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gitweb.michael.orlitzky.com - sage.d.git/blob - mjo/cone/cone.py
1ab6b97c128cde3d1e176032cf0d90f601057166
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: set_random_seed()
180 sage: K = random_cone(max_ambient_dim = 5)
181 sage: pi_of_K = positive_operator_gens(K)
182 sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
185 The dimension of the cone of positive operators is given by the
186 corollary in my paper::
188 sage: set_random_seed()
189 sage: K = random_cone(max_ambient_dim = 5)
190 sage: n = K.lattice_dim()
192 sage: l = K.lineality()
193 sage: pi_of_K = positive_operator_gens(K)
194 sage: L = ToricLattice(n**2)
195 sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
196 sage: expected = n**2 - l*(m - l) - (n - m)*m
197 sage: actual == expected
200 The lineality of the cone of positive operators is given by the
201 corollary in my paper::
203 sage: set_random_seed()
204 sage: K = random_cone(max_ambient_dim = 5)
205 sage: n = K.lattice_dim()
206 sage: pi_of_K = positive_operator_gens(K)
207 sage: L = ToricLattice(n**2)
208 sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
209 sage: expected = n**2 - K.dim()*K.dual().dim()
210 sage: actual == expected
213 # Matrices are not vectors in Sage, so we have to convert them
214 # to vectors explicitly before we can find a basis. We need these
215 # two values to construct the appropriate "long vector" space.
216 F
= K
.lattice().base_field()
219 tensor_products
= [ s
.tensor_product(x
) for x
in K
for s
in K
.dual() ]
221 # Convert those tensor products to long vectors.
222 W
= VectorSpace(F
, n
**2)
223 vectors
= [ W(tp
.list()) for tp
in tensor_products
]
225 # Create the *dual* cone of the positive operators, expressed as
227 pi_dual
= Cone(vectors
, ToricLattice(W
.dimension()))
229 # Now compute the desired cone from its dual...
230 pi_cone
= pi_dual
.dual()
232 # And finally convert its rays back to matrix representations.
233 M
= MatrixSpace(F
, n
)
234 return [ M(v
.list()) for v
in pi_cone
.rays() ]
237 def Z_transformation_gens(K
):
239 Compute generators of the cone of Z-transformations on this cone.
243 A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
244 Each matrix ``L`` in the list should have the property that
245 ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
246 discrete complementarity set of ``K``. Moreover, any nonnegative
247 linear combination of these matrices shares the same property.
251 Z-transformations on the nonnegative orthant are just Z-matrices.
252 That is, matrices whose off-diagonal elements are nonnegative::
254 sage: K = Cone([(1,0),(0,1)])
255 sage: Z_transformation_gens(K)
257 [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
258 [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
260 sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
261 sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
262 ....: for i in range(z.nrows())
263 ....: for j in range(z.ncols())
267 The trivial cone in a trivial space has no Z-transformations::
269 sage: K = Cone([], ToricLattice(0))
270 sage: Z_transformation_gens(K)
273 Z-transformations on a subspace are Lyapunov-like and vice-versa::
275 sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
276 sage: K.is_full_space()
278 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
279 sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
285 The Z-property is possessed by every Z-transformation::
287 sage: set_random_seed()
288 sage: K = random_cone(max_ambient_dim = 6)
289 sage: Z_of_K = Z_transformation_gens(K)
290 sage: dcs = K.discrete_complementarity_set()
291 sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
292 ....: for (x,s) in dcs])
295 The lineality space of Z is LL::
297 sage: set_random_seed()
298 sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
299 sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
300 sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
301 sage: z_cone.linear_subspace() == lls
304 And thus, the lineality of Z is the Lyapunov rank::
306 sage: set_random_seed()
307 sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
308 sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ])
309 sage: z_cone.lineality() == K.lyapunov_rank()
312 # Matrices are not vectors in Sage, so we have to convert them
313 # to vectors explicitly before we can find a basis. We need these
314 # two values to construct the appropriate "long vector" space.
315 F
= K
.lattice().base_field()
318 # These tensor products contain generators for the dual cone of
319 # the cross-positive transformations.
320 tensor_products
= [ s
.tensor_product(x
)
321 for (x
,s
) in K
.discrete_complementarity_set() ]
323 # Turn our matrices into long vectors...
324 W
= VectorSpace(F
, n
**2)
325 vectors
= [ W(m
.list()) for m
in tensor_products
]
327 # Create the *dual* cone of the cross-positive operators,
328 # expressed as long vectors..
329 Sigma_dual
= Cone(vectors
, lattice
=ToricLattice(W
.dimension()))
331 # Now compute the desired cone from its dual...
332 Sigma_cone
= Sigma_dual
.dual()
334 # And finally convert its rays back to matrix representations.
335 # But first, make them negative, so we get Z-transformations and
336 # not cross-positive ones.
337 M
= MatrixSpace(F
, n
)
338 return [ -M(v
.list()) for v
in Sigma_cone
.rays() ]