X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=48b4061748281e0ebcb3ef19e9872be824666ba5;hb=af3e2ce56ad6561c5c9b1b6cf3df22d690550618;hp=b9e930e6819643710b82c06faa0b72b934298d96;hpb=6bd30534d5aa984c73f511121efa8fda4386c51a;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index b9e930e..48b4061 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,23 +8,55 @@ addsitedir(abspath('../../')) from sage.all import * -def drop_dependent(vs): +def _basically_the_same(K1, K2): r""" - Return the largest linearly-independent subset of ``vs``. - """ - result = [] - m = matrix(vs).echelon_form() - for idx in range(0, m.nrows()): - if not m[idx].is_zero(): - result.append(m[idx]) + Test whether or not ``K1`` and ``K2`` are "basically the same." - return result + 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: -def basically_the_same(K1,K2): - r""" ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` 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 @@ -35,7 +67,7 @@ def basically_the_same(K1,K2): if K1.dim() != K2.dim(): return False - if lineality(K1) != lineality(K2): + if K1.lineality() != K2.lineality(): return False if K1.is_solid() != K2.is_solid(): @@ -59,13 +91,14 @@ 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. INPUT: - - ``K2`` -- another cone whose lattice has the same rank as this cone. + - ``K2`` -- another cone whose lattice has the same rank as this + cone. OUTPUT: @@ -74,18 +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: @@ -93,8 +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 @@ -102,136 +135,73 @@ 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: lineality(K) == lineality(rho(K)) + 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: lineality(K) >= lineality(rho(K)) + sage: K = random_cone(max_ambient_dim = 8) + sage: K.lineality() >= _rho(K).lineality() True - sage: lineality(K) >= lineality(rho(K, K.dual())) + 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: P = rho(K_S.dual()).dual() - sage: P.is_proper() - True - sage: P = rho(K_S, K_S.dual()) - sage: P.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: P = rho(K_S.dual()).dual() - sage: P.is_proper() - True - sage: P = rho(K_S, K_S.dual()) - sage: P.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: P = rho(K_S.dual()).dual() - sage: P.is_proper() - True - sage: P = rho(K_S, K_S.dual()) - sage: P.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: P = rho(K_S.dual()).dual() - sage: P.is_proper() - True - sage: P = rho(K_S, K_S.dual()) - sage: P.is_proper() + sage: K = random_cone(max_ambient_dim = 8) + sage: K_S = _rho(K) + sage: K_SP = _rho(K_S.dual()).dual() + sage: K_SP.is_proper() True - - Test the proposition in our paper concerning the duals, where the - subspace `W` is the span of `K^{*}`:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False) - sage: K_W = rho(K, K.dual()) - sage: K_star_W_star = rho(K.dual()).dual() - sage: basically_the_same(K_W, K_star_W_star) - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False) - sage: K_W = rho(K, K.dual()) - sage: K_star_W_star = rho(K.dual()).dual() - sage: basically_the_same(K_W, K_star_W_star) + 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 + 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: K = random_cone(max_dim = 8, solid=False, strictly_convex=True) - sage: K_W = rho(K, K.dual()) - sage: K_star_W_star = rho(K.dual()).dual() - sage: basically_the_same(K_W, K_star_W_star) - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True) - sage: K_W = rho(K, K.dual()) - sage: K_star_W_star = rho(K.dual()).dual() - sage: basically_the_same(K_W, K_star_W_star) + sage: J = random_cone(max_ambient_dim = 8) + sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) + 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 """ if K2 is None: K2 = K - # First we project K onto the span of K2. This can be done with - # cones (i.e. without converting to vector spaces), but it's - # annoying to deal with lattice mismatches. + # First we project K onto the span of K2. This will explode if the + # rank of ``K2.lattice()`` doesn't match ours. span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice()) K = K.intersection(span_K2) - V = K.lattice().vector_space() - - # Create the space W \times W^{\perp} isomorphic to V. - # First we get an orthogonal (but not normal) basis... - W_basis = drop_dependent(K2.rays()) - W = V.subspace_with_basis(W_basis) + # Cheat a little to get the subspace span(K2). The paper uses the + # rays of K2 as a basis, but everything is invariant under linear + # isomorphism (i.e. a change of basis), and this is a little + # faster. + W = span_K2.linear_subspace() # We've already intersected K with the span of K2, so every # generator of K should belong to W now. @@ -242,96 +212,33 @@ def rho(K, K2=None): -def lineality(K): - r""" - Compute the lineality of this cone. - - The lineality of a cone is the dimension of the largest linear - subspace contained in that cone. - - OUTPUT: - - A nonnegative integer; the dimension of the largest subspace - contained within this cone. - - REFERENCES: - - .