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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (NConvex) - Chapter 4: Cones</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap4" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <p id="mathjaxlink" class="pcenter"><a href="chap4_mj.html">[MathJax on]</a></p> <p><a id="X8524A7567BA4FFA6" name="X8524A7567BA4FFA6"></a></p> <div class="ChapSects"><a 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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7DC52F6B8490C5C4">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B0DA052877CE7BC">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8267BFA47CD282E1">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A807C828273D4ED">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X849EFCA77CE04DF9">4 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8490C19A78ED287A">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X831EBE117FC07C40">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X810956D0823BDA51">4 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X782E6C6684E1514D">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80715AC378616F14">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7946B20D834F31D2">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X825A7CBD79653961">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81FBC83B87053C56">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C52373C83B9282D">4 </div></div> </div> <h3>4 <span class="Heading">Cones</span></h3> <p><a id="X837F56037886A1EF" name="X837F56037886A1EF"></a></p> <h4>4.1 <span class="Heading">Creating cones</span></h4> <p><a id="X7E6FB9EC7DFF5403" name="X7E6FB9EC7DFF5403"></a></p> <h5>4.1-1 ConeByInequalities</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <code class="code">Cone</code> Object</p> <p>The function takes a list of lists <span class="Math">[L_1, L_2, ...]</span> where each <span clas <p><a id="X8711F345805C5FBD" name="X8711F345805C5FBD"></a></p> <h5>4.1-2 ConeByEqualitiesAndInequalities</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <code class="code">Cone</code> Object</p> <p>The function takes two lists. The first list is the equalities and the second is the inequaliti <p><a id="X7F0634207B3A4EC6" name="X7F0634207B3A4EC6"></a></p> <h5>4.1-3 Cone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <code class="code">Cone</code> Object</p> <p>The function takes a list in which every entry represents a ray in the ambient vector space and r <p><a id="X7CD248238093A748" name="X7CD248238093A748"></a></p> <h5>4.1-4 Cone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a <code class="code">Cone</code> Object</p> <p>This function takes a cone defined in <em>CddInterface</em> and converts it to a cone in <em>NConvex< <p><a id="X85EE90A37DE1E3AC" name="X85EE90A37DE1E3AC"></a></p> <h4>4.2 <span class="Heading">Attributes of Cones</span></h4> <p><a id="X80468DD7834C2EFD" name="X80468DD7834C2EFD"></a></p> <h5>4.2-1 DefiningInequalities</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list</p> <p>Returns the list of the defining inequalities of the cone <code class="code">C</code>.</p> <p><a id="X782F9A6E82AC6EA4" name="X782F9A6E82AC6EA4"></a></p> <h5>4.2-2 EqualitiesOfCone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list</p> <p>Returns the list of the equalities in the defining inequalities of the cone <code class="code"> <p><a id="X80DB81B58590FC36" name="X80DB81B58590FC36"></a></p> <h5>4.2-3 DualCone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a cone</p> <p>Returns the dual cone of the cone <code class="code">C</code>.</p> <p><a id="X825F3C9380C4E2EE" name="X825F3C9380C4E2EE"></a></p> <h5>4.2-4 FacesOfCone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list of cones</p> <p>Returns the list of all faces of the cone <code class="code">C</code>.</p> <p><a id="X82C77B2A8235567E" name="X82C77B2A8235567E"></a></p> <h5>4.2-5 Facets</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list of cones</p> <p>Returns the list of all facets of the cone <code class="code">C</code>.</p> <p><a id="X817CAD1E84C0757A" name="X817CAD1E84C0757A"></a></p> <h5>4.2-6 FVector</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list</p> <p>Returns a list whose <span class="Math">i</span>'th entry is the number of faces of dimension <spa <p><a id="X81488179780BE42F" name="X81488179780BE42F"></a></p> <h5>4.2-7 RelativeInteriorRay</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list</p> <p>Returns a relative interior point (or ray) in the cone <code class="code">C</code>.</p> <p><a id="X7EB6BB087BE2FFD6" name="X7EB6BB087BE2FFD6"></a></p> <h5>4.2-8 HilbertBasis</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list</p> <p>Returns the Hilbert basis of the cone <code class="code">C</code></p> <p><a id="X8445477A83A137CB" name="X8445477A83A137CB"></a></p> <h5>4.2-9 HilbertBasisOfDualCone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list</p> <p>Returns the Hilbert basis of the dual cone of the cone <code class="code">C</code></p> <p><a id="X815F2FC4821A961D" name="X815F2FC4821A961D"></a></p> <h5>4.2-10 LinealitySpaceGenerators</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list</p> <p>Returns a basis of the lineality space of the cone <code class="code">C</code>.