]>
gitweb.michael.orlitzky.com - sage.d.git/blob - cone.py
777d45e1a3e7a72a5ebbb167fd1ef6540c745282
1 # Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
2 # have to explicitly mangle our sitedir here so that "mjo.cone"
4 from os
.path
import abspath
5 from site
import addsitedir
6 addsitedir(abspath('../../'))
11 def discrete_complementarity_set(K
):
13 Compute the discrete complementarity set of this cone.
15 The complementarity set of this cone is the set of all orthogonal
16 pairs `(x,s)` such that `x` is in this cone, and `s` is in its
17 dual. The discrete complementarity set restricts `x` and `s` to be
18 generators of their respective cones.
22 A list of pairs `(x,s)` such that,
24 * `x` is in this cone.
25 * `x` is a generator of this cone.
26 * `s` is in this cone's dual.
27 * `s` is a generator of this cone's dual.
28 * `x` and `s` are orthogonal.
32 The discrete complementarity set of the nonnegative orthant consists
33 of pairs of standard basis vectors::
35 sage: K = Cone([(1,0),(0,1)])
36 sage: discrete_complementarity_set(K)
37 [((1, 0), (0, 1)), ((0, 1), (1, 0))]
39 If the cone consists of a single ray, the second components of the
40 discrete complementarity set should generate the orthogonal
41 complement of that ray::
43 sage: K = Cone([(1,0)])
44 sage: discrete_complementarity_set(K)
45 [((1, 0), (0, 1)), ((1, 0), (0, -1))]
46 sage: K = Cone([(1,0,0)])
47 sage: discrete_complementarity_set(K)
48 [((1, 0, 0), (0, 1, 0)),
49 ((1, 0, 0), (0, -1, 0)),
50 ((1, 0, 0), (0, 0, 1)),
51 ((1, 0, 0), (0, 0, -1))]
53 When the cone is the entire space, its dual is the trivial cone, so
54 the discrete complementarity set is empty::
56 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
57 sage: discrete_complementarity_set(K)
62 The complementarity set of the dual can be obtained by switching the
63 components of the complementarity set of the original cone::
65 sage: K1 = random_cone(max_dim=10, max_rays=10)
67 sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
68 sage: actual = discrete_complementarity_set(K1)
69 sage: actual == expected
73 V
= K
.lattice().vector_space()
75 # Convert the rays to vectors so that we can compute inner
77 xs
= [V(x
) for x
in K
.rays()]
78 ss
= [V(s
) for s
in K
.dual().rays()]
80 return [(x
,s
) for x
in xs
for s
in ss
if x
.inner_product(s
) == 0]
85 Compute the Lyapunov (or bilinearity) rank of this cone.
87 The Lyapunov rank of a cone can be thought of in (mainly) two ways:
89 1. The dimension of the Lie algebra of the automorphism group of the
92 2. The dimension of the linear space of all Lyapunov-like
93 transformations on the cone.
97 A closed, convex polyhedral cone.
101 An integer representing the Lyapunov rank of the cone. If the
102 dimension of the ambient vector space is `n`, then the Lyapunov rank
103 will be between `1` and `n` inclusive; however a rank of `n-1` is
104 not possible (see the first reference).
108 In the references, the cones are always assumed to be proper. We
109 do not impose this restriction.
117 The codimension formula from the second reference is used. We find
118 all pairs `(x,s)` in the complementarity set of `K` such that `x`
119 and `s` are rays of our cone. It is known that these vectors are
120 sufficient to apply the codimension formula. Once we have all such
121 pairs, we "brute force" the codimension formula by finding all
122 linearly-independent `xs^{T}`.
126 1. M.S. Gowda and J. Tao. On the bilinearity rank of a proper cone
127 and Lyapunov-like transformations, Mathematical Programming, 147
130 2. G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
131 optimality constraints for the cone of positive polynomials,
132 Mathematical Programming, Series B, 129 (2011) 5-31.
136 The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`::
138 sage: positives = Cone([(1,)])
139 sage: lyapunov_rank(positives)
141 sage: quadrant = Cone([(1,0), (0,1)])
142 sage: lyapunov_rank(quadrant)
144 sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
145 sage: lyapunov_rank(octant)
148 The `L^{3}_{1}` cone is known to have a Lyapunov rank of one::
150 sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
151 sage: lyapunov_rank(L31)
154 Likewise for the `L^{3}_{\infty}` cone::
156 sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
157 sage: lyapunov_rank(L3infty)
160 The Lyapunov rank should be additive on a product of cones::
162 sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
163 sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
164 sage: K = L31.cartesian_product(octant)
165 sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant)
168 Two isomorphic cones should have the same Lyapunov rank. The cone
169 ``K`` in the following example is isomorphic to the nonnegative
170 octant in `\mathbb{R}^{3}`::
172 sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
173 sage: lyapunov_rank(K)
176 The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
179 sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
180 sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
185 The Lyapunov rank should be additive on a product of cones::
187 sage: K1 = random_cone(max_dim=10, max_rays=10)
188 sage: K2 = random_cone(max_dim=10, max_rays=10)
189 sage: K = K1.cartesian_product(K2)
190 sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
193 The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
196 sage: K = random_cone(max_dim=10, max_rays=10)
197 sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
201 V
= K
.lattice().vector_space()
203 C_of_K
= discrete_complementarity_set(K
)
205 matrices
= [x
.tensor_product(s
) for (x
,s
) in C_of_K
]
207 # Sage doesn't think matrices are vectors, so we have to convert
208 # our matrices to vectors explicitly before we can figure out how
209 # many are linearly-indepenedent.
211 # The space W has the same base ring as V, but dimension
212 # dim(V)^2. So it has the same dimension as the space of linear
213 # transformations on V. In other words, it's just the right size
214 # to create an isomorphism between it and our matrices.
215 W
= VectorSpace(V
.base_ring(), V
.dimension()**2)
219 Convert a matrix to a vector isomorphically.
223 vectors
= [phi(m
) for m
in matrices
]
225 return (W
.dimension() - W
.span(vectors
).rank())