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton - University Press, Princeton, 1970. - - EXAMPLES: - - The lineality of the nonnegative orthant is zero, since it clearly - contains no lines:: - - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 0 - - However, if we add another ray so that the entire `x`-axis belongs - to the cone, then the resulting cone will have lineality one:: - - sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 1 - - If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal - to the dimension of the ambient space (i.e. two):: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: lineality(K) - 2 - - Per the definition, the lineality of the trivial cone in a trivial - space is zero:: - - sage: K = Cone([], lattice=ToricLattice(0)) - sage: lineality(K) - 0 - - TESTS: - - The lineality of a cone should be an integer between zero and the - dimension of the ambient space, inclusive:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: l = lineality(K) - sage: l in ZZ - True - sage: (0 <= l) and (l <= K.lattice_dim()) - True - - A strictly convex cone should have lineality zero:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex = True) - sage: lineality(K) - 0 - - """ - return K.linear_subspace().dimension() - - 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, - * `x` is in this cone. + * Both `x` and `s` are vectors (not rays). * `x` is a generator of this cone. - * `s` is in this cone's dual. * `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 @@ -362,25 +269,43 @@ 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] @@ -461,7 +386,7 @@ 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)) @@ -473,7 +398,7 @@ 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) ] @@ -626,7 +551,7 @@ def lyapunov_rank(K): sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z]) sage: lyapunov_rank(K) 19 - sage: K.lattice_dim()**2 - K.dim()*codim(K) + sage: K.lattice_dim()**2 - K.dim()*K.codim() 19 The Lyapunov rank should be additive on a product of proper cones @@ -659,45 +584,30 @@ 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 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: 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: 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: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - :: + The Lyapunov rank is invariant under a linear isomorphism + [Orlitzky/Gowda]_:: - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + 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 - :: + 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, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True @@ -708,7 +618,9 @@ 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 @@ -720,7 +632,7 @@ 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 @@ -730,65 +642,53 @@ 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: P = rho(K_S.dual()).dual() - sage: l = lineality(K) - sage: c = codim(K) - sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2 + 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 sage: actual == expected True - The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: + The Lyapunov rank of any 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) sage: lyapunov_rank(K) == len(LL(K)) True - In fact the same can be said of any cone. These additional tests - just increase our confidence that the reduction scheme works:: + Test Theorem 3 in [Orlitzky/Gowda]_:: sage: set_random_seed() - sage: K = random_cone(max_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: lyapunov_rank(K) == len(LL(K)) - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) - sage: lyapunov_rank(K) == len(LL(K)) + 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 """ - K_orig = K beta = 0 m = K.dim() n = K.lattice_dim() - l = lineality(K) + l = K.lineality() if m < n: - # K is not solid, project onto its span. - K = rho(K) + # K is not solid, restrict to its span. + K = _rho(K) # Lemma 2 beta += m*(n - m) + (n - m)**2 if l > 0: - # K is not pointed, project its dual onto its span. - # Uses a proposition from our paper, i.e. this is - # equivalent to K = rho(K.dual()).dual() - K = rho(K, K.dual()) + # 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()) # Lemma 3 beta += m * l