</p> <p><a id="X7CFFCB5C7F4C183D" name="X7CFFCB5C7F4C183D"></a></p> <h5>4.2-11 ExternalCddCone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a cdd object</p> <p>Converts the cone to a cdd object. The operations of CddInterface can then be applied on this co <p><a id="X79B4436C8051BA08" name="X79B4436C8051BA08"></a></p> <h5>4.2-12 ExternalNmzCone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: an normaliz object</p> <p>Converts the cone to a normaliz object. The operations of NormalizInterface can then be appli <p><a id="X83709F0380B097BD" name="X83709F0380B097BD"></a></p> <h5>4.2-13 AmbientSpaceDimension</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: an integer</p> <p>The dimension of the ambient space of the cone, i.e., the space that contains the cone.</p> <p><a id="X7DC52F6B8490C5C4" name="X7DC52F6B8490C5C4"></a></p> <h5>4.2-14 LatticePointsGenerators</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list</p> <p>See <code class="code">LatticePointsGenerators</code> for polyhedrons. Please note that any <p><a id="X7B0DA052877CE7BC" name="X7B0DA052877CE7BC"></a></p> <h5>4.2-15 GridGeneratedByCone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a homalg module</p> <p>Returns the homalg <span class="Math">\mathbb{Z}</span>-module that is generated by the ray ge <p><a id="X8267BFA47CD282E1" name="X8267BFA47CD282E1"></a></p> <h5>4.2-16 FactorGrid</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a homalg module</p> <p>Returns the homalg <span class="Math">\mathbb{Z}</span>-module that is presented by the matri <p><a id="X7A807C828273D4ED" name="X7A807C828273D4ED"></a></p> <h5>4.2-17 FactorGridMorphism</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a homalg morphism</p> <p>Returns an epimorphism from a free <span class="Math">\mathbb{Z}</span>-module to <code class= <p><a id="X849EFCA77CE04DF9" name="X849EFCA77CE04DF9"></a></p> <h5>4.2-18 GridGeneratedByOrthogonalCone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a homalg module</p> <p>Returns the homalg <span class="Math">\mathbb{Z}</span>-module that is by generated the ray ge <p><a id="X7DD2805F7DF61F97" name="X7DD2805F7DF61F97"></a></p> <h4>4.3 <span class="Heading">Properties of Cones</span></h4> <p><a id="X8490C19A78ED287A" name="X8490C19A78ED287A"></a></p> <h5>4.3-1 IsRegularCone</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: true or false</p> <p>Returns if the cone <code class="code">C</code> is regular or not.</p> <p><a id="X831EBE117FC07C40" name="X831EBE117FC07C40"></a></p> <h5>4.3-2 IsRay</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: true or false</p> <p>Returns if the cone <code class="code">C</code> is ray or not.</p> <p><a id="X810956D0823BDA51" name="X810956D0823BDA51"></a></p> <h5>4.3-3 IsZero</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: true or false</p> <p>Returns whether the cone is the zero cone or not.</p> <p><a id="X843CBBD37FC2827B" name="X843CBBD37FC2827B"></a></p> <h4>4.4 <span class="Heading">Operations on cones</span></h4> <p><a id="X782E6C6684E1514D" name="X782E6C6684E1514D"></a></p> <h5>4.4-1 FourierProjection</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a cone</p> <p>Returns the projection of the cone on the space (O, <span class="Math">x_1,...,x_{m-1}, x_{m+ <p><a id="X80715AC378616F14" name="X80715AC378616F14"></a></p> <h5>4.4-2 IntersectionOfCones</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a cone</p> <p>Returns the intersection.</p> <p><a id="X7946B20D834F31D2" name="X7946B20D834F31D2"></a></p> <h5>4.4-3 IntersectionOfCones</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a cone</p> <p>The input is a list of cones and the output is their intersection.</p> <p><a id="X825A7CBD79653961" name="X825A7CBD79653961"></a></p> <h5>4.4-4 Contains</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a true or false</p> <p>Returns if the cone <code class="code">C1</code> contains the cone <code class="code">C2</code>.</ <p><a id="X81FBC83B87053C56" name="X81FBC83B87053C56"></a></p> <h5>4.4-5 IsRelativeInteriorRay</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a true or false</p> <p>Checks whether the input point (or ray) <code class="code">L</code> is in the relative interior o <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">P:= Cone( [ [ 2, 7 ], [ 0, 12 ], [ -2, 5 ] ] );< <A cone in |R^2> <span class="GAPprompt">gap></span> <span class="GAPinput">d:= DefiningInequalities( P );</spa [ [ -7, 2 ], [ 5, 2 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">Q:= ConeByInequalities( d );</span> <A cone in |R^2> <span class="GAPprompt">gap></span> <span class="GAPinput">P=Q;</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsPointed( P );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( P );</span> [ [ -2, 5 ], [ 2, 7 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">HilbertBasis( P );</span> [ [ -2, 5 ], [ -1, 3 ], [ 0, 1 ], [ 1, 4 ], [ 2, 7 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">HilbertBasis( Q );</span> [ [ -2, 5 ], [ -1, 3 ], [ 0, 1 ], [ 1, 4 ], [ 2, 7 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">P_dual:= DualCone( P );</span> <A cone in |R^2> <span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( P_dual );</span> [ [ -7, 2 ], [ 5, 2 