<?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>2304-0106</journal-id>
<journal-title><![CDATA[Anales de la Academia de Ciencias de Cuba]]></journal-title>
<abbrev-journal-title><![CDATA[Anales de la ACC]]></abbrev-journal-title>
<issn>2304-0106</issn>
<publisher>
<publisher-name><![CDATA[Academia de Ciencias de Cuba]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S2304-01062023000100011</article-id>
<title-group>
<article-title xml:lang="es"><![CDATA[Caracterización y cálculo de soluciones de problemas de optimización con múltiples funciones objetivos]]></article-title>
<article-title xml:lang="en"><![CDATA[Characterization and computation of the solutions of optimization problems with multiple objective functions]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[Gemayqzel]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Quintana Aparicio]]></surname>
<given-names><![CDATA[Ernest]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Tammer]]></surname>
<given-names><![CDATA[Christiane]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad de La Habana Facultad de Matemática y Computación ]]></institution>
<addr-line><![CDATA[ La Habana]]></addr-line>
<country>Cuba</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Universidad Técnica de Ilmenau Instituto de Matemática ]]></institution>
<addr-line><![CDATA[Ilmenau ]]></addr-line>
<country>Alemania</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Universidad Martin Luther, Halle-Wittenberg Instituto de Matemática ]]></institution>
<addr-line><![CDATA[Halle ]]></addr-line>
<country>Alemania</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>04</month>
<year>2023</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>04</month>
<year>2023</year>
</pub-date>
<volume>13</volume>
<numero>1</numero>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://scielo.sld.cu/scielo.php?script=sci_arttext&amp;pid=S2304-01062023000100011&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://scielo.sld.cu/scielo.php?script=sci_abstract&amp;pid=S2304-01062023000100011&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://scielo.sld.cu/scielo.php?script=sci_pdf&amp;pid=S2304-01062023000100011&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="es"><p><![CDATA[RESUMEN  Introducción:  En muchas aplicaciones se tiene que uno o varios decisores optimizan varias funciones objetivos. Los paradigmas que han descrito este tipo de situaciones han sido 3: los problemas de 2 niveles, en los cuales la mejor opción ha dependido de la decisión que han tomado otros agentes, simultáneamente o no; los problemas multiobjetivo con orden variable, aquellos en que un decisor ha tenido varios criterios de optimización combinados y la comparación entre 2 puntos dado por un cono que no varía y los problemas de optimización conjunto evaluada que corresponde al caso en que los valores de la función sean conjuntos. El objetivo de este trabajo fue presentar nuevas caracterizaciones de las soluciones de estos problemas y proponer algoritmos que hallen estos puntos.  Métodos: Demostraciones matemáticas y experimentación numérica.  Resultados y discusión: Caracterización de las soluciones del problema de múltiples líderes y seguidores disjuntos en el caso genérico. Propuesta de un algoritmo inexacto de gradiente proyectado para problemas multiobjetivo con orden variable, estudio de su convergencia. Generación de un conjunto de problemas prueba para modelos multiobjetivo con orden variable. Caracterización mediante una regla tipo Fermat de las soluciones de problemas de optimización conjunto evaluada. Propuesta de algoritmo de máximo descenso para problemas de optimización conjunto evaluada y estudio de su convergencia. Estimación del subdiferencial de Clarke para funciones marginales. En conclusión, al caracterizar las soluciones de los problemas tratados bajo hipótesis más suaves, se tuvo un mejor conocimiento de la estructura del conjunto solución. Este hecho fue vital para la solución numérica de dichos modelos. Los algoritmos ya propuestos en este trabajo han arrojado resultados que en muchos casos han superado los conocidos en la literatura al requerir un menor número de operaciones y alcanzar soluciones de buena calidad. Su convergencia fue analizada también teóricamente, lográndose bajo hipótesis no muy restrictiva.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[ABSTRACT  Introduction:  In many applications, one or many decision makers optimize several objective functions. Three important paradigms that describe these situations are bi-level problems, in which the best option depends on the decision that other agents make simultaneously or not; multiobjective problems with variable order, that is, a decision maker has several combined optimality criteria and the comparison between 2 points is given by a cone that is, in general, non-constant, and set-valued optimization problems, which correspond to the case in which the function values are sets. In this work we present new characterizations of the solutions to these problems and propose algorithms to find these points.  Methods:  Mathematical demonstrations, numerical experimentation.  Results and discussion: For the problem of multiple disjoint leaders and followers a characterization of the solutions in the generic case was provided. An inexact projected gradient algorithm for multiobjective problems with variable order was proposed and its convergence was studied. A way of generating test problems for multiobjective models with variable order was obtained. Characterization of the solutions of set-operation optimization problems via a Fermat-type rule. A steepest descent algorithm for set-test optimization problems was proposed and its convergence was studied. Estimation of the Clarke subdifferential for marginal functions. It is concluded that the obtained characterizations of the solutions of the problems improve the results of the literature. This better knowledge of the structure of the solution set is very helpful also for an algorithmic viewpoint. The algorithms already proposed have yielded results that in many cases are better than those of the literature. Indeed, they computed well using fewer operations. Their convergence was also theoretically analyzed. It shall be remarked that not very restrictive hypotheses were needed.]]></p></abstract>
