X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=e40579fa634dc51afc8eaf83ff5b3e68025180b5;hb=98637c981445d35a061878923baf3ae4651ecb0b;hp=f4b2244c3486ad6d9ec417bdcdddcc9ffd0f2a62;hpb=7c71dbc3454b5211269b879462f3530d76ad6991;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index f4b2244..e40579f 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,12 +8,55 @@ addsitedir(abspath('../../')) from sage.all import * -def basically_the_same(K1,K2): +def _basically_the_same(K1, K2): r""" + Test whether or not ``K1`` and ``K2`` are "basically the same." + + This is a hack to get around the fact that it's difficult to tell + when two cones are linearly isomorphic. We have a proposition that + equates two cones, but represented over `\mathbb{Q}`, they are + merely linearly isomorphic (not equal). So rather than test for + equality, we test a list of properties that should be preserved + under an invertible linear transformation. + + OUTPUT: + ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` - otherwise. This is intended as a lazy way to check whether or not - ``K1`` and ``K2`` are linearly isomorphic (i.e. ``A(K1) == K2`` for - some invertible linear transformation ``A``). + otherwise. + + EXAMPLES: + + Any proper cone with three generators in `\mathbb{R}^{3}` is + basically the same as the nonnegative orthant:: + + sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)]) + sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)]) + sage: _basically_the_same(K1, K2) + True + + Negating a cone gives you another cone that is basically the same:: + + sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)]) + sage: _basically_the_same(K, -K) + True + + TESTS: + + Any cone is basically the same as itself:: + + sage: K = random_cone(max_ambient_dim = 8) + sage: _basically_the_same(K, K) + True + + After applying an invertible matrix to the rows of a cone, the + result should be basically the same as the cone we started with:: + + sage: K1 = random_cone(max_ambient_dim = 8) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: _basically_the_same(K1, K2) + True + """ if K1.lattice_dim() != K2.lattice_dim(): return False @@ -48,7 +91,7 @@ def basically_the_same(K1,K2): -def rho(K, K2=None): +def _rho(K, K2=None): r""" Restrict ``K`` into its own span, or the span of another cone. @@ -64,18 +107,18 @@ def rho(K, K2=None): EXAMPLES:: sage: K = Cone([(1,)]) - sage: rho(K) == K + sage: _rho(K) == K True sage: K2 = Cone([(1,0)]) - sage: rho(K2).rays() + sage: _rho(K2).rays() N(1) in 1-d lattice N sage: K3 = Cone([(1,0,0)]) - sage: rho(K3).rays() + sage: _rho(K3).rays() N(1) in 1-d lattice N - sage: rho(K2) == rho(K3) + sage: _rho(K2) == _rho(K3) True TESTS: @@ -83,8 +126,8 @@ def rho(K, K2=None): The projected cone should always be solid:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K_S = rho(K) + sage: K = random_cone(max_ambient_dim = 8) + sage: K_S = _rho(K) sage: K_S.is_solid() True @@ -92,123 +135,139 @@ def rho(K, K2=None): dimension as the space we restricted it to:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K_S = rho(K, K.dual() ) + sage: K = random_cone(max_ambient_dim = 8) + sage: K_S = _rho(K, K.dual() ) sage: K_S.lattice_dim() == K.dual().dim() True This function should not affect the dimension of a cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K.dim() == rho(K).dim() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.dim() == _rho(K).dim() True Nor should it affect the lineality of a cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K.lineality() == rho(K).lineality() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.lineality() == _rho(K).lineality() True No matter which space we restrict to, the lineality should not increase:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K.lineality() >= rho(K).lineality() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.lineality() >= _rho(K).lineality() True - sage: K.lineality() >= rho(K, K.dual()).lineality() + sage: K.lineality() >= _rho(K, K.dual()).lineality() True If we do this according to our paper, then the result is proper:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False) - sage: K_S = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K = random_cone(max_ambient_dim = 8, + ....: strictly_convex=False, + ....: solid=False) + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() True - sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP = _rho(K_S, K_S.dual()) sage: K_SP.is_proper() True :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False) - sage: K_S = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K = random_cone(max_ambient_dim = 8, + ....: strictly_convex=True, + ....: solid=False) + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() True - sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP = _rho(K_S, K_S.dual()) sage: K_SP.is_proper() True :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True) - sage: K_S = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K = random_cone(max_ambient_dim = 8, + ....: strictly_convex=False, + ....: solid=True) + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() True - sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP = _rho(K_S, K_S.dual()) sage: K_SP.is_proper() True :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True) - sage: K_S = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K = random_cone(max_ambient_dim = 8, + ....: strictly_convex=True, + ....: solid=True) + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: K_SP.is_proper() True - sage: K_SP = rho(K_S, K_S.dual()) + sage: K_SP = _rho(K_S, K_S.dual()) sage: K_SP.is_proper() True - Test the proposition in our paper concerning the duals and + Test Proposition 7 in our paper concerning the duals and restrictions. Generate a random cone, then create a subcone of it. The operation of dual-taking should then commute with rho:: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False) + sage: J = random_cone(max_ambient_dim = 8, + ....: solid=False, + ....: strictly_convex=False) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W = rho(K, J) - sage: K_star_W_star = rho(K.dual(), J).dual() - sage: basically_the_same(K_W, K_star_W_star) + sage: K_W_star = _rho(K, J).dual() + sage: K_star_W = _rho(K.dual(), J) + sage: _basically_the_same(K_W_star, K_star_W) True :: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False) + sage: J = random_cone(max_ambient_dim = 8, + ....: solid=True, + ....: strictly_convex=False) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W = rho(K, J) - sage: K_star_W_star = rho(K.dual(), J).dual() - sage: basically_the_same(K_W, K_star_W_star) + sage: K_W_star = _rho(K, J).dual() + sage: K_star_W = _rho(K.dual(), J) + sage: _basically_the_same(K_W_star, K_star_W) True :: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=True) + sage: J = random_cone(max_ambient_dim = 8, + ....: solid=False, + ....: strictly_convex=True) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W = rho(K, J) - sage: K_star_W_star = rho(K.dual(), J).dual() - sage: basically_the_same(K_W, K_star_W_star) + sage: K_W_star = _rho(K, J).dual() + sage: K_star_W = _rho(K.dual(), J) + sage: _basically_the_same(K_W_star, K_star_W) True :: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=True) + sage: J = random_cone(max_ambient_dim = 8, + ....: solid=True, + ....: strictly_convex=True) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W = rho(K, J) - sage: K_star_W_star = rho(K.dual(), J).dual() - sage: basically_the_same(K_W, K_star_W_star) + sage: K_W_star = _rho(K, J).dual() + sage: K_star_W = _rho(K.dual(), J) + sage: _basically_the_same(K_W_star, K_star_W) True """ @@ -239,19 +298,29 @@ def discrete_complementarity_set(K): r""" Compute the discrete complementarity set of this cone. - The complementarity set of this cone is the set of all orthogonal - pairs `(x,s)` such that `x` is in this cone, and `s` is in its - dual. The discrete complementarity set restricts `x` and `s` to be - generators of their respective cones. + The complementarity set of a cone is the set of all orthogonal pairs + `(x,s)` such that `x` is in the cone, and `s` is in its dual. The + discrete complementarity set is a subset of the complementarity set + where `x` and `s` are required to be generators of their respective + cones. + + For polyhedral cones, the discrete complementarity set is always + finite. OUTPUT: A list of pairs `(x,s)` such that, + * Both `x` and `s` are vectors (not rays). * `x` is a generator of this cone. * `s` is a generator of this cone's dual. * `x` and `s` are orthogonal. + REFERENCES: + + .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an + Improper Cone. Work in-progress. + EXAMPLES: The discrete complementarity set of the nonnegative orthant consists @@ -282,25 +351,43 @@ def discrete_complementarity_set(K): sage: discrete_complementarity_set(K) [] + Likewise when this cone is trivial (its dual is the entire space):: + + sage: L = ToricLattice(0) + sage: K = Cone([], ToricLattice(0)) + sage: discrete_complementarity_set(K) + [] + TESTS: The complementarity set of the dual can be obtained by switching the components of the complementarity set of the original cone:: sage: set_random_seed() - sage: K1 = random_cone(max_dim=6) + sage: K1 = random_cone(max_ambient_dim=6) sage: K2 = K1.dual() sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)] sage: actual = discrete_complementarity_set(K1) sage: sorted(actual) == sorted(expected) True + The pairs in the discrete complementarity set are in fact + complementary:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=6) + sage: