]>
gitweb.michael.orlitzky.com - sage.d.git/blob - cone/cone.py
3 def is_lyapunov_like(L
,K
):
5 Determine whether or not ``L`` is Lyapunov-like on ``K``.
7 We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
8 L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
9 `\left\langle x,s \right\rangle` in the complementarity set of
10 ``K``. It is known [Orlitzky]_ that this property need only be
11 checked for generators of ``K`` and its dual.
15 - ``L`` -- A linear transformation or matrix.
17 - ``K`` -- A polyhedral closed convex cone.
21 ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
22 and ``False`` otherwise.
26 If this function returns ``True``, then ``L`` is Lyapunov-like
27 on ``K``. However, if ``False`` is returned, that could mean one
28 of two things. The first is that ``L`` is definitely not
29 Lyapunov-like on ``K``. The second is more of an "I don't know"
30 answer, returned (for example) if we cannot prove that an inner
35 M. Orlitzky. The Lyapunov rank of an improper cone.
36 http://www.optimization-online.org/DB_HTML/2015/10/5135.html
40 The identity is always Lyapunov-like in a nontrivial space::
42 sage: set_random_seed()
43 sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
44 sage: L = identity_matrix(K.lattice_dim())
45 sage: is_lyapunov_like(L,K)
48 As is the "zero" transformation::
50 sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
51 sage: R = K.lattice().vector_space().base_ring()
52 sage: L = zero_matrix(R, K.lattice_dim())
53 sage: is_lyapunov_like(L,K)
56 Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
59 sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
60 sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
64 return all([(L
*x
).inner_product(s
) == 0
65 for (x
,s
) in K
.discrete_complementarity_set()])
68 def random_element(K
):
70 Return a random element of ``K`` from its ambient vector space.
74 The cone ``K`` is specified in terms of its generators, so that
75 ``K`` is equal to the convex conic combination of those generators.
76 To choose a random element of ``K``, we assign random nonnegative
77 coefficients to each generator of ``K`` and construct a new vector
80 A vector, rather than a ray, is returned so that the element may
81 have non-integer coordinates. Thus the element may have an
82 arbitrarily small norm.
86 A random element of the trivial cone is zero::
88 sage: set_random_seed()
89 sage: K = Cone([], ToricLattice(0))
90 sage: random_element(K)
92 sage: K = Cone([(0,)])
93 sage: random_element(K)
95 sage: K = Cone([(0,0)])
96 sage: random_element(K)
98 sage: K = Cone([(0,0,0)])
99 sage: random_element(K)
104 Any cone should contain an element of itself::
106 sage: set_random_seed()
107 sage: K = random_cone(max_rays = 8)
108 sage: K.contains(random_element(K))
112 V
= K
.lattice().vector_space()
114 coefficients
= [ F
.random_element().abs() for i
in range(K
.nrays()) ]
115 vector_gens
= map(V
, K
.rays())
116 scaled_gens
= [ coefficients
[i
]*vector_gens
[i
]
117 for i
in range(len(vector_gens
)) ]
119 # Make sure we return a vector. Without the coercion, we might
120 # return ``0`` when ``K`` has no rays.
121 v
= V(sum(scaled_gens
))
125 def positive_operator_gens(K
):
127 Compute generators of the cone of positive operators on this cone.
131 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
132 Each matrix ``P`` in the list should have the property that ``P*x``
133 is an element of ``K`` whenever ``x`` is an element of
134 ``K``. Moreover, any nonnegative linear combination of these
135 matrices shares the same property.
139 The trivial cone in a trivial space has no positive operators::
141 sage: K = Cone([], ToricLattice(0))
142 sage: positive_operator_gens(K)
145 Positive operators on the nonnegative orthant are nonnegative matrices::
147 sage: K = Cone([(1,)])
148 sage: positive_operator_gens(K)
151 sage: K = Cone([(1,0),(0,1)])
152 sage: positive_operator_gens(K)
154 [1 0] [0 1] [0 0] [0 0]
155 [0 0], [0 0], [1 0], [0 1]
158 Every operator is positive on the ambient vector space::
160 sage: K = Cone([(1,),(-1,)])
161 sage: K.is_full_space()
163 sage: positive_operator_gens(K)
166 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
167 sage: K.is_full_space()
169 sage: positive_operator_gens(K)
171 [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
172 [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
177 A positive operator on a cone should send its generators into the cone::
179 sage: K = random_cone(max_ambient_dim = 6)
180 sage: pi_of_K = positive_operator_gens(K)
181 sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
184 The dimension of the cone of positive operators is given by the
185 corollary in my paper::
187 sage: K = random_cone(max_ambient_dim = 6)
188 sage: n = K.lattice_dim()
190 sage: l = K.lineality()
191 sage: pi_of_K = positive_operator_gens(K)
192 sage: actual = Cone([p.list() for p in pi_of_K]).dim()
193 sage: expected = n**2 - l*(n - l) - (n - m)*m
194 sage: actual == expected
198 # Matrices are not vectors in Sage, so we have to convert them
199 # to vectors explicitly before we can find a basis. We need these
200 # two values to construct the appropriate "long vector" space.
201 F
= K
.lattice().base_field()
204 tensor_products
= [ s
.tensor_product(x
) for x
in K
for s
in K
.dual() ]
206 # Convert those tensor products to long vectors.
207 W
= VectorSpace(F
, n
**2)
208 vectors
= [ W(tp
.list()) for tp
in tensor_products
]
210 # Create the *dual* cone of the positive operators, expressed as
212 pi_dual
= Cone(vectors
, ToricLattice(W
.dimension()))
214 # Now compute the desired cone from its dual...
215 pi_cone
= pi_dual
.dual()
217 # And finally convert its rays back to matrix representations.
218 M
= MatrixSpace(F
, n
)
219 return [ M(v
.list()) for v
in pi_cone
.rays() ]
222 def Z_transformation_gens(K
):
224 Compute generators of the cone of Z-transformations on this cone.
228 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
229 Each matrix ``L`` in the list should have the property that
230 ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
231 discrete complementarity set of ``K``. Moreover, any nonnegative
232 linear combination of these matrices shares the same property.
236 Z-transformations on the nonnegative orthant are just Z-matrices.
237 That is, matrices whose off-diagonal elements are nonnegative::
239 sage: K = Cone([(1,0),(0,1)])
240 sage: Z_transformation_gens(K)
242 [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
243 [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
245 sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
246 sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
247 ....: for i in range(z.nrows())
248 ....: for j in range(z.ncols())
252 The trivial cone in a trivial space has no Z-transformations::
254 sage: K = Cone([], ToricLattice(0))
255 sage: Z_transformation_gens(K)
258 Z-transformations on a subspace are Lyapunov-like and vice-versa::
260 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
261 sage: K.is_full_space()
263 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
264 sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
270 The Z-property is possessed by every Z-transformation::
272 sage: set_random_seed()
273 sage: K = random_cone(max_ambient_dim = 6)
274 sage: Z_of_K = Z_transformation_gens(K)
275 sage: dcs = K.discrete_complementarity_set()
276 sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
277 ....: for (x,s) in dcs])
280 The lineality space of Z is LL::
282 sage: set_random_seed()
283 sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
284 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
285 sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
286 sage: z_cone.linear_subspace() == lls
290 # Matrices are not vectors in Sage, so we have to convert them
291 # to vectors explicitly before we can find a basis. We need these
292 # two values to construct the appropriate "long vector" space.
293 F
= K
.lattice().base_field()
296 # These tensor products contain generators for the dual cone of
297 # the cross-positive transformations.
298 tensor_products
= [ s
.tensor_product(x
)
299 for (x
,s
) in K
.discrete_complementarity_set() ]
301 # Turn our matrices into long vectors...
302 W
= VectorSpace(F
, n
**2)
303 vectors
= [ W(m
.list()) for m
in tensor_products
]
305 # Create the *dual* cone of the cross-positive operators,
306 # expressed as long vectors..
307 Sigma_dual
= Cone(vectors
, lattice
=ToricLattice(W
.dimension()))
309 # Now compute the desired cone from its dual...
310 Sigma_cone
= Sigma_dual
.dual()
312 # And finally convert its rays back to matrix representations.
313 # But first, make them negative, so we get Z-transformations and
314 # not cross-positive ones.
315 M
= MatrixSpace(F
, n
)
316 return [ -M(v
.list()) for v
in Sigma_cone
.rays() ]