<kwd-group>
<kwd lng="es"><![CDATA[algoritmos tipo máximo descenso]]></kwd>
<kwd lng="es"><![CDATA[genericidad]]></kwd>
<kwd lng="es"><![CDATA[problemas de optimización con varios objetivos]]></kwd>
<kwd lng="es"><![CDATA[reglas tipo Fermat]]></kwd>
<kwd lng="en"><![CDATA[steepest descent type algorithm]]></kwd>
<kwd lng="en"><![CDATA[genericity]]></kwd>
<kwd lng="en"><![CDATA[optimization problems with multiple objective functions]]></kwd>
<kwd lng="en"><![CDATA[Fermat rule]]></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[Dempe]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Annotated bibliography on bilevel programming and mathematical programs with equilibrium constrains]]></article-title>
<source><![CDATA[Optimization]]></source>
<year>2003</year>
<volume>52</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>333-59</page-range></nlm-citation>
</ref>
<ref id="B2">
<label>2</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Dempe]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Zemkoho]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
</person-group>
<source><![CDATA[Bilevel Optimization]]></source>
<year>2020</year>
<publisher-name><![CDATA[Springer Optim]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B3">
<label>3</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Eichfelder]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
</person-group>
<source><![CDATA[Variable ordering structures in vector optimization]]></source>
<year>2014</year>
<publisher-loc><![CDATA[Berlin ]]></publisher-loc>
<publisher-name><![CDATA[Springer]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B4">
<label>4</label><nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Köbis]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
<name>
<surname><![CDATA[Köbis]]></surname>
<given-names><![CDATA[MA]]></given-names>
</name>
</person-group>
<source><![CDATA[Treatment of set order relations by means of a nonlinear scalarization functional: a full characterization, Optimization]]></source>
<year>2016</year>
<volume>65</volume>
<page-range>1805-27</page-range></nlm-citation>
</ref>
<ref id="B5">
<label>5</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Köbis]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
<name>
<surname><![CDATA[Thanh Le]]></surname>
<given-names><![CDATA[T]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Numerical procedures for obtaining strong, strict and ideal minimal solutions of set optimization problems Anal]]></article-title>
<source><![CDATA[Optim]]></source>
<year>2018</year>
<volume>2</volume>
<page-range>423-40</page-range></nlm-citation>
</ref>
<ref id="B6">
<label>6</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Aussel]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Van]]></surname>
<given-names><![CDATA[KC]]></given-names>
</name>
<name>
<surname><![CDATA[Salas]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
</person-group>
<source><![CDATA[A single-leader-disjoint-follower approach of electricity contract model]]></source>
<year>2021</year>
<publisher-name><![CDATA[Preprint]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B7">
<label>7</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ramos]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<name>
<surname><![CDATA[Boix]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<name>
<surname><![CDATA[Aussel]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Montastrucy]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
<name>
<surname><![CDATA[Domenech]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Water integration in eco-industrial parks using a multi-leader-follower approach]]></article-title>
<source><![CDATA[Computers Chemical Engineering]]></source>
<year>2016</year>
<volume>87</volume>
<page-range>190-207</page-range></nlm-citation>
</ref>
<ref id="B8">
<label>8</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Engau]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Variable preference modeling with ideal-symmetric convex cones]]></article-title>
<source><![CDATA[J Global Optim]]></source>
<year>2008</year>
<volume>42</volume>
<page-range>295-311</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[Wiecek]]></surname>
<given-names><![CDATA[MM]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Advances in cone-based preference modeling for decision making with multiple criteria]]></article-title>
<source><![CDATA[Decis Mak Manuf Serv]]></source>
<year>2007</year>
<volume>1</volume>
<page-range>153-73</page-range></nlm-citation>
</ref>
<ref id="B10">
<label>10</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bao]]></surname>
<given-names><![CDATA[T]]></given-names>
</name>
<name>
<surname><![CDATA[Mordukhovich]]></surname>
<given-names><![CDATA[B]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Set-valued optimization in welfare economics]]></article-title>
<source><![CDATA[Advances in mathematical economics]]></source>
<year>2010</year>
<volume>13</volume>
<page-range>113-53</page-range><publisher-loc><![CDATA[Tokyo ]]></publisher-loc>
<publisher-name><![CDATA[Adv. Math. Econ]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B11">
<label>11</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Jongen]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
<name>
<surname><![CDATA[Jonker]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
<name>
<surname><![CDATA[Twilt]]></surname>
<given-names><![CDATA[F]]></given-names>
</name>
</person-group>
<source><![CDATA[Nonlinear Optimization in Finite Dimensions. Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects, Nonconvex Optim]]></source>
<year>2000</year>
<publisher-name><![CDATA[Kluwer, Dordrecht]]></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[Aussel]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Dempe]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
<name>
