@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: a skos:ConceptScheme .
psr:-VH4F4166-V
  skos:definition """Farkas' lemma is a solvability theorem for a finite system of linear inequalities in mathematics. It was originally proven by the Hungarian mathematician Gyula Farkas. Farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization (alternatively, mathematical programming). It is used amongst other things in the proof of the Karush–Kuhn–Tucker theorem in nonlinear programming. Remarkably, in the area of the foundations of quantum theory, the lemma also underlies the complete set of Bell inequalities in the form of necessary and sufficient conditions for the existence of a local hidden-variable theory, given data from any specific set of measurements. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Farkas%27_lemma">https://en.wikipedia.org/wiki/Farkas%27_lemma</a>)"""@en, """Le <b>lemme de Farkas</b> est un résultat d'analyse convexe (une branche des mathématiques) qui s'exprime et s'interprète de multiples manières. Sous une forme assez générale, il donne une expression duale de l'adhérence de l'image d'un cône convexe <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 K}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>K</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle K}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="K"></span> par une application linéaire <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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>A</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span>&nbsp;:
<br/>
<br/><center>
<br/><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 {\\\\overline {A(K)}}=[{A^{*}}^{-1}(K^{*})]^{*},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mover>
<br/>            <mrow>
<br/>              <mi>A</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>K</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mo accent="false">¯<!-- ¯ --></mo>
<br/>          </mover>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <msup>
<br/>              <mi>A</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mo>∗<!-- ∗ --></mo>
<br/>              </mrow>
<br/>            </msup>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msup>
<br/>          <mi>K</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">)</mo>
<br/>        <msup>
<br/>          <mo stretchy="false">]</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\overline {A(K)}}=[{A^{*}}^{-1}(K^{*})]^{*},}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b92b8628f755090394e009daf6fbb9b1ad17898" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:21.914ex; height:3.676ex;" alt="{\\\\displaystyle {\\\\overline {A(K)}}=[{A^{*}}^{-1}(K^{*})]^{*},}"></span>
<br/>
<br/></center>
<br/>où <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^{*}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A^{*}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="A^{*}"></span> est l'adjointe de <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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>A</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> 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 P^{*}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>P</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle P^{*}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24635cb75ac6b04bbdb6054de6df7eca458d3ef9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.876ex; height:2.343ex;" alt="P^{*}"></span> désigne le cône dual d'une partie <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 P}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>P</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle P}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"></span>.
<br/>Lorsque <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(K)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>A</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>K</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A(K)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c52cff79814a88bbbfecaeac38710b651c95cd35" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.618ex; height:2.843ex;" alt="A(K)"></span> est fermé, on obtient ainsi des conditions nécessaires et suffisantes pour qu'un système d'équations linéaires (ou affines) ait une solution dans <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 K}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>K</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle K}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="K"></span>. Sous la forme ci-dessus, le lemme de Farkas généralise la relation d'algèbre linéaire reliant l'image d'une application linéaire <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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>A</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> entre deux espaces euclidiens et le noyau de <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^{*}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A^{*}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="A^{*}"></span>, à savoir
<br/>
<br/><center>
<br/><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 \\\\operatorname {Im} {(A)}={\\\\operatorname {Ker} {(A^{*})}}^{\\\\perp }.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>Im</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>A</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>Ker</mi>
<br/>            <mo>⁡<!-- ⁡ --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mo stretchy="false">(</mo>
<br/>              <msup>
<br/>                <mi>A</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mo>∗<!-- ∗ --></mo>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>⊥<!-- ⊥ --></mo>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\operatorname {Im} {(A)}={\\\\operatorname {Ker} {(A^{*})}}^{\\\\perp }.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/575989722fe2589ec1d48b344333a46fdf35ac3a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.717ex; height:3.343ex;" alt="{\\\\displaystyle \\\\operatorname {Im} {(A)}={\\\\operatorname {Ker} {(A^{*})}}^{\\\\perp }.}"> </center>
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Lemme_de_Farkas">https://fr.wikipedia.org/wiki/Lemme_de_Farkas</a>)"""@fr ;
  skos:broader psr:-ZTD7VMDS-3, psr:-XJ7K95G7-L ;
  skos:prefLabel "Farkas' lemma"@en, "lemme de Farkas"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Lemme_de_Farkas>, <https://en.wikipedia.org/wiki/Farkas%27_lemma> ;
  dc:modified "2023-08-16"^^xsd:date ;
  skos:inScheme psr: ;
  a skos:Concept ;
  dc:created "2023-08-16"^^xsd:date .

psr:-ZTD7VMDS-3
  skos:prefLabel "analyse convexe"@fr, "convex analysis"@en ;
  a skos:Concept ;
  skos:narrower psr:-VH4F4166-V .

psr:-XJ7K95G7-L
  skos:prefLabel "optimization"@en, "optimisation"@fr ;
  a skos:Concept ;
  skos:narrower psr:-VH4F4166-V .