X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=f2e8b2e9ee104e6cbd216e7473fd65dd76947d98;hb=44802773ad9e5151890ed37e7bb2463ff9fc4135;hp=ff7d195d134c15943dbb75c1f26b741bb4a0afba;hpb=10142e85f34c47fa35df002f519d1d58a79a74f4;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index ff7d195..f2e8b2e 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,55 +8,246 @@ addsitedir(abspath('../../')) from sage.all import * -def project_span(K): +def iso_space(K): r""" - Project ``K`` into its own span. + Construct the space `W \times W^{\perp}` isomorphic to the ambient space + of ``K`` where `W` is equal to the span of ``K``. + """ + V = K.lattice().vector_space() + + # Create the space W \times W^{\perp} isomorphic to V. + # First we get an orthogonal (but not normal) basis... + M = matrix(V.base_field(), K.rays()) + W_basis,_ = M.gram_schmidt() + + W = V.subspace_with_basis(W_basis) + W_perp = W.complement() + + return W.cartesian_product(W_perp) + + +def ips_iso(K): + r""" + Construct the IPS isomorphism and its inverse from our paper. + + Given a cone ``K``, the returned isomorphism will split its ambient + vector space `V` into a cartesian product `W \times W^{\perp}` where + `W` equals the span of ``K``. + """ + V = K.lattice().vector_space() + V_iso = iso_space(K) + (W, W_perp) = V_iso.cartesian_factors() + + # A space equivalent to V, but using our basis. + V_user = V.subspace_with_basis( W.basis() + W_perp.basis() ) + + def phi(v): + # Write v in terms of our custom basis, where the first dim(W) + # coordinates are for the W-part of the basis. + cs = V_user.coordinates(v) + + w1 = sum([ V_user.basis()[idx]*cs[idx] + for idx in range(0, W.dimension()) ]) + w2 = sum([ V_user.basis()[idx]*cs[idx] + for idx in range(W.dimension(), V.dimension()) ]) + + return V_iso( (w1, w2) ) + + + def phi_inv( pair ): + # Crash if the arguments are in the wrong spaces. + V_iso(pair) + + #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ]) + #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx] + # for idx in range(0,n-m) ]) + + return sum( pair.cartesian_factors() ) + + + return (phi,phi_inv) + + + +def unrestrict_span(K, K2=None): + if K2 is None: + K2 = K + + _,phi_inv = ips_iso(K2) + V_iso = iso_space(K2) + (W, W_perp) = V_iso.cartesian_factors() + + rays = [] + for r in K.rays(): + w = sum([ r[idx]*W.basis()[idx] for idx in range(0,len(r)) ]) + pair = V_iso( (w, W_perp.zero()) ) + rays.append( phi_inv(pair) ) + + L = ToricLattice(W.dimension() + W_perp.dimension()) + + return Cone(rays, lattice=L) + + + +def intersect_span(K1, K2): + r""" + Return a new cone obtained by intersecting ``K1`` with the span of ``K2``. + """ + L = K1.lattice() + + if L.rank() != K2.lattice().rank(): + raise ValueError('K1 and K2 must belong to lattices of the same rank.') + + SL_gens = list(K2.rays()) + span_K2_gens = SL_gens + [ -g for g in SL_gens ] + + # The lattices have the same rank (see above) so this should work. + span_K2 = Cone(span_K2_gens, L) + return K1.intersection(span_K2) + + + +def restrict_span(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. + + OUTPUT: + + A new cone in a sublattice. EXAMPLES:: sage: K = Cone([(1,)]) - sage: project_span(K) == K + sage: restrict_span(K) == K True sage: K2 = Cone([(1,0)]) - sage: project_span(K2).rays() + sage: restrict_span(K2).rays() N(1) in 1-d lattice N sage: K3 = Cone([(1,0,0)]) - sage: project_span(K3).rays() + sage: restrict_span(K3).rays() N(1) in 1-d lattice N - sage: project_span(K2) == project_span(K3) + sage: restrict_span(K2) == restrict_span(K3) True TESTS: The projected cone should always be solid:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10) - sage: K_S = project_span(K) + sage: K_S = restrict_span(K) sage: K_S.is_solid() True + And the resulting cone should live in a space having the same + dimension as the space we restricted it to:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10) + sage: K_S = restrict_span( intersect_span(K, K.dual()), K.dual() ) + sage: K_S.lattice_dim() == K.dual().dim() + True + + This function has ``unrestrict_span()`` as