X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=dbe369a8c2feabaee32bade898106641c3a3c3f0;hb=88f2aa54da4207051441d84551b3a704f3604a54;hp=23043381bca83acd6d7f17479ef4769980b9960f;hpb=012175f3a2591586099b4e955bb440958f2d63d7;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 2304338..dbe369a 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,6 +8,92 @@ addsitedir(abspath('../../')) from sage.all import * +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 LL(K): + r""" + Compute the space `\mathbf{LL}` of all Lyapunov-like transformations + on this cone. + + OUTPUT: + + A ``MatrixSpace`` object `M` such that every matrix `L \in M` is + Lyapunov-like on this cone. + + """ + pass # implement me lol + + def lyapunov_rank(K): r""" Compute the Lyapunov (or bilinearity) rank of this cone. @@ -51,17 +137,18 @@ 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 + .. [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) @@ -69,23 +156,25 @@ 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)]) @@ -93,8 +182,8 @@ 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)]) @@ -102,23 +191,49 @@ 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()) True + TESTS: + + The Lyapunov rank should be additive on a product of cones + [Rudolf et al.]_:: + + 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 [Rudolf et al.]_:: + + sage: K = random_cone(max_dim=10, max_rays=10) + 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, this holds for the trivial cone in a + trivial space as well):: + + sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) + sage: b = lyapunov_rank(K) + sage: n = K.lattice_dim() + sage: 1 <= b and b <= n + True + sage: b == n-1 + False + """ 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