dcs = discrete_complementarity_set(K) + sage: sum([x.inner_product(s).abs() for (x,s) in dcs]) + 0 + """ V = K.lattice().vector_space() - # Convert the rays to vectors so that we can compute inner - # products. + # Convert rays to vectors so that we can compute inner products. xs = [V(x) for x in K.rays()] + + # We also convert the generators of the dual cone so that we + # return pairs of vectors and not (vector, ray) pairs. ss = [V(s) for s in K.dual().rays()] return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] @@ -381,7 +468,7 @@ def LL(K): of the cone:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: C_of_K = discrete_complementarity_set(K) sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ] sage: sum(map(abs, l)) @@ -393,7 +480,7 @@ def LL(K): \right)` sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: LL2 = [ L.transpose() for L in LL(K.dual()) ] sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2) sage: LL1_vecs = [ V(m.list()) for m in LL(K) ] @@ -579,45 +666,106 @@ def lyapunov_rank(K): [Rudolf et al.]_:: sage: set_random_seed() - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: K2 = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) + sage: K2 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: K = K1.cartesian_product(K2) sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) True + The Lyapunov rank is invariant under a linear isomorphism + [Orlitzky/Gowda]_:: + + sage: K1 = random_cone(max_ambient_dim = 8) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + + Just to be sure, test a few more:: + + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + + :: + + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=False) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + + :: + + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=True) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + + :: + + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=False) + sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') + sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) + sage: lyapunov_rank(K1) == lyapunov_rank(K2) + True + The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` itself [Rudolf et al.]_:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True Make sure we exercise the non-strictly-convex/non-solid case:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=False) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True Let's check the other permutations as well, just to be sure:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=True) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True :: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=False) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True :: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True @@ -628,7 +776,9 @@ def lyapunov_rank(K): the Lyapunov rank of the trivial cone will be zero:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: (n == 0 or 1 <= b) and b <= n @@ -640,7 +790,7 @@ def lyapunov_rank(K): Lyapunov rank `n-1` in `n` dimensions:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: b == n-1 @@ -650,10 +800,10 @@ def lyapunov_rank(K): reduced to that of a proper cone [Orlitzky/Gowda]_:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: actual = lyapunov_rank(K) - sage: K_S = rho(K) - sage: K_SP = rho(K_S.dual()).dual() + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() sage: l = K.lineality() sage: c = K.codim() sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 @@ -663,7 +813,9 @@ def lyapunov_rank(K): The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: lyapunov_rank(K) == len(LL(K)) True @@ -671,24 +823,41 @@ def lyapunov_rank(K): just increase our confidence that the reduction scheme works:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=False) sage: lyapunov_rank(K) == len(LL(K)) True :: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=True) sage: lyapunov_rank(K) == len(LL(K)) True :: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=False, + ....: solid=False) sage: lyapunov_rank(K) == len(LL(K)) True + Test Theorem 3 in [Orlitzky/Gowda]_:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) + sage: L = ToricLattice(K.lattice_dim() + 1) + sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L) + sage: lyapunov_rank(K) >= K.lattice_dim() + True + """ beta = 0 @@ -698,7 +867,7 @@ def lyapunov_rank(K): if m < n: # K is not solid, restrict to its span. - K = rho(K) + K = _rho(K) # Lemma 2 beta += m*(n - m) + (n - m)**2 @@ -706,8 +875,8 @@ def lyapunov_rank(K): if l > 0: # K is not pointed, restrict to the span of its dual. Uses a # proposition from our paper, i.e. this is equivalent to K = - # rho(K.dual()).dual(). - K = rho(K, K.dual()) + # _rho(K.dual()).dual(). + K = _rho(K, K.dual()) # Lemma 3 beta += m * l