its inverse:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10, solid=True) + sage: J = restrict_span(K) + sage: K == unrestrict_span(J,K) + True + + This function should not affect the dimension of a cone:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10) + sage: K.dim() == restrict_span(K).dim() + True + + Nor should it affect the lineality of a cone:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10) + sage: lineality(K) == lineality(restrict_span(K)) + True + + No matter which space we restrict to, the lineality should not + increase:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10) + sage: J = intersect_span(K, K.dual()) + sage: lineality(K) >= lineality(restrict_span(J, K.dual())) + True + If we do this according to our paper, then the result is proper:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10) - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() + sage: K_S = restrict_span(K) + sage: P = restrict_span(K_S.dual()).dual() sage: P.is_proper() True + If ``K`` is strictly convex, then both ``K_W`` and + ``K_star_W.dual()`` should equal ``K`` (after we unrestrict):: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 10, strictly_convex=True) + sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual()) + sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: j1 = unrestrict_span(K_W, K.dual()) + sage: j2 = unrestrict_span(K_star_W_star, K.dual()) + sage: j1 == j2 + True + sage: j1 == K + True + sage: K; [ list(r) for r in K.rays() ] + + 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 = 10, solid=False, strictly_convex=False) + sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual()) + sage: K_star_W_star = restrict_span(K.dual(), K.dual()).dual() + sage: K_W.nrays() == K_star_W_star.nrays() + True + sage: K_W.dim() == K_star_W_star.dim() + True + sage: lineality(K_W) == lineality(K_star_W_star) + True + sage: K_W.is_solid() == K_star_W_star.is_solid() + True + sage: K_W.is_strictly_convex() == K_star_W_star.is_strictly_convex() + True + """ - L = K.lattice() - F = L.base_field() - Q = L.quotient(K.sublattice_complement()) - vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ] + if K2 is None: + K2 = K + + phi,_ = ips_iso(K2) + (W, W_perp) = iso_space(K2).cartesian_factors() + + ray_pairs = [ phi(r) for r in K.rays() ] + + if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]): + msg = 'Cone has nonzero components in W-perp!' + raise ValueError(msg) + + # Represent the cone in terms of a basis for W, i.e. with smaller + # vectors. + ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ] - newL = None - if len(vecs) == 0: - newL = ToricLattice(0) + L = ToricLattice(W.dimension()) - return Cone(vecs, lattice=newL) + return Cone(ws, lattice=L) @@ -112,6 +303,7 @@ def lineality(K): 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 = 10) sage: l = lineality(K) sage: l in ZZ @@ -121,6 +313,7 @@ def lineality(K): A strictly convex cone should have lineality zero:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10, strictly_convex = True) sage: lineality(K) 0 @@ -191,6 +384,7 @@ def codim(K): The codimension 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 = 10) sage: c = codim(K) sage: c in ZZ @@ -200,12 +394,14 @@ def codim(K): A solid cone should have codimension zero:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10, solid = True) sage: codim(K) 0 The codimension of a cone is equal to the lineality of its dual:: + sage: set_random_seed() sage: K = random_cone(max_dim = 10, solid = True) sage: codim(K) == lineality(K.dual()) True @@ -268,11 +464,12 @@ def discrete_complementarity_set(K): The complementarity set of the dual can be obtained by switching the components of the complementarity set of the original cone:: - sage: K1 = random_cone(max_dim=10, max_rays=10) + sage: set_random_seed() + sage: K1 = random_cone(max_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: actual == expected + sage: sorted(actual) == sorted(expected) True """ @@ -352,18 +549,33 @@ def LL(K): every pair `\left( x,s \right)` in the discrete