X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=e9d0f1e643e3a0d40d8a754faf551ac317e36f37;hb=d27a3f72ac6b898534615de9111ace4082a4c55e;hp=8adc51cdd4836e8d1a405fb342f6c4bb8fa50fb6;hpb=c4fdc3d232c1c6c179f7919ebb4fd169017edafd;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 8adc51c..e9d0f1e 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,30 +8,12 @@ addsitedir(abspath('../../')) from sage.all import * -def drop_dependent(vs): - r""" - Return the largest linearly-independent subset of ``vs``. - """ - if len(vs) == 0: - # ...for lazy enough definitions of linearly-independent - return vs - - result = [] - old_V = VectorSpace(vs[0].parent().base_field(), 0) - - for v in vs: - new_V = span(result + [v]) - if new_V.dimension() > old_V.dimension(): - result.append(v) - old_V = new_V - - return result - - def basically_the_same(K1,K2): r""" ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` - otherwise. + 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``). """ if K1.lattice_dim() != K2.lattice_dim(): return False @@ -42,7 +24,7 @@ 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(): @@ -66,94 +48,14 @@ def basically_the_same(K1,K2): -def iso_space(K): - r""" - 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 = drop_dependent(K.rays()) - - 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 restrict_span(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: @@ -162,18 +64,18 @@ def restrict_span(K, K2=None): EXAMPLES:: sage: K = Cone([(1,)]) - sage: restrict_span(K) == K + sage: rho(K) == K True sage: K2 = Cone([(1,0)]) - sage: restrict_span(K2).rays() + sage: rho(K2).rays() N(1) in 1-d lattice N sage: K3 = Cone([(1,0,0)]) - sage: restrict_span(K3).rays() + sage: rho(K3).rays() N(1) in 1-d lattice N - sage: restrict_span(K2) == restrict_span(K3) + sage: rho(K2) == rho(K3) True TESTS: @@ -182,7 +84,7 @@ def restrict_span(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: K_S = restrict_span(K) + sage: K_S = rho(K) sage: K_S.is_solid() True @@ -191,30 +93,22 @@ def restrict_span(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: K_S = restrict_span(K, K.dual() ) + sage: K_S = rho(K, 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 = 8, 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 = 8) - sage: K.dim() == restrict_span(K).dim() + 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(restrict_span(K)) + sage: K.lineality() == rho(K).lineality() True No matter which space we restrict to, the lineality should not @@ -222,93 +116,98 @@ def restrict_span(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_dim = 8) - sage: lineality(K) >= lineality(restrict_span(K)) + sage: K.lineality() >= rho(K).lineality() True - sage: lineality(K) >= lineality(restrict_span(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 = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: P.is_proper() + sage: K_S = rho(K) + sage: K_SP = rho(K_S.dual()).dual() + sage: K_SP.is_proper() True - sage: P = restrict_span(K_S, K_S.dual()) - sage: P.is_proper() + 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 = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: P.is_proper() + sage: K_S = rho(K) + sage: K_SP = rho(K_S.dual()).dual() + sage: K_SP.is_proper() True - sage: P = restrict_span(K_S, K_S.dual()) - sage: P.is_proper() + 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 = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: P.is_proper() + sage: K_S = rho(K) + sage: K_SP = rho(K_S.dual()).dual() + sage: K_SP.is_proper() True - sage: P = restrict_span(K_S, K_S.dual()) - sage: P.is_proper() + 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 = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: P.is_proper() + sage: K_S = rho(K) + sage: K_SP = rho(K_S.dual()).dual() + sage: K_SP.is_proper() True - sage: P = restrict_span(K_S, K_S.dual()) - sage: P.is_proper() + sage: K_SP = rho(K_S, K_S.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^{*}`:: + 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=False) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: J = random_cone(max_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) True :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: J = random_cone(max_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) True :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: J = random_cone(max_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) True :: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() + sage: J = random_cone(max_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) True @@ -316,190 +215,24 @@ def restrict_span(K, K2=None): 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() ] - - # Shouldn't matter? - # - #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 ] - - L = ToricLattice(W.dimension()) - - return Cone(ws, lattice=L) - - - -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):: + # 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) - 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 codim(K): - r""" - Compute the codimension of this cone. - - The codimension of a cone is the dimension of the space of all - elements perpendicular to every element of the cone. In other words, - the codimension is the difference between the dimension of the - ambient space and the dimension of the cone itself. - - OUTPUT: - - A nonnegative integer representing the dimension of the space of all - elements perpendicular to this cone. - - .. seealso:: + # 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() - :meth:`dim`, :meth:`lattice_dim` + # We've already intersected K with the span of K2, so every + # generator of K should belong to W now. + W_rays = [ W.coordinate_vector(r) for r in K.rays() ] - EXAMPLES: + L = ToricLattice(K2.dim()) + return Cone(W_rays, lattice=L) - The codimension of the nonnegative orthant is zero, since the span of - its generators equals the entire ambient space:: - - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: codim(K) - 0 - - However, if we remove a ray so that the entire cone is contained - within the `x-y`-plane, then the resulting cone will have - codimension one, because the `z`-axis is perpendicular to every - element of the cone:: - - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: codim(K) - 1 - - If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero:: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: codim(K) - 0 - - And if the cone is trivial in any space, then its codimension is - equal to the dimension of the ambient space:: - - sage: K = Cone([], lattice=ToricLattice(0)) - sage: K.lattice_dim() - 0 - sage: codim(K) - 0 - - sage: K = Cone([(0,)]) - sage: K.lattice_dim() - 1 - sage: codim(K) - 1 - - sage: K = Cone([(0,0)]) - sage: K.lattice_dim() - 2 - sage: codim(K) - 2 - - TESTS: - - 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 = 8) - sage: c = codim(K) - sage: c in ZZ - True - sage: (0 <= c) and (c <= K.lattice_dim()) - True - - A solid cone should have codimension zero:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, 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 = 8, solid = True) - sage: codim(K) == lineality(K.dual()) - True - - """ - return (K.lattice_dim() - K.dim()) def discrete_complementarity_set(K): @@ -515,9 +248,7 @@ def discrete_complementarity_set(K): A list of pairs `(x,s)` such that, - * `x` is in this cone. * `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. @@ -815,7 +546,7 @@ 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 @@ -921,11 +652,11 @@ def lyapunov_rank(K): sage: set_random_seed() sage: K = random_cone(max_dim=8) sage: actual = lyapunov_rank(K) - 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 + 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 @@ -958,27 +689,34 @@ def lyapunov_rank(K): sage: lyapunov_rank(K) == len(LL(K)) True + Test Theorem 3 in [Orlitzky/Gowda]_:: + + sage: set_random_seed() + sage: K = random_cone(max_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 = restrict_span(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 = restrict_span(K.dual()).dual() - #K = restrict_span(intersect_span(K,K.dual()), K.dual()) - K = restrict_span(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