<?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>0370-3908</journal-id>
<journal-title><![CDATA[Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales]]></journal-title>
<abbrev-journal-title><![CDATA[Rev. acad. colomb. cienc. exact. fis. nat.]]></abbrev-journal-title>
<issn>0370-3908</issn>
<publisher>
<publisher-name><![CDATA[Academia Colombiana de Ciencias Exactas, Físicas y Naturales]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0370-39082022000200325</article-id>
<article-id pub-id-type="doi">10.18257/raccefyn.1623</article-id>
<title-group>
<article-title xml:lang="es"><![CDATA[Resolviendo el problema de valores propios complementarios mediante un algoritmo cuasi-Newton]]></article-title>
<article-title xml:lang="en"><![CDATA[Solving the eigenvalue complementarity problem using a quasi-Newton algorithm]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Arenas]]></surname>
<given-names><![CDATA[Favián]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Arias]]></surname>
<given-names><![CDATA[Carlos]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Pérez]]></surname>
<given-names><![CDATA[Rosana]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad del Cauca Departamento de Matemáticas ]]></institution>
<addr-line><![CDATA[Popayán ]]></addr-line>
<country>Colombia</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>06</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>06</month>
<year>2022</year>
</pub-date>
<volume>46</volume>
<numero>179</numero>
<fpage>325</fpage>
<lpage>338</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_arttext&amp;pid=S0370-39082022000200325&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_abstract&amp;pid=S0370-39082022000200325&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_pdf&amp;pid=S0370-39082022000200325&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen En este artículo consideramos el problema de valores propios complementarios, de gran interés para muchos investigadores por sus numerosas aplicaciones en Ingeniería y Física, y abordamos su solution como un problema de complementariedad no lineal usando mi método cuasi-Newton, un tipo de método que, hasta donde conocemos, no ha sido utilizad para dicho fin. Verificamos que el problema satisface ciertas hipótesis que permiten utilizar un algoritmo global cuasi-Newton y realizamos pruebas numéricas que muestran la eficacia del algoritmo utilizado y lo hacen una buena alternativa de solución para problema de rotores propios complementarios.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract In this article we consider the complementarity eigenvalue problem, which is of great interest to many researchers due to its numerous applications in Engineering and Physics. We approach its solution as a nonlinear complementarity problem using a quasi-Newton method, a type of method that, as far as we know, has not been used for this purpose. We verify that the problem satisfies certain hypotheses that allow the use of a global quasi-Newton algorithm and we analyze its numerical performance. Numerical tests show the efficiency of the algorithm used and make it a good alternative to solve complementary eigenvalue problems]]></p></abstract>
<kwd-group>
<kwd lng="es"><![CDATA[valores propios complementarios]]></kwd>
<kwd lng="es"><![CDATA[problema de complementariedad no lineal]]></kwd>
<kwd lng="es"><![CDATA[programmation no lineal]]></kwd>
<kwd lng="es"><![CDATA[cuasi-Newton]]></kwd>
<kwd lng="en"><![CDATA[eigenvalue complementarity]]></kwd>
<kwd lng="en"><![CDATA[nonlinear complementarity problem]]></kwd>
<kwd lng="en"><![CDATA[nonlinear programming]]></kwd>
<kwd lng="en"><![CDATA[quasi-Newton, AMS Subject Classification: 93B60]]></kwd>
<kwd lng="en"><![CDATA[90C30]]></kwd>
<kwd lng="en"><![CDATA[90C33]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Arenas]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
<name>
<surname><![CDATA[Martínez]]></surname>
<given-names><![CDATA[H. J.]]></given-names>
</name>
<name>
<surname><![CDATA[Pérez]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Redefinición de la función de complementariedad de Kanzow]]></article-title>
<source><![CDATA[Revista de Ciencias]]></source>
<year>2014</year>
<volume>18</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>111-22</page-range></nlm-citation>
</ref>
<ref id="B2">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Arenas]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
<name>
<surname><![CDATA[Martínez]]></surname>
<given-names><![CDATA[H. J.]]></given-names>
</name>
<name>
<surname><![CDATA[Pérez]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Least change secant update methods for nonlinear complementarity problem]]></article-title>
<source><![CDATA[Ingeniería y Ciencia]]></source>
<year>2015</year>
<volume>11</volume>
<numero>21</numero>
<issue>21</issue>
<page-range>11-36</page-range></nlm-citation>
</ref>
<ref id="B3">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Arenas]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
<name>
<surname><![CDATA[Martínez]]></surname>
<given-names><![CDATA[H.]]></given-names>
</name>
<name>
<surname><![CDATA[Pérez]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A local jacobian smoothing method for solving nonlinear complementarity problems]]></article-title>
<source><![CDATA[Universitas Scientiarum]]></source>
<year>2020</year>
<month>,</month>
