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Polyhedra Vertices, Faces, Facets Illustration Discrete Math for Bioinformatics WS 10/11, by A. Bockmayr/K. Reinert, 30. November 2010, 15:52 2001 Polyhedra • Hyperplane H = fx 2 Rn j aT x = bg, a 2 Rn n f0g,b 2 R • Closed halfspace H = fx 2 Rn j aT x ≤ bg n m×n m • Polyhedron P = fx 2 R j Ax ≤ bg,A 2 R ,b 2 R • Polytope P = fx 2 Rn j Ax ≤ b, l ≤ x ≤ ug,l,u 2 Rn • Polyhedral cone P = fx 2 Rn j Ax ≤ 0g The feasible set n P = fx 2 R j Ax ≤ bg of a linear optimization problem is a polyhedron. Vertices, Faces, Facets • P ⊆ H,H \ P 6= 0/ (Supporting hyperplane) • F = P \ H (Face) • dim(F) = 0 (Vertex) • dim(F) = 1 (Edge) • dim(F) = dim(P) − 1 (Facet) • P pointed: P has at least one vertex. Illustration 2002 Linear Programming, by Alexander Bockmayr, 30. November 2010, 15:52 Rays and extreme rays • r 2 Rn is a ray of the polyhedron P if for each x 2 P the set fx + lr j l ≥ 0g is contained in P. • A ray r of P is extreme if there do not exist two linearly independent rays r 1,r 2 of P 1 1 2 such that r = 2 (r + r ). Hull operations • x 2 Rn is a linear combination of x1,...,xk 2 Rn if 1 k x = l1x + ··· + lk x , for some l1,...,lk 2 R. • If, in addition 8 9 8 9 < l1,...,lk ≥ 0, = < conic = l1 + ··· + lk = 1, x is a affine combination. : l1,...,lk ≥ 0, l1 + ··· + lk = 1, ; : convex ; • For S ⊆ Rn,S 6= 0/, the set lin(S) (resp. cone(S),aff(S),conv(S)) of all linear (resp. conic, affine, convex) combinations of finitely many vectors of S is called the linear (resp. conic, affine, convex) hull of S. Outer and inner descriptions • A subset P ⊆ Rn is a H-polytope, i.e., a bounded set of the form Outer n m×n m P = fx 2 R j Ax ≤ bg, for some A 2 R ,b 2 R . if and only if it is a V -polytope, i.e., Inner n P = conv(V ), for some finite V ⊂ R • A subset C ⊆ Rn is a H-cone, i.e., Outer n m×n C = fx 2 R j Ax ≤ 0g, for some A 2 R . if and only if it is a V -cone, i.e., Inner n C = cone(Y ), for some finite Y ⊂ R Minkowski sum • X,Y ⊆ Rn • X + Y = fx + y j x 2 X,y 2 Y g (Minkowski sum) Linear Programming, by Alexander Bockmayr, 30. November 2010, 15:52 2003 + = Main theorem for polyhedra A subset P ⊆ Rn is a H-polyhedron, i.e., Outer n m×n m P = fx 2 R j Ax ≤ bg, for some A 2 R ,b 2 R . if and only if it is a V -polyhedron, i.e., Inner n P = conv(V ) + cone(Y ), for some finite V ,Y ⊂ R Theorem of Minkowski • For each polyhedron P ⊆ Rn there exist finitely many points p1,...,pk in P and finitely many rays r 1,...,r l of P such that P = conv(p1,...,pk ) + cone(r 1,...,r l ). • If the polyhedron P is pointed, then p1,...,pk may be chosen as the uniquely determined vertices of P, and r 1,...,r l as representatives of the up to scalar multiplication uniquely determined extreme rays of P. • Special cases – A polytope is the convex hull of its vertices. – A pointed polyhedral cone is the conic hull of its extreme rays. Application: Metabolic networks ¦ ¦ ¦ ¦ § § § § Reversible reaction ¦ ¦ ¦ ¦ § § § Internal metabolite § X3 ¦ ¦ ¦ ¦ § § § § "System boundary" 3 F X12 External metabolite 12 11 5 B D ¤ ¤ ¤ ¥ ¥ ¥ 9 ¨ ¨ ¨ © © 1 © ¤ ¤ ¤ ¥ ¥ ¥ A 28 10 ¨ ¨ ¨ © © X1 © ¤ ¤ ¤ ¥ ¥ ¥ G X10 ¨ ¨ ¨ © © © C 6 E External reaction 4 7 ¡ ¡ ¡ ¢ ¢ ¢ ¢ £ £ £ £ ¡ ¡ ¡ ¢ ¢ ¢ ¢ £ £ £ £ X7 Internal reaction ¡ ¡ X4 ¡ ¢ ¢ ¢ ¢ £ £ £ Irreversible reaction £ Stoichiometric matrix • Metabolites (internal) rows 2004 Linear Programming, by Alexander Bockmayr, 30. November 2010, 15:52 • Biochemical reactions columns A 0 ... −k ... 1 B B 0 C B C C B −l C B C j D B 0 C kA + lC −! mE + nH B C E B m C B C F B 0 C B C G @ 0 A H ... n ... Example network F 3 12 11 5 B D 9 1 A 28 10 G C 6 E 4 7 0 1 −10000000000 1 B 0 1 −1 0 −1 0 0 0 0 0 0 0 C B C B 0 1 0 −1 0 −1 0 0 0 0 0 0 C B C S = B 0 0 0 0 1 0 0 1 −1 0 −1 0 C. B C B 0 0 0 0 0 1 −1 −1 0 0 0 0 C B C @ 00000000001 −1 A 0 0 0 0 0 0 0 0 1 −1 0 0 Flux cone • Flux balance: Sv = 0 • Irreversibiliy of some reactions: vi ≥ 0,i 2 Irr. • Steady-state flux cone n C = fv 2 R j Sv = 0,vi ≥ 0, for i 2 Irrg 1 k 1 k • Metabolic network analysis find p ,...,p 2 C with C = conefp ,...,p g..
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