X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=60f9c34ec8bc271d65812859f51ca77636c8cbbc;hb=874e3ce831e0b1901b3c280a32ffe18e36f54959;hp=4b0193692edd7655f2880408d143bf141e6d567c;hpb=3a312d50b5aba08e72039f1ebcde7b12c62a1e9f;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 4b01936..60f9c34 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -7,6 +7,116 @@ addsitedir(abspath('../../')) from sage.all import * +def project_span(K): + r""" + Project ``K`` into its own span. + + EXAMPLES:: + + sage: K = Cone([(1,)]) + sage: project_span(K) == K + True + + sage: K2 = Cone([(1,0)]) + sage: project_span(K2).rays() + N(1) + in 1-d lattice N + sage: K3 = Cone([(1,0,0)]) + sage: project_span(K3).rays() + N(1) + in 1-d lattice N + sage: project_span(K2) == project_span(K3) + True + + TESTS: + + The projected cone should always be solid:: + + sage: K = random_cone() + sage: K_S = project_span(K) + sage: K_S.is_solid() + True + + If we do this according to our paper, then the result is proper:: + + sage: K = random_cone() + sage: K_S = project_span(K) + sage: P = project_span(K_S.dual()).dual() + sage: P.is_proper() + True + + """ + F = K.lattice().base_field() + Q = K.lattice().quotient(K.sublattice_complement()) + vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ] + + L = None + if len(vecs) == 0: + L = ToricLattice(0) + + return Cone(vecs, lattice=L) + + +def rename_lattice(L,s): + r""" + Change all names of the given lattice to ``s``. + """ + L._name = s + L._dual_name = s + L._latex_name = s + L._latex_dual_name = s + +def span_iso(K): + r""" + Return an isomorphism (and its inverse) that will send ``K`` into a + lower-dimensional space isomorphic to its span (and back). + + EXAMPLES: + + The inverse composed with the isomorphism should be the identity:: + + sage: K = random_cone(max_dim=10) + sage: (phi, phi_inv) = span_iso(K) + sage: phi_inv(phi(K)) == K + True + + The image of ``K`` under the isomorphism should have full dimension:: + + sage: K = random_cone(max_dim=10) + sage: (phi, phi_inv) = span_iso(K) + sage: phi(K).dim() == phi(K).lattice_dim() + True + + """ + phi_domain = K.sublattice().vector_space() + phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension()) + + # S goes from the new space to the cone space. + S = linear_transformation(phi_codo, phi_domain, phi_domain.basis()) + + # phi goes from the cone space to the new space. + def phi(J_orig): + r""" + Takes a cone ``J`` and sends it into the new space. + """ + newrays = map(S.inverse(), J_orig.rays()) + L = None + if len(newrays) == 0: + L = ToricLattice(0) + + return Cone(newrays, lattice=L) + + def phi_inverse(J_sub): + r""" + The inverse to phi which goes from the new space to the cone space. + """ + newrays = map(S, J_sub.rays()) + return Cone(newrays, lattice=K.lattice()) + + + return (phi, phi_inverse) + + def discrete_complementarity_set(K): r""" @@ -233,6 +343,9 @@ def lyapunov_rank(K): cone and Lyapunov-like transformations, Mathematical Programming, 147 (2014) 155-170. + .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an + Improper Cone. Work in-progress. + .. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear optimality constraints for the cone of positive polynomials, Mathematical Programming, Series B, 129 (2011) 5-31. @@ -321,5 +434,91 @@ def lyapunov_rank(K): sage: b == n-1 False + In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have + Lyapunov rank `n-1` in `n` dimensions:: + + sage: K = random_cone(max_dim=10) + sage: b = lyapunov_rank(K) + sage: n = K.lattice_dim() + sage: b == n-1 + False + + The calculation of the Lyapunov rank of an improper cone can be + reduced to that of a proper cone [Orlitzky/Gowda]_:: + + sage: K = random_cone(max_dim=15, solid=False, strictly_convex=False) + sage: actual = lyapunov_rank(K) + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + + Repeat the previous test with different ``random_cone()`` params:: + + sage: K = random_cone(max_dim=15, solid=False, strictly_convex=True) + sage: actual = lyapunov_rank(K) + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + + sage: K = random_cone(max_dim=15, solid=True, strictly_convex=False) + sage: actual = lyapunov_rank(K) + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + + sage: K = random_cone(max_dim=15, solid=True, strictly_convex=True) + sage: actual = lyapunov_rank(K) + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + + sage: K = random_cone(max_dim=15) + sage: actual = lyapunov_rank(K) + sage: (phi1, _) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, _) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + + And test with the project_span function:: + + sage: K = random_cone(max_dim=15) + sage: actual = lyapunov_rank(K) + sage: K_S = project_span(K) + sage: P = project_span(K_S.dual()).dual() + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True + """ return len(LL(K))