<volume>25</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>149-74</page-range></nlm-citation>
</ref>
<ref id="B4">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Arias]]></surname>
<given-names><![CDATA[C.]]></given-names>
</name>
<name>
<surname><![CDATA[Martínez]]></surname>
<given-names><![CDATA[H.]]></given-names>
</name>
<name>
<surname><![CDATA[Pérez]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A nonsmooth global cuasi-newton method for nonlinear complementarity problems]]></article-title>
<source><![CDATA[Computer Methods in Applied Mechanics and Engineering]]></source>
<year>2017</year>
<volume>13</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>1-15</page-range></nlm-citation>
</ref>
<ref id="B5">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Brown]]></surname>
<given-names><![CDATA[P. N.]]></given-names>
</name>
<name>
<surname><![CDATA[Saad]]></surname>
<given-names><![CDATA[Y]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Convergence theory of nonlinear Newton-Krylov algorithms]]></article-title>
<source><![CDATA[SIAM Journal on Optimization]]></source>
<year>1994</year>
<volume>4</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>297-330</page-range></nlm-citation>
</ref>
<ref id="B6">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Broyden]]></surname>
<given-names><![CDATA[C. G]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A Class of Methods for Solving Nonlinear Simultaneous Equations]]></article-title>
<source><![CDATA[Mathematics of Computation]]></source>
<year>1965</year>
<volume>19</volume>
<numero>92</numero>
<issue>92</issue>
<page-range>577-93</page-range></nlm-citation>
</ref>
<ref id="B7">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Clarke]]></surname>
<given-names><![CDATA[F. H]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Generalized gradients and applications]]></article-title>
<source><![CDATA[Transactions of the american society]]></source>
<year>1975</year>
<volume>205</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>247-62</page-range></nlm-citation>
</ref>
<ref id="B8">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Dennis]]></surname>
<given-names><![CDATA[J. E.]]></given-names>
</name>
<name>
<surname><![CDATA[Schnabel]]></surname>
<given-names><![CDATA[R. B]]></given-names>
</name>
</person-group>
<source><![CDATA[Numerical methods for unconstrained optimization and nonlinear equations]]></source>
<year>1996</year>
<publisher-name><![CDATA[Society for Industrial and Applied Mathematics]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B9">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Facchinei]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
<name>
<surname><![CDATA[Pang]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
</person-group>
<source><![CDATA[Finite-dimensional variational inequalities and complementarity problems (Vol. II; Springer, Ed.)]]></source>
<year>2003</year>
<publisher-loc><![CDATA[Springer ]]></publisher-loc>
<publisher-name><![CDATA[Verlag]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B10">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ferris]]></surname>
<given-names><![CDATA[M. C.]]></given-names>
</name>
<name>
<surname><![CDATA[Pang]]></surname>
<given-names><![CDATA[J. S]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Engineering and economic applications of complementarity problems]]></article-title>
<source><![CDATA[SIAM Review]]></source>
<year>1997</year>
<volume>39</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>669-713</page-range></nlm-citation>
</ref>
<ref id="B11">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Iusem]]></surname>
<given-names><![CDATA[A. N.]]></given-names>
</name>
<name>
<surname><![CDATA[Júdice]]></surname>
<given-names><![CDATA[J. J.]]></given-names>
</name>
<name>
<surname><![CDATA[Sessa]]></surname>
<given-names><![CDATA[V.]]></given-names>
</name>
<name>
<surname><![CDATA[Sarabando]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Splitting methods for the eigenvalue complementarity problem]]></article-title>
<source><![CDATA[Optimization Methods and Software]]></source>
<year>2019</year>
<volume>34</volume>
<numero>6</numero>
<issue>6</issue>
<page-range>1184-212</page-range></nlm-citation>
</ref>
<ref id="B12">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Júdice]]></surname>
<given-names><![CDATA[J. J.]]></given-names>
</name>
<name>
<surname><![CDATA[Sherali]]></surname>
<given-names><![CDATA[H. D.]]></given-names>
</name>
<name>
<surname><![CDATA[Ribeiro]]></surname>
<given-names><![CDATA[I]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The eigenvalue complementarity problem]]></article-title>
<source><![CDATA[Computational Optimization and Applications]]></source>
<year>2007</year>
<volume>37</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>139-56</page-range></nlm-citation>
</ref>
<ref id="B13">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kanzow]]></surname>
<given-names><![CDATA[C.]]></given-names>
</name>
<name>
<surname><![CDATA[Kleinmichel]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A new class of semismooth Newton-type methods for nonlinear complementarity problems.]]></article-title>
<source><![CDATA[Computational Optimization and Applications]]></source>
<year>1998</year>
<volume>11</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>227-51</page-range></nlm-citation>
