From b966ec383942ef1ff7837786c29d1f3edc33b84e Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 31 May 2015 20:32:51 -0400 Subject: [PATCH] Commit the busted version of cone.py. --- mjo/cone/cone.py | 84 +++++++++++++++++++++++++++++++++++++++++++++++- 1 file changed, 83 insertions(+), 1 deletion(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 4b01936..81698e4 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,6 +8,58 @@ addsitedir(abspath('../../')) from sage.all import * +def project_span(K, K2 = None): + r""" + Return a "copy" of ``K`` embeded in a lower-dimensional space. + + 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. + + EXAMPLES:: + + 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 = 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 + + """ + # Allow us to use a second cone to generate the subspace into + # which we're "projecting." + if K2 is None: + K2 = K + + # Use these to generate the new cone. + cs1 = K.rays().matrix().columns() + + # And use these to figure out which indices to drop. + cs2 = K2.rays().matrix().columns() + + perp_idxs = [] + + 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) ] + + m = matrix(solid_cols) + L = ToricLattice(len(m.rows())) + J = Cone(m.transpose(), lattice=L) + return J + + def discrete_complementarity_set(K): r""" Compute the discrete complementarity set of this cone. @@ -207,7 +259,7 @@ def lyapunov_rank(K): An integer representing the Lyapunov rank of the cone. If the dimension of the ambient vector space is `n`, then the Lyapunov rank will be between `1` and `n` inclusive; however a rank of `n-1` is - not possible (see the first reference). + not possible for any cone. .. note:: @@ -233,6 +285,9 @@ 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 +376,32 @@ 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, max_rays=16) + 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, max_rays=25) + sage: actual = lyapunov_rank(K) + sage: K_S = project_span(K) + sage: J_T1 = project_span(K_S.dual()).dual() + sage: J_T2 = project_span(K, K_S.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 + """ return len(LL(K)) -- 2.44.2