X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=3b724a7c0fe8d0305351aae778b23978ed79e8f4;hb=b1f56b9fd21ce64f0d4613d1d07e090e89eba621;hp=23043381bca83acd6d7f17479ef4769980b9960f;hpb=012175f3a2591586099b4e955bb440958f2d63d7;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 2304338..3b724a7 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,6 +8,266 @@ addsitedir(abspath('../../')) from sage.all import * +def random_cone(min_dim=0, max_dim=None, min_rays=0, max_rays=None): + r""" + Generate a random rational convex polyhedral 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. If a + lower bound is left unspecified, it defaults to zero. Unspecified + upper bounds will be chosen randomly. + + The lower bound on the number of rays is limited to twice the + maximum dimension of the ambient vector space. To see why, consider + the space $\mathbb{R}^{2}$, and suppose we have generated four rays, + $\left\{ \pm e_{1}, \pm e_{2} \right\}$. Clearly any other ray in + the space is a nonnegative linear combination of those four, + so it is hopeless to generate more. It is therefore an error + to request more in the form of ``min_rays``. + + .. NOTE: + + If you do not explicitly request more than ``2 * max_dim`` rays, + a larger number may still be randomly generated. In that case, + the returned cone will simply be equal to the entire space. + + INPUT: + + - ``min_dim`` (default: zero) -- A nonnegative integer representing the + minimum dimension of the ambient lattice. + + - ``max_dim`` (default: random) -- A nonnegative integer representing + the maximum dimension of the ambient + lattice. + + - ``min_rays`` (default: zero) -- A nonnegative integer representing the + minimum number of generating rays of the + cone. + + - ``max_rays`` (default: random) -- A nonnegative integer representing the + maximum number of generating rays of + the cone. + + OUTPUT: + + A new, randomly generated cone. + + A ``ValueError` will be thrown under the following conditions: + + * Any of ``min_dim``, ``max_dim``, ``min_rays``, or ``max_rays`` + are negative. + + * ``max_dim`` is less than ``min_dim``. + + * ``max_rays`` is less than ``min_rays``. + + * ``min_rays`` is greater than twice ``max_dim``. + + EXAMPLES: + + If we set the lower/upper bounds to zero, then our result is + predictable:: + + sage: random_cone(0,0,0,0) + 0-d cone in 0-d lattice N + + We can predict the dimension when ``min_dim == max_dim``:: + + sage: random_cone(min_dim=4, max_dim=4, min_rays=0, max_rays=0) + 0-d cone in 4-d lattice N + + Likewise for the number of rays when ``min_rays == max_rays``:: + + sage: random_cone(min_dim=10, max_dim=10, min_rays=10, max_rays=10) + 10-d cone in 10-d lattice N + + TESTS: + + It's hard to test the output of a random process, but we can at + least make sure that we get a cone back:: + + sage: from sage.geometry.cone import is_Cone # long time + sage: K = random_cone() # long time + sage: is_Cone(K) # long time + True + + The upper/lower bounds are respected:: + + sage: K = random_cone(min_dim=5, max_dim=10, min_rays=3, max_rays=4) + sage: 5 <= K.lattice_dim() and K.lattice_dim() <= 10 + True + sage: 3 <= K.nrays() and K.nrays() <= 4 + True + + Ensure that an exception is raised when either lower bound is greater + than its respective upper bound:: + + sage: random_cone(min_dim=5, max_dim=2) + Traceback (most recent call last): + ... + ValueError: max_dim cannot be less than min_dim. + + sage: random_cone(min_rays=5, max_rays=2) + Traceback (most recent call last): + ... + ValueError: max_rays cannot be less than min_rays. + + And if we request too many rays:: + + sage: random_cone(min_rays=5, max_dim=1) + Traceback (most recent call last): + ... + ValueError: min_rays cannot be larger than twice max_dim. + + """ + + # Catch obvious mistakes so that we can generate clear error + # messages. + + if min_dim < 0: + raise ValueError('min_dim must be nonnegative.') + + if min_rays < 0: + raise ValueError('min_rays must be nonnegative.') + + if max_dim is not None: + if max_dim < 0: + raise ValueError('max_dim must be nonnegative.') + if (max_dim < min_dim): + raise ValueError('max_dim cannot be less than min_dim.') + if min_rays > 2*max_dim: + raise ValueError('min_rays cannot be larger than twice max_dim.') + + if max_rays is not None: + if max_rays < 0: + raise ValueError('max_rays must be nonnegative.') + if (max_rays < min_rays): + raise ValueError('max_rays cannot be less than min_rays.') + + + def random_min_max(l,u): + r""" + We need to handle two cases for the upper bounds, and we need to do + the same thing for max_dim/max_rays. So we consolidate the logic here. + """ + if u is None: + # The upper bound is unspecified; return a random integer + # in [l,infinity). + return l + ZZ.random_element().abs() + else: + # We have an upper bound, and it's greater than or equal + # to our lower bound. So we generate a random integer in + # [0,u-l], and then add it to l to get something in + # [l,u]. To understand the "+1", check the + # ZZ.random_element() docs. + return l + ZZ.random_element(u - l + 1) + + def is_full_space(K): + r""" + Is this cone equivalent to the full ambient vector space? + """ + return K.lines().dim() == K.lattice_dim() + + d = random_min_max(min_dim, max_dim) + r = random_min_max(min_rays, max_rays) + + L = ToricLattice(d) + + # The rays are trickier to generate, since we could generate v and + # 2*v as our "two rays." In that case, the resuting cone would + # have one generating ray. To avoid such a situation, we start by + # generating ``r`` rays where ``r`` is the number we want to end + # up with... + rays = [L.random_element() for i in range(0, r)] + + # (The lattice parameter is required when no rays are given, so we + # pass it just in case ``r == 0``). + K = Cone(rays, lattice=L) + + # Now if we generated two of the "same" rays, we'll have fewer + # generating rays than ``r``. In that case, we keep making up new + # rays and recreating the cone until we get the right number of + # independent generators. We can obviously stop if ``K`` is the + # entire ambient vector space. + while r > K.nrays() and not is_full_space(K): + rays.append(L.random_element()) + K = Cone(rays) + + return K + + +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. + + OUTPUT: + + 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. + + EXAMPLES: + + The discrete complementarity set of the nonnegative orthant consists + of pairs of standard basis vectors:: + + sage: K = Cone([(1,0),(0,1)]) + sage: discrete_complementarity_set(K) + [((1, 0), (0, 1)), ((0, 1), (1, 0))] + + 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: 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:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + 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: 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 + + """ + V = K.lattice().vector_space() + + # 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()] + + return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] + + def lyapunov_rank(K): r""" Compute the Lyapunov (or bilinearity) rank of this cone. @@ -108,17 +368,29 @@ def lyapunov_rank(K): sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True + TESTS: + + The Lyapunov rank should be additive on a product of cones:: + + 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 + + The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` + itself:: + + 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