</ref>
<ref id="B14">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Martínez]]></surname>
<given-names><![CDATA[J. M]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[On the relation between two local convergence theories of least change secant update methods]]></article-title>
<source><![CDATA[Mathematics of Computation]]></source>
<year>1992</year>
<volume>59</volume>
<numero>200</numero>
<issue>200</issue>
<page-range>457-81</page-range></nlm-citation>
</ref>
<ref id="B15">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Martínez]]></surname>
<given-names><![CDATA[J. M]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Practical quasi-Newton methods for solving nonlinear systems]]></article-title>
<source><![CDATA[Journal of Computational and Applied Mathematics]]></source>
<year>2000</year>
<volume>124</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>97-121</page-range></nlm-citation>
</ref>
<ref id="B16">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Pérez]]></surname>
<given-names><![CDATA[R.]]></given-names>
</name>
<name>
<surname><![CDATA[Arenas]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
<name>
<surname><![CDATA[Martínez]]></surname>
<given-names><![CDATA[H.]]></given-names>
</name>
<name>
<surname><![CDATA[Arias]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
</person-group>
<source><![CDATA[El problema de complementariedad no lineal]]></source>
<year>2019</year>
<volume>1</volume>
<publisher-name><![CDATA[Programa editorial Universidad del Valle]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B17">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Pinto da Costa]]></surname>
<given-names><![CDATA[A.]]></given-names>
</name>
<name>
<surname><![CDATA[Martins]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Figueiredo]]></surname>
<given-names><![CDATA[I. N.]]></given-names>
</name>
<name>
<surname><![CDATA[Júdice]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The directional instability problem in systems with frictional contacts]]></article-title>
<source><![CDATA[Computer Methods in Applied Mechanics and Engineering]]></source>
<year>2004</year>
<volume>193</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>357-84</page-range></nlm-citation>
</ref>
<ref id="B18">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Queiroz]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
<name>
<surname><![CDATA[Júdice]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Humes]]></surname>
<given-names><![CDATA[C. J]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The symmetric eigenvalue complementarity problem]]></article-title>
<source><![CDATA[Mathematics of Computation]]></source>
<year>2004</year>
<volume>73</volume>
<numero>248</numero>
<issue>248</issue>
<page-range>1849-63</page-range></nlm-citation>
</ref>
<ref id="B19">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Schubert]]></surname>
<given-names><![CDATA[L. K]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Modification of a quasi-Newton method for nonlinear equations with a sparse jacobian]]></article-title>
<source><![CDATA[Mathematics of Computation]]></source>
<year>1970</year>
<volume>24</volume>
<page-range>27-30</page-range></nlm-citation>
</ref>
<ref id="B20">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Seeger]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions]]></article-title>
<source><![CDATA[Linear Algebra and its Applications]]></source>
<year>1999</year>
<volume>292</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>1-14</page-range></nlm-citation>
</ref>
<ref id="B21">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Sherali]]></surname>
<given-names><![CDATA[H. D.]]></given-names>
</name>
<name>
<surname><![CDATA[Tuncbilek]]></surname>
<given-names><![CDATA[C. H]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A global optimization algorithm for polyno- mial programming problems using a reformulation-linearization technique]]></article-title>
<source><![CDATA[Journal of Global Optimization]]></source>
<year>1992</year>
<volume>2</volume>
<page-range>101-12</page-range></nlm-citation>
</ref>
<ref id="B22">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Sánchez]]></surname>
<given-names><![CDATA[W.]]></given-names>
</name>
<name>
<surname><![CDATA[Pérez]]></surname>
<given-names><![CDATA[R.]]></given-names>
</name>
<name>
<surname><![CDATA[Martínez]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Un algoritmo global con jacobiano suavizado para problemas de complementariedad no lineal]]></article-title>
<source><![CDATA[Revista integración, temas de matemáticas]]></source>
<year>2021</year>
<volume>39</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>191-215</page-range></nlm-citation>
</ref>
<ref id="B23">
<nlm-citation citation-type="confpro">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Yong]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<source><![CDATA[Nonlinear complementarity problem and solution methods]]></source>
<year>2010</year>
<conf-name><![CDATA[ Proceedings of the 2010 international conference on artificial intelligence and computational intelligence: Part i]]></conf-name>
<conf-loc> </conf-loc>
<page-range>461-9</page-range><publisher-loc><![CDATA[Springer ]]></publisher-loc>
<publisher-name><![CDATA[Verlag]]></publisher-name>
</nlm-citation>
</ref>
</ref-list>
</back>
</article>
