on Evolutionary Computation]]></source>
<year>2012</year>
<volume>16</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>504-22</page-range></nlm-citation>
</ref>
<ref id="B10">
<label>[10]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Siwel]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
<name>
<surname><![CDATA[Yew-Soon]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<name>
<surname><![CDATA[Jie]]></surname>
<given-names><![CDATA[Z]]></given-names>
</name>
<name>
<surname><![CDATA[Liang]]></surname>
<given-names><![CDATA[F]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Consistencies and contradictions of performance metrics in multiobjective optimization]]></article-title>
<source><![CDATA[IEEE Transactions on Evolutionary Computation]]></source>
<year>2014</year>
<volume>44</volume>
<numero>12</numero>
<issue>12</issue>
<page-range>2329-404</page-range></nlm-citation>
</ref>
<ref id="B11">
<label>[11]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Sun]]></surname>
<given-names><![CDATA[JQ]]></given-names>
</name>
<name>
<surname><![CDATA[Xiong]]></surname>
<given-names><![CDATA[FR]]></given-names>
</name>
<name>
<surname><![CDATA[Schütze]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<name>
<surname><![CDATA[Hernández]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Cell Mapping Methods - Algorithmic Approaches and Applications]]></article-title>
<source><![CDATA[Nonlinear Systems and Complexity]]></source>
<year>2019</year>
<numero>99</numero>
<issue>99</issue>
<publisher-name><![CDATA[Springer]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B12">
<label>[12]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Vargas]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Bogoya]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A generalization of the averaged Hausdorff distance]]></article-title>
<source><![CDATA[Computación y Sistemas]]></source>
<year>2018</year>
<volume>22</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>331-45</page-range></nlm-citation>
</ref>
<ref id="B13">
<label>[13]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Veldhuizen]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
</person-group>
<source><![CDATA[Multiobjective evolutionary algorithms: classifications, analyses, and new innovations]]></source>
<year>1999</year>
<page-range>249</page-range><publisher-name><![CDATA[Air Force Institute of Technology]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B14">
<label>[14]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Zitzler]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
<name>
<surname><![CDATA[Thiele]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach]]></article-title>
<source><![CDATA[IEEE Transactions on Evolutionary Computation]]></source>
<year>1999</year>
<volume>3</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>257-71</page-range></nlm-citation>
</ref>
<ref id="B15">
<label>[15]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Zitzler]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
<name>
<surname><![CDATA[Thiele]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
<name>
<surname><![CDATA[Laumanns]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<name>
<surname><![CDATA[Fonseca]]></surname>
<given-names><![CDATA[CM]]></given-names>
</name>
<name>
<surname><![CDATA[Da Fonseca]]></surname>
<given-names><![CDATA[VG]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Performance assessment of multiobjective optimizers: An analysis and review]]></article-title>
<source><![CDATA[IEEE Transactions on Evolutionary Computation]]></source>
<year>2003</year>
<volume>7</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>117-32</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