<surname><![CDATA[Lepaul]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Genericity analysis of multi-leader-disjoint-followers game SIAM J]]></article-title>
<source><![CDATA[OPTIM]]></source>
<year>2021</year>
<volume>31</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>2055-79</page-range></nlm-citation>
</ref>
<ref id="B13">
<label>13</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Bello]]></surname>
<given-names><![CDATA[Y]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[On inexact Projected Gradient Methods for solving Variable Vector Optimization problems]]></article-title>
<source><![CDATA[Optimization and Engineering]]></source>
<year>2020</year>
</nlm-citation>
</ref>
<ref id="B14">
<label>14</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Hernández Escobar]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Rückmann]]></surname>
<given-names><![CDATA[JJ]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Some properties of K-convex mappings in variable ordering settings]]></article-title>
<source><![CDATA[Optimization]]></source>
<year>2021</year>
<page-range>1-22</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[Kuroiwa]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Tanaka]]></surname>
<given-names><![CDATA[T]]></given-names>
</name>
<name>
<surname><![CDATA[Ha]]></surname>
<given-names><![CDATA[TXD]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[On cone convexity of set-valued maps]]></article-title>
<source><![CDATA[Nonlinear Anal]]></source>
<year>1997</year>
<volume>30</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>1487-96</page-range></nlm-citation>
</ref>
<ref id="B16">
<label>16</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Quintana]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
<name>
<surname><![CDATA[Tammer]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
<name>
<surname><![CDATA[Tuan]]></surname>
<given-names><![CDATA[VA]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The Fermat rule for set Optimization problems with Lipschitzian set valued mappings, joint with and Journal of Non Linear and Convex]]></article-title>
<source><![CDATA[Analysis]]></source>
<year>2020</year>
<volume>21</volume>
<numero>5</numero>
<issue>5</issue>
<page-range>1137-74</page-range></nlm-citation>
</ref>
<ref id="B17">
<label>17</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Quintana]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
<name>
<surname><![CDATA[Tammer]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A Steepest Descent Method for Set .OptimizationProblems with Set-Valued Mappings of Finite Cardinality]]></article-title>
<source><![CDATA[Journal of OptimizationTheory and Applications]]></source>
<year>2021</year>
<page-range>1-33</page-range></nlm-citation>
</ref>
<ref id="B18">
<label>18</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Quintana]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
<name>
<surname><![CDATA[Tammer]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[On Clarke subdifferential of marginal functions]]></article-title>
<source><![CDATA[Applied Set Valued Analysis and Optimization]]></source>
<year>2021</year>
<volume>13</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>281-92</page-range></nlm-citation>
</ref>
<ref id="B19">
<label>19</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Scholtes]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
<name>
<surname><![CDATA[Stöhr]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[How stringent is the linear independence assumption for mathematical programs with stationary constraints?]]></article-title>
<source><![CDATA[Math. Oper. Res]]></source>
<year>2001</year>
<volume>26</volume>
<page-range>851-63</page-range></nlm-citation>
</ref>
<ref id="B20">
<label>20</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Jongen]]></surname>
<given-names><![CDATA[HT]]></given-names>
</name>
<name>
<surname><![CDATA[Shikhman]]></surname>
<given-names><![CDATA[V]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Bilevel optimization on the structure of the feasible set, Math]]></article-title>
<source><![CDATA[Program]]></source>
<year>2012</year>
<volume>136</volume>
<page-range>65-89</page-range></nlm-citation>
</ref>
<ref id="B21">
<label>21</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Still]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Solving bilevel programs with the KKT-approach Math]]></article-title>
<source><![CDATA[Program]]></source>
<year>2013</year>
<volume>138</volume>
<page-range>309-32</page-range></nlm-citation>
</ref>
<ref id="B22">
<label>22</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Fukuda]]></surname>
<given-names><![CDATA[EH]]></given-names>
</name>
<name>
<surname><![CDATA[Graña Drummond]]></surname>
<given-names><![CDATA[LM]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Inexact projected gradient method for vector optimization]]></article-title>
<source><![CDATA[Comput Optim Appl]]></source>
<year>2013</year>
<volume>54</volume>
<page-range>473-93</page-range></nlm-citation>
</ref>
<ref id="B23">
<label>23</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bello Cruz]]></surname>
<given-names><![CDATA[JY]]></given-names>
</name>
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A steepest descent-like method for variable order vector optimization problems]]></article-title>
<source><![CDATA[J Optim Theory Appl]]></source>
<year>2014</year>
<volume>162</volume>
<page-range>371-91</page-range></nlm-citation>
</ref>
<ref id="B24">
<label>24</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bento]]></surname>
<given-names><![CDATA[GC]]></given-names>
</name>
<name>
<surname><![CDATA[Bouza Allende]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Pereira]]></surname>
<given-names><![CDATA[YR]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A Newton-like method for variable order vector optimization problems]]></article-title>
<source><![CDATA[J Optim Theory Appl]]></source>
<year>2018</year>
<volume>177</volume>
<page-range>201-21</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
