X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=777d45e1a3e7a72a5ebbb167fd1ef6540c745282;hb=81a763e35b3e4322be6c60a815064be1f0dfcc3c;hp=2296e3fc010db71091e530b211c73ada05962f81;hpb=3b659b1d0440daf3ff7bd8cf3cf53f90523a1609;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 2296e3f..777d45e 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,82 +8,76 @@ addsitedir(abspath('../../')) from sage.all import * -def random_cone(min_dim=None, max_dim=None, min_rays=None, max_rays=None): +def discrete_complementarity_set(K): r""" - Generate a random rational convex polyhedral cone. + Compute the discrete complementarity set of this cone. - Lower and upper bounds may be provided for both the dimension of the - ambient space and the number of generating rays of the cone. Any - parameters left unspecified will be chosen randomly. + 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. - INPUT: - - - ``min_dim`` (default: random) -- The minimum dimension of the ambient - lattice. - - - ``max_dim`` (default: random) -- The maximum dimension of the ambient - lattice. + OUTPUT: - - ``min_rays`` (default: random) -- The minimum number of generating rays - of the cone. + A list of pairs `(x,s)` such that, - - ``max_rays`` (default: random) -- The maximum number of generating rays - of the cone. + * `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. - OUTPUT: + EXAMPLES: - A new, randomly generated cone. + The discrete complementarity set of the nonnegative orthant consists + of pairs of standard basis vectors:: - TESTS: + sage: K = Cone([(1,0),(0,1)]) + sage: discrete_complementarity_set(K) + [((1, 0), (0, 1)), ((0, 1), (1, 0))] - It's hard to test the output of a random process, but we can at - least make sure that we get a cone back:: + If the cone consists of a single ray, the second components of the + discrete complementarity set should generate the orthogonal + complement of that ray:: - sage: from sage.geometry.cone import is_Cone - sage: K = random_cone() - sage: is_Cone(K) # long time - True + sage: K = Cone([(1,0)]) + sage: discrete_complementarity_set(K) + [((1, 0), (0, 1)), ((1, 0), (0, -1))] + sage: K = Cone([(1,0,0)]) + sage: discrete_complementarity_set(K) + [((1, 0, 0), (0, 1, 0)), + ((1, 0, 0), (0, -1, 0)), + ((1, 0, 0), (0, 0, 1)), + ((1, 0, 0), (0, 0, -1))] - """ + When the cone is the entire space, its dual is the trivial cone, so + the discrete complementarity set is empty:: - def random_min_max(l,u): - r""" - We need to handle four cases to prevent us from doing - something stupid like having an upper bound that's lower than - our lower bound. And we would need to repeat all of that logic - for the dimension/rays, so we consolidate it here. - """ - if l is None and u is None: - # They're both random, just return a random nonnegative - # integer. - return ZZ.random_element().abs() + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: discrete_complementarity_set(K) + [] - if l is not None and u is not None: - # Both were specified. Again, just make up a number and - # return it. If the user wants to give us u < l then he - # can have an exception. - return ZZ.random_element(l,u) + TESTS: - if l is not None and u is None: - # In this case, we're generating the upper bound randomly - # GIVEN A LOWER BOUND. So we add a random nonnegative - # integer to the given lower bound. - u = l + ZZ.random_element().abs() - return ZZ.random_element(l,u) + The complementarity set of the dual can be obtained by switching the + components of the complementarity set of the original cone:: - # Here we must be in the only remaining case, where we are - # given an upper bound but no lower bound. We might as well - # use zero. - return ZZ.random_element(0,u) + sage: K1 = random_cone(max_dim=10, max_rays=10) + 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 + True - d = random_min_max(min_dim, max_dim) - r = random_min_max(min_rays, max_rays) + """ + V = K.lattice().vector_space() - L = ToricLattice(d) - rays = [L.random_element() for i in range(0,r)] + # Convert the rays to vectors so that we can compute inner + # products. + xs = [V(x) for x in K.rays()] + ss = [V(s) for s in K.dual().rays()] - # We pass the lattice in case there are no rays. - return Cone(rays, lattice=L) + return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] def lyapunov_rank(K): @@ -190,8 +184,8 @@ def lyapunov_rank(K): The Lyapunov rank should be additive on a product of cones:: - sage: K1 = random_cone(0,10,0,10) - sage: K2 = random_cone(0,10,0,10) + sage: K1 = random_cone(max_dim=10, max_rays=10) + sage: K2 = random_cone(max_dim=10, max_rays=10) sage: K = K1.cartesian_product(K2) sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) True @@ -199,21 +193,16 @@ def lyapunov_rank(K): The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` itself:: - sage: K = random_cone(0,10,0,10) + sage: K = random_cone(max_dim=10, max_rays=10) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True """ V = K.lattice().vector_space() - xs = [V(x) for x in K.rays()] - ss = [V(s) for s in K.dual().rays()] - - # WARNING: This isn't really C(K), it only contains the pairs - # (x,s) in C(K) where x,s are extreme in their respective cones. - C_of_K = [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] + C_of_K = discrete_complementarity_set(K) - matrices = [x.column() * s.row() for (x,s) in C_of_K] + matrices = [x.tensor_product(s) 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