X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=81698e467c603df3e57cb764c256a2e082df5ec6;hb=b966ec383942ef1ff7837786c29d1f3edc33b84e;hp=2296e3fc010db71091e530b211c73ada05962f81;hpb=3b659b1d0440daf3ff7bd8cf3cf53f90523a1609;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 2296e3f..81698e4 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,82 +8,234 @@ addsitedir(abspath('../../')) from sage.all import * -def random_cone(min_dim=None, max_dim=None, min_rays=None, max_rays=None): +def project_span(K, K2 = None): r""" - Generate a random rational convex polyhedral cone. + Return a "copy" of ``K`` embeded in a lower-dimensional space. - 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. + By default, we will project ``K`` into the subspace spanned by its + rays. However, if ``K2`` is not ``None``, we will project into the + space spanned by the rays of ``K2`` instead. - INPUT: + EXAMPLES:: + + sage: K = Cone([(1,0,0), (0,1,0)]) + sage: project_span(K) + 2-d cone in 2-d lattice N + sage: project_span(K).rays() + N(1, 0), + N(0, 1) + in 2-d lattice N + + sage: K = Cone([(1,0,0), (0,1,0)]) + sage: K2 = Cone([(0,1)]) + sage: project_span(K, K2).rays() + N(1) + in 1-d lattice N + + """ + # Allow us to use a second cone to generate the subspace into + # which we're "projecting." + if K2 is None: + K2 = K + + # Use these to generate the new cone. + cs1 = K.rays().matrix().columns() - - ``min_dim`` (default: random) -- The minimum dimension of the ambient - lattice. + # And use these to figure out which indices to drop. + cs2 = K2.rays().matrix().columns() - - ``max_dim`` (default: random) -- The maximum dimension of the ambient - lattice. + perp_idxs = [] - - ``min_rays`` (default: random) -- The minimum number of generating rays - of the cone. + for idx in range(0, len(cs2)): + if cs2[idx].is_zero(): + perp_idxs.append(idx) - - ``max_rays`` (default: random) -- The maximum number of generating rays - of the cone. + solid_cols = [ cs1[idx] for idx in range(0,len(cs1)) + if not idx in perp_idxs + and not idx >= len(cs2) ] + + m = matrix(solid_cols) + L = ToricLattice(len(m.rows())) + J = Cone(m.transpose(), lattice=L) + return J + + +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 new, randomly generated cone. + 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: - It's hard to test the output of a random process, but we can at - least make sure that we get a cone back:: + The complementarity set of the dual can be obtained by switching the + components of the complementarity set of the original cone:: - sage: from sage.geometry.cone import is_Cone - sage: K = random_cone() - sage: is_Cone(K) # long time + 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 LL(K): + r""" + Compute the space `\mathbf{LL}` of all Lyapunov-like transformations + on this cone. + + OUTPUT: + + A list of matrices forming a basis for the space of all + Lyapunov-like transformations on the given cone. + + EXAMPLES: + + The trivial cone has no Lyapunov-like transformations:: + + sage: L = ToricLattice(0) + sage: K = Cone([], lattice=L) + sage: LL(K) + [] + + The Lyapunov-like transformations on the nonnegative orthant are + simply diagonal matrices:: + + sage: K = Cone([(1,)]) + sage: LL(K) + [[1]] + + sage: K = Cone([(1,0),(0,1)]) + sage: LL(K) + [ + [1 0] [0 0] + [0 0], [0 1] + ] + + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: LL(K) + [ + [1 0 0] [0 0 0] [0 0 0] + [0 0 0] [0 1 0] [0 0 0] + [0 0 0], [0 0 0], [0 0 1] + ] + + Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and + `L^{3}_{\infty}` cones [Rudolf et al.]_:: + + sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) + sage: LL(L31) + [ + [1 0 0] + [0 1 0] + [0 0 1] + ] + + sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) + sage: LL(L3infty) + [ + [1 0 0] + [0 1 0] + [0 0 1] + ] + + TESTS: + + The inner product `\left< L\left(x\right), s \right>` is zero for + every pair `\left( x,s \right)` in the discrete complementarity set + of the cone:: + + 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 + + """ + 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] + + # Sage doesn't think matrices are vectors, so we have to convert + # our matrices to vectors explicitly before we can figure out how + # many are linearly-indepenedent. + # + # The space W has the same base ring as V, but dimension + # dim(V)^2. So it has the same dimension as the space of linear + # transformations on V. In other words, it's just the right size + # to create an isomorphism between it and our matrices. + W = VectorSpace(V.base_ring(), V.dimension()**2) + + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] + + # Vector space representation of Lyapunov-like matrices + # (i.e. vec(L) where L is Luapunov-like). + LL_vector = W.span(vectors).complement() + + # Now construct an ambient MatrixSpace in which to stick our + # transformations. + M = MatrixSpace(V.base_ring(), V.dimension()) + + matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] + + return matrix_basis - 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() - - 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) - - 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) - - # 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) - - d = random_min_max(min_dim, max_dim) - r = random_min_max(min_rays, max_rays) - - L = ToricLattice(d) - rays = [L.random_element() for i in range(0,r)] - - # We pass the lattice in case there are no rays. - return Cone(rays, lattice=L) def lyapunov_rank(K): @@ -107,7 +259,7 @@ def lyapunov_rank(K): An integer representing the Lyapunov rank of the cone. If the dimension of the ambient vector space is `n`, then the Lyapunov rank will be between `1` and `n` inclusive; however a rank of `n-1` is - not possible (see the first reference). + not possible for any cone. .. note:: @@ -129,17 +281,21 @@ def lyapunov_rank(K): REFERENCES: - 1. M.S. Gowda and J. Tao. On the bilinearity rank of a proper cone - and Lyapunov-like transformations, Mathematical Programming, 147 + .. [Gowda/Tao] M.S. Gowda and J. Tao. On the bilinearity rank of a proper + cone and Lyapunov-like transformations, Mathematical Programming, 147 (2014) 155-170. - 2. G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear + .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an + Improper Cone. Work in-progress. + + .. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear optimality constraints for the cone of positive polynomials, Mathematical Programming, Series B, 129 (2011) 5-31. EXAMPLES: - The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`:: + The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n` + [Rudolf et al.]_:: sage: positives = Cone([(1,)]) sage: lyapunov_rank(positives) @@ -147,23 +303,25 @@ def lyapunov_rank(K): sage: quadrant = Cone([(1,0), (0,1)]) sage: lyapunov_rank(quadrant) 2 - sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) + sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) sage: lyapunov_rank(octant) 3 - The `L^{3}_{1}` cone is known to have a Lyapunov rank of one:: + The `L^{3}_{1}` cone is known to have a Lyapunov rank of one + [Rudolf et al.]_:: sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) sage: lyapunov_rank(L31) 1 - Likewise for the `L^{3}_{\infty}` cone:: + Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_:: sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) sage: lyapunov_rank(L3infty) 1 - The Lyapunov rank should be additive on a product of cones:: + The Lyapunov rank should be additive on a product of cones + [Rudolf et al.]_:: sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) @@ -171,8 +329,8 @@ def lyapunov_rank(K): sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant) True - Two isomorphic cones should have the same Lyapunov rank. The cone - ``K`` in the following example is isomorphic to the nonnegative + Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_. + The cone ``K`` in the following example is isomorphic to the nonnegative octant in `\mathbb{R}^{3}`:: sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) @@ -180,7 +338,7 @@ def lyapunov_rank(K): 3 The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself:: + itself [Rudolf et al.]_:: sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) @@ -188,49 +346,62 @@ def lyapunov_rank(K): TESTS: - The Lyapunov rank should be additive on a product of cones:: + The Lyapunov rank should be additive on a product of cones + [Rudolf et al.]_:: - 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 The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself:: + itself [Rudolf et al.]_:: - 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] - - matrices = [x.column() * s.row() for (x,s) in C_of_K] + 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 doesn't think matrices are vectors, so we have to convert - # our matrices to vectors explicitly before we can figure out how - # many are linearly-indepenedent. - # - # The space W has the same base ring as V, but dimension - # dim(V)^2. So it has the same dimension as the space of linear - # transformations on V. In other words, it's just the right size - # to create an isomorphism between it and our matrices. - W = VectorSpace(V.base_ring(), V.dimension()**2) - - def phi(m): - r""" - Convert a matrix to a vector isomorphically. - """ - return W(m.list()) - - vectors = [phi(m) for m in matrices] + sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) + sage: b = lyapunov_rank(K) + sage: n = K.lattice_dim() + sage: (n == 0 or 1 <= b) and b <= n + True + sage: b == n-1 + False + + In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have + Lyapunov rank `n-1` in `n` dimensions:: + + sage: K = random_cone(max_dim=10, max_rays=16) + sage: b = lyapunov_rank(K) + sage: n = K.lattice_dim() + sage: b == n-1 + False + + The calculation of the Lyapunov rank of an improper cone can be + reduced to that of a proper cone [Orlitzky/Gowda]_:: + + sage: K = random_cone(max_dim=15, max_rays=25) + sage: actual = lyapunov_rank(K) + sage: K_S = project_span(K) + sage: J_T1 = project_span(K_S.dual()).dual() + sage: J_T2 = project_span(K, K_S.dual()) + sage: J_T2 = Cone(J_T2.rays(), lattice=J_T1.lattice()) + sage: J_T1 == J_T2 + True + sage: J_T = J_T1 + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True - return (W.dimension() - W.span(vectors).rank()) + """ + return len(LL(K))