From 3e6f51aa1f2d6f300cb22281701901add3631904 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 1 Jun 2015 01:33:58 -0400 Subject: [PATCH] Try the span_iso approach to fix my isomorphism tests. --- mjo/cone/cone.py | 154 ++++++++++++++++++++++++++++++----------------- 1 file changed, 100 insertions(+), 54 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 3a1e190..3f5a4fe 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -7,57 +7,91 @@ addsitedir(abspath('../../')) from sage.all import * +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 project_span(K, K2 = None): +def span_iso(K): r""" - Return a "copy" of ``K`` embeded in a lower-dimensional space. + 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:: - By default, we will project ``K`` into the subspace spanned by its - rays. However, if ``K2`` is not ``None``, we will project into the - space spanned by the rays of ``K2`` instead. + sage: K = random_cone(max_dim=10) + sage: (phi, phi_inv) = span_iso(K) + sage: phi_inv(phi(K)) == K + True - EXAMPLES:: + The image of ``K`` under the isomorphism should have full dimension:: - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: project_span(K) - 2-d cone in 2-d lattice N - sage: project_span(K).rays() - N(1, 0), - N(0, 1) - in 2-d lattice N + sage: K = random_cone(max_dim=10) + sage: (phi, phi_inv) = span_iso(K) + sage: phi(K).dim() == phi(K).lattice_dim() + True - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: K2 = Cone([(0,1)]) - sage: project_span(K, K2).rays() - N(1) - in 1-d lattice N + The isomorphism should be an inner product space isomorphism, and + thus it should preserve dual cones (and commute with the "dual" + operation). But beware the automatic renaming of the dual lattice. + OH AND YOU HAVE TO SORT THE CONES:: + + sage: K = random_cone(max_dim=10, strictly_convex=False, solid=True) + sage: L = K.lattice() + sage: rename_lattice(L, 'L') + sage: (phi, phi_inv) = span_iso(K) + sage: sorted(phi_inv( phi(K).dual() )) == sorted(K.dual()) + True + + We may need to isomorph twice to make sure we stop moving down to + smaller spaces. (Once you've done this on a cone and its dual, the + result should be proper.) OH AND YOU HAVE TO SORT THE CONES:: + + sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False) + sage: L = K.lattice() + sage: rename_lattice(L, 'L') + sage: (phi, phi_inv) = span_iso(K) + sage: K_S = phi(K) + sage: (phi_dual, phi_dual_inv) = span_iso(K_S.dual()) + sage: J_T = phi_dual(K_S.dual()).dual() + sage: phi_inv(phi_dual_inv(J_T)) == K + True """ - # Allow us to use a second cone to generate the subspace into - # which we're "projecting." - if K2 is None: - K2 = K + 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()) - # Use these to generate the new cone. - cs1 = K.rays().matrix().columns() + # 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) - # And use these to figure out which indices to drop. - cs2 = K2.rays().matrix().columns() + return Cone(newrays, lattice=L) - perp_idxs = [] + 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()) - for idx in range(0, len(cs2)): - if cs2[idx].is_zero(): - perp_idxs.append(idx) - solid_cols = [ cs1[idx] for idx in range(0,len(cs1)) - if not idx in perp_idxs - and not idx >= len(cs2) ] + return (phi, phi_inverse) - m = matrix(solid_cols) - L = ToricLattice(len(m.rows())) - J = Cone(m.transpose(), lattice=L) - return J def discrete_complementarity_set(K): @@ -204,23 +238,6 @@ def LL(K): sage: sum(map(abs, l)) 0 - Try the formula in my paper:: - - sage: K = random_cone(max_dim=15, max_rays=25) - sage: actual = lyapunov_rank(K) - sage: K_S = project_span(K) - sage: J_T1 = project_span(K, K_S.dual()) - sage: J_T2 = project_span(K_S.dual()).dual() - sage: J_T2 = Cone(J_T2.rays(), lattice=J_T1.lattice()) - sage: J_T1 == J_T2 - True - sage: J_T = J_T1 - 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 - """ V = K.lattice().vector_space() @@ -302,6 +319,9 @@ 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. @@ -390,5 +410,31 @@ 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, phi1_inv) = span_iso(K) + sage: K_S = phi1(K) + sage: (phi2, phi2_inv) = span_iso(K_S.dual()) + sage: J_T = phi2(K_S.dual()).dual() + sage: phi1_inv(phi2_inv(J_T)) == K + True + 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 + """ return len(LL(K)) -- 2.43.2