@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-FWTTZ9R7-X
  skos:prefLabel "fonction numérique à plusieurs variables réelles"@fr, "function of several real variables"@en ;
  a skos:Concept ;
  skos:narrower psr:-QL0GMPR2-2 .

psr: a skos:ConceptScheme .
psr:-XJ7K95G7-L
  skos:prefLabel "optimization"@en, "optimisation"@fr ;
  a skos:Concept ;
  skos:narrower psr:-QL0GMPR2-2 .

psr:-QL0GMPR2-2
  dc:modified "2023-07-26"^^xsd:date ;
  skos:inScheme psr: ;
  skos:broader psr:-XJ7K95G7-L, psr:-FWTTZ9R7-X ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Rastrigin_function>, <https://fr.wikipedia.org/wiki/Fonction_de_Rastrigin> ;
  skos:prefLabel "Rastrigin function"@en, "fonction de Rastrigin"@fr ;
  dc:created "2023-07-26"^^xsd:date ;
  skos:definition """In mathematical optimization, the <b>Rastrigin function</b> is a non-convex function used as a performance test problem for optimization algorithms. It is a typical example of non-linear multimodal function. It was first proposed in 1974 by Rastrigin as a 2-dimensional function and has been generalized by Rudolph. The generalized version was popularized by Hoffmeister &amp; Bäck and Mühlenbein et al. Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima.
<br/>On an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>n</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span>-dimensional domain it is defined by:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(\\\\mathbf {x} )=An+\\\\sum _{i=1}^{n}\\\\left[x_{i}^{2}-A\\\\cos(2\\\\pi x_{i})\\
ight]}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">x</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>A</mi>
<br/>        <mi>n</mi>
<br/>        <mo>+</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>=</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow>
<br/>          <mo>[</mo>
<br/>          <mrow>
<br/>            <msubsup>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>i</mi>
<br/>              </mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msubsup>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>A</mi>
<br/>            <mi>cos</mi>
<br/>            <mo>⁡<!-- ⁡ --></mo>
<br/>            <mo stretchy="false">(</mo>
<br/>            <mn>2</mn>
<br/>            <mi>π<!-- π --></mi>
<br/>            <msub>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>i</mi>
<br/>              </mrow>
<br/>            </msub>
<br/>            <mo stretchy="false">)</mo>
<br/>          </mrow>
<br/>          <mo>]</mo>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(\\\\mathbf {x} )=An+\\\\sum _{i=1}^{n}\\\\left[x_{i}^{2}-A\\\\cos(2\\\\pi x_{i})\\
ight]}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa1c38ee739ca9cf4582867d74d469df4676cbc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:36.156ex; height:6.843ex;" alt="f(\\\\mathbf {x} )=An+\\\\sum _{i=1}^{n}\\\\left[x_{i}^{2}-A\\\\cos(2\\\\pi x_{i})\\
ight]"></span></dd></dl>
<br/>where  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle A=10}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>A</mi>
<br/>        <mo>=</mo>
<br/>        <mn>10</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A=10}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c9d6b464d89297f7f0ba6ea13536d0c2646cad8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.166ex; height:2.176ex;" alt="A=10"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle x_{i}\\\\in [-5.12,5.12]}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>5.12</mn>
<br/>        <mo>,</mo>
<br/>        <mn>5.12</mn>
<br/>        <mo stretchy="false">]</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x_{i}\\\\in [-5.12,5.12]}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3560b2d6c10fff0a26c994d45374c4dc70f98e5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.374ex; height:2.843ex;" alt="x_{i}\\\\in [-5.12,5.12]"></span>. There are many extrema:
<br/>
<br/><ul><li>The global minimum is at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbf {x} =\\\\mathbf {0} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">x</mi>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mn mathvariant="bold">0</mn>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbf {x} =\\\\mathbf {0} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecef8f00bf3524507cbbde1aec694e2237c9a0d9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="\\\\mathbf {x} =\\\\mathbf {0} "></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(\\\\mathbf {x} )=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">x</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(\\\\mathbf {x} )=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99e397fb3dd89767631f50bb9dbd89f35baa4234" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.76ex; height:2.843ex;" alt="f(\\\\mathbf {x} )=0"></span>.</li> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Rastrigin_function">https://en.wikipedia.org/wiki/Rastrigin_function</a>)"""@en, """La <b>fonction de Rastrigin</b> est une fonction mathématique souvent utilisée pour évaluer la performance d'algorithmes d’optimisation. Elle présente des pièges intéressants, sous la forme de ses nombreux minima et maxima locaux. Elle a été proposée, en 1974, par Rastrigin en deux dimensions et a été généralisée par Mühlenbein <i><abbr class="abbr" title="et alii (« et d’autres »)" lang="la">et al.</abbr></i>.
<br/>Sa définition, en dimension <i>n</i>, est&nbsp;:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(\\\\mathbf {x} )=\\\\mathrm {A} \\\\cdot n+\\\\sum _{i=1}^{n}\\\\left[x_{i}^{2}-\\\\mathrm {A} \\\\cdot \\\\cos(2\\\\pi x_{i})\\
ight]}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="bold">x</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="normal">A</mi>
<br/>        </mrow>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mi>n</mi>
<br/>        <mo>+</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>=</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow>
<br/>          <mo>[</mo>
<br/>          <mrow>
<br/>            <msubsup>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>i</mi>
<br/>              </mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>            </msubsup>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">A</mi>
<br/>            </mrow>
<br/>            <mo>⋅<!-- ⋅ --></mo>
<br/>            <mi>cos</mi>
<br/>            <mo>⁡<!-- ⁡ --></mo>
<br/>            <mo stretchy="false">(</mo>
<br/>            <mn>2</mn>
<br/>            <mi>π<!-- π --></mi>
<br/>            <msub>
<br/>              <mi>x</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>i</mi>
<br/>              </mrow>
<br/>            </msub>
<br/>            <mo stretchy="false">)</mo>
<br/>          </mrow>
<br/>          <mo>]</mo>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(\\\\mathbf {x} )=\\\\mathrm {A} \\\\cdot n+\\\\sum _{i=1}^{n}\\\\left[x_{i}^{2}-\\\\mathrm {A} \\\\cdot \\\\cos(2\\\\pi x_{i})\\
ight]}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488d528149712fa82ab3638e1da72928bfd51579" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:39.127ex; height:6.843ex;" alt="{\\\\displaystyle f(\\\\mathbf {x} )=\\\\mathrm {A} \\\\cdot n+\\\\sum _{i=1}^{n}\\\\left[x_{i}^{2}-\\\\mathrm {A} \\\\cdot \\\\cos(2\\\\pi x_{i})\\
ight]}"></span></dd></dl>
<br/>où A = 10 et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle x_{i}\\\\in [-5,12\\\\ ;\\\\ 5,12]}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>5</mn>
<br/>        <mo>,</mo>
<br/>        <mn>12</mn>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mo>;</mo>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mn>5</mn>
<br/>        <mo>,</mo>
<br/>        <mn>12</mn>
<br/>        <mo stretchy="false">]</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x_{i}\\\\in [-5,12\\\\ ;\\\\ 5,12]}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1d9d16918b8ee805f6d68e74ba2f8ce7e5f970" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:19.31ex; height:2.843ex;" alt="{\\\\displaystyle x_{i}\\\\in [-5,12\\\\ ;\\\\ 5,12]}"></span>. Son minimum global se trouve à l'origine, où sa valeur est nulle.
<br/> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_de_Rastrigin">https://fr.wikipedia.org/wiki/Fonction_de_Rastrigin</a>)"""@fr ;
  a skos:Concept .