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( P );</span> 2 <span class="GAPprompt">gap></span> <span class="GAPinput">List( Facets( P ), RayGenerators );</ [ [ [ -2, 5 ] ], [ [ 2, 7 ] ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">faces := FacesOfCone( P );</span> [ <A cone in |R^2>, <A cone in |R^2>, <A ray in |R^2>, <A ray in |R^2> ] <span class="GAPprompt">gap></span> <span class="GAPinput">RelativeInteriorRay( P );</span> [ -2, 41 ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsRelativeInteriorRay( [ -2, 41 ], P true <span class="GAPprompt">gap></span> <span class="GAPinput">IsRelativeInteriorRay( [ 2, 7 ], P );< false <span class="GAPprompt">gap></span> <span class="GAPinput">LinealitySpaceGenerators( P );</sp [ ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularCone( P );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">IsRay( P );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">proj_x1:= FourierProjection( P, 2 ) <A cone in |R^1> <span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( proj_x1 );</span> [ [ -1 ], [ 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">DefiningInequalities( proj_x1 );</ [ [ 0 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">R:= Cone( [ [ 4, 5 ], [ -2, 1 ] ] );</span> <A cone in |R^2> <span class="GAPprompt">gap></span> <span class="GAPinput">T:= IntersectionOfCones( P, R );</sp <A cone in |R^2> <span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( T );</span> [ [ -2, 5 ], [ 2, 7 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">W:= Cone( [ [-3,-4 ] ] );</span> <A ray in |R^2> <span class="GAPprompt">gap></span> <span class="GAPinput">I:= IntersectionOfCones( P, W );</sp <A cone in |R^2> <span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( I );</span> [ ] <span class="GAPprompt">gap></span> <span class="GAPinput">Contains( P, I );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">Contains( W, I );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">Contains( P, R );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">Contains( R, P );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">cdd_cone:= ExternalCddCone( P );</s < Polyhedron given by its V-representation > <span class="GAPprompt">gap></span> <span class="GAPinput">Display( cdd_cone );</span> V-representation begin 3 X 3 rational 0 2 7 0 0 12 0 -2 5 end <span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Dimension( cdd_cone );</span> 2 <span class="GAPprompt">gap></span> <span class="GAPinput">H:= Cdd_H_Rep( cdd_cone );</span> < Polyhedron given by its H-representation > <span class="GAPprompt">gap></span> <span class="GAPinput">Display( H );</span> H-representation begin 2 X 3 rational 0 5 2 0 -7 2 end <span class="GAPprompt">gap></span> <span class="GAPinput">P:= Cone( [ [ 1, 1, -3 ], [ -1, -1, 3 ], [ 1, < A cone in |R^3> <span class="GAPprompt">gap></span> <span class="GAPinput">IsPointed( P );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( P );</span> 3 <span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularCone( P );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">P;</span> < A cone in |R^3 of dimension 3 with 4 ray generators> <span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( P );</span> [ [ -1, -1, 3 ], [ 1, 1, -3 ], [ 1, 2, 1 ], [ 2, 1, 2 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">d:= DefiningInequalities( P );</spa [ [ -5, 8, 1 ], [ 7, -4, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">facets:= Facets( P );</span> [ <A cone in |R^3>, <A cone in |R^3> ] <span class="GAPprompt">gap></span> <span class="GAPinput">faces := FacesOfCone( P );</span> [ <A cone in |R^3>, <A cone in |R^3>, <A cone in |R^3>, <A cone in |R^3>, <A cone in |R^3> ] <span class="GAPprompt">gap></span> <span class="GAPinput">FVector( P );</span> [ 1, 2, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">List( faces, Dimension );</span> [ 0, 3, 2, 1, 2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">L_using_4ti2 := [ [ [ 0, 0, 0 ] ], [ [ -2, - <span class="GAPprompt">></span> <span class="GAPinput">[ 0, 0, 1 ], [ 2, 1, 2 ] ], [ [ 1, 1, -3 ] ] ];;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">L_using_Normaliz := [ [ [ 0, 0, 0 ] ], [ [ <span class="GAPprompt">></span> <span class="GAPinput">[ 0, 0, 1 ], [ 1, 0, 5 ] ], [ [ 1, 1, -3 ] ] ];;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">L := LatticePointsGenerators( P );;< <span class="GAPprompt">gap></span> <span class="GAPinput">L = L_using_4ti2 or L = L_using_Norma true <span class="GAPprompt">gap></span> <span class="GAPinput">DualCone( P );</span> < A cone in |R^3> <span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( DualCone( P ) );</spa [ [ -5, 8, 1 ], [ 7, -4, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">Q_x1x3:= FourierProjection(P, 2 );< <A cone in |R^2> <span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( Q_x1x3 );</span> [ [ -1, 3 ], [ 1, -3 ], [ 1, 1 ] ] </pre></div> <p><a id="X7C52373C83B9282D" name="X7C52373C83B9282D"></a></p> <h5>4.4-6 NonReducedInequalities</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list</p> <p>It returns a list of inequalities that define the cone.</p> <div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html>
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