complementarity set of the cone:: + sage: set_random_seed() sage: K = random_cone(max_dim=8, max_rays=10) 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)) 0 + The Lyapunov-like transformations on a cone and its dual are related + by transposition, but we're not guaranteed to compute transposed + elements of `LL\left( K \right)` as our basis for `LL\left( K^{*} + \right)` + + sage: set_random_seed() + sage: K = random_cone(max_dim=8, max_rays=10) + 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) ] + sage: LL2_vecs = [ V(m.list()) for m in LL2 ] + sage: V.span(LL1_vecs) == V.span(LL2_vecs) + True + """ V = K.lattice().vector_space() C_of_K = discrete_complementarity_set(K) - tensor_products = [s.tensor_product(x) for (x,s) in C_of_K] + tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] # Sage doesn't think matrices are vectors, so we have to convert # our matrices to vectors explicitly before we can figure out how @@ -465,8 +677,8 @@ def lyapunov_rank(K): [Orlitzky/Gowda]_:: sage: R5 = VectorSpace(QQ, 5) - sage: gens = R5.basis() + [ -r for r in R5.basis() ] - sage: K = Cone(gens) + sage: gs = R5.basis() + [ -r for r in R5.basis() ] + sage: K = Cone(gs) sage: lyapunov_rank(K) 25 @@ -499,8 +711,8 @@ def lyapunov_rank(K): sage: neg_e1 = (-1,0,0,0,0) sage: e2 = (0,1,0,0,0) sage: neg_e2 = (0,-1,0,0,0) - sage: zero = (0,0,0,0,0) - sage: K = Cone([e1, neg_e1, e2, neg_e2, zero, zero, zero]) + sage: z = (0,0,0,0,0) + 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) @@ -535,6 +747,7 @@ def lyapunov_rank(K): The Lyapunov rank should be additive on a product of proper cones [Rudolf et al.]_:: + sage: set_random_seed() sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True) sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True) sage: K = K1.cartesian_product(K2) @@ -544,16 +757,25 @@ def lyapunov_rank(K): 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=10, max_rays=10) 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=10, strictly_convex=False, solid=False) + sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + True + The Lyapunov rank of a proper polyhedral cone in `n` dimensions can be any number between `1` and `n` inclusive, excluding `n-1` [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the trivial cone in a trivial space as well. However, in zero dimensions, the Lyapunov rank of the trivial cone will be zero:: + sage: set_random_seed() sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() @@ -565,6 +787,7 @@ def lyapunov_rank(K): In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have Lyapunov rank `n-1` in `n` dimensions:: + sage: set_random_seed() sage: K = random_cone(max_dim=10) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() @@ -574,10 +797,11 @@ def lyapunov_rank(K): The calculation of the Lyapunov rank of an improper cone can be reduced to that of a proper cone [Orlitzky/Gowda]_:: + sage: set_random_seed() sage: K = random_cone(max_dim=10) sage: actual = lyapunov_rank(K) - sage: K_S = project_span(K) - sage: P = project_span(K_S.dual()).dual() + sage: K_S = restrict_span(K) + sage: P = restrict_span(K_S.dual()).dual() sage: l = lineality(K) sage: c = codim(K) sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2 @@ -586,11 +810,13 @@ 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=10, strictly_convex=True, solid=True) sage: lyapunov_rank(K) == len(LL(K)) True """ + K_orig = K beta = 0 m = K.dim() @@ -599,14 +825,23 @@ def lyapunov_rank(K): if m < n: # K is not solid, project onto its span. - K = project_span(K) + K = restrict_span(K) # Lemma 2 beta += m*(n - m) + (n - m)**2 if l > 0: # K is not pointed, project its dual onto its span. - K = project_span(K.dual()).dual() + # Uses a proposition from our paper, i.e. this is + # equivalent to K = restrict_span(K.dual()).dual() + K = restrict_span(intersect_span(K,K.dual()), K.dual()) + #K = restrict_span(K.dual()).dual() + + #Ks = [ list(r) for r in sorted(K.rays()) ] + #Js = [ list(r) for r in sorted(J.rays()) ] + + #if Ks != Js: + # print [ list(r) for r in K_orig.rays() ] # Lemma